On families of rational curves in the Hilbert square of a surface (with
  an Appendix by Edoardo Sernesi)
ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A
SURFACE
(WITH AN APPENDIX BY EDOARDO SERNESI)
FLAMINIO FLAMINI*, ANDREAS LEOPOLD KNUTSEN** AND GIANLUCA PACIENZA***
Abstra
t. Under natural hypotheses we give an upper bound on the dimension of families of
singular 
urves with hyperellipti
 normalizations on a surfa
e S with pg > 0 via the study of the
asso
iated families of rational 
urves in S
. We use this result to prove the existen
e of nodal 
urves
of geometri
 genus 3 with hyperellipti
 normalizations, on a general K3 surfa
e, thus obtaining
spe
i�
 2-dimensional families of rational 
urves in S
. We give two in�nite series of examples
of general, primitively polarized K3s su
h that their Hilbert squares 
ontain a P
or a threefold
birational to a P
-bundle over a K3. We dis
uss the 
onsequen
es on the Mori 
one of the Hilbert
square.
1. Introdu
tion
For any smooth surfa
e S, the Hilbert s
heme S[n] parametrizing 0-dimensional length n sub-
s
hemes of S is a smooth 2n-dimensional variety whose inner geometry is naturally related to that
of S. For instan
e, if ∆ ⊂ S[n] is the ex
eptional divisor, that is, the ex
eptional lo
us of the
Hilbert-Chow morphism µ : S[n] → Symn(S), then irredu
ible (possibly singular) rational 
urves
not 
ontained in ∆ roughly 
orrespond to irredu
ible (possibly singular) 
urves on S with a g1n′ on
their normalizations, for some n′ ≤ n (see � 2.1 for the pre
ise 
orresponden
e when n = 2). One
of the features of this paper is to show how ideas and te
hniques from one of the two sides of the

orresponden
e makes it possible to shed light on problems naturally arising on the other side.
If S is a K3 surfa
e, S[n] is a hyperkähler manifold (
f. [31, 2.2℄) and rational 
urves play a
fundamental r�le in the study of the (birational) geometry of S[n]. Indeed a result due to Huybre
hts
and Bou
ksom [32, 11℄ implies in parti
ular that these 
urves govern the ample 
one of S[n] (we will
re
all the pre
ise statement below and in � 6.1). The presen
e of a Pn ⊂ S[n] gives rise to a birational
map (the so-
alledMukai �op [41℄) to another hyperkähler manifold and, for n = 2, all birational maps
between hyperkähler fourfolds fa
tor through a sequen
e of Mukai �ops [12, 30, 60, 62℄. Moreover, as
shown by Huybre
hts [32℄, uniruled divisors allow to des
ribe the birational Kähler 
one of S[n] (see
� 7 for the pre
ise statement). For hyperkähler fourfolds pre
ise numeri
al and geometri
 properties
of the rational 
urves that are extremal in the Mori 
one have been 
onje
tured by Hassett and
Ts
hinkel [25℄.
The s
ope of this paper, and the stru
ture of it as well, is twofold: we �rst devise general methods
and tools to study families of 
urves with hyperellipti
 normalizations on a surfa
e S, mostly under
the additional hypothesis that pg(S) > 0, in � 2-� 4. Then we apply these to obtain 
on
rete results
in the 
ase of K3 surfa
es, in � 5-� 7. In parti
ular, we have tried to develop a systemati
 way to
2000 Mathemati
s Subje
t Classi�
ation : Primary 14H10, 14H51, 14J28. Se
ondary 14C05, 14C25, 14D15, 14E30.
(*) and (***) Member of MIUR-GNSAGA at INdAM "F. Severi".
(**) Resear
h supported by a Marie Curie Intra-European Fellowship within the 6th European Community Frame-
work Programme.
(***) During the last part of the work the author bene�tted from an "a

ueil en délégation au CNRS".
http://arxiv.org/abs/0704.1367v1
2 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA
produ
e rational 
urves on S[2] by showing the existen
e of nodal 
urves on S with hyperellipti
normalizations.
To give an overview of the paper, we 
hoose to start with the se
ond part.
Let (S,H) be a general, smooth, primitively polarizedK3 surfa
e of genus p = pa(H) ≥ 2. We have
[2])R ≃ R[Y ]⊕ R[P1∆], where P1∆ is the 
lass of a rational 
urve in the ruling of the ex
eptional
divisor ∆ ⊂ S[2], and Y := {ξ ∈ S[2]|Supp(ξ) = {p0, y}, with p0 ∈ S and y ∈ C ∈ |H|}, where
p0 and C are 
hosen. One has that P
∆ lies on the boundary of the Mori 
one and by the result of
Huybre
hts and Bou
ksom [32, 11℄ mentioned above, if the Mori 
one is 
losed, then also the other
boundary is generated by the 
lass of a rational 
urve. If X ∼alg aY − bP1∆ is an irredu
ible 
urve
in S[2], di�erent from a �ber of ∆, then we de�ne a/b to be the slope of the 
urve. Thus, the lower
the slope is, the 
loser is X to the boundary of the Mori 
one. Des
ribing the Mori 
one NE(S[2])
amounts to 
omputing
slope(NE(S[2])) := inf
slope(X) | X is an irredu
ible 
urve in S[2]
and, if the Mori 
one is 
losed, then slope(NE(S[2])) = sloperat(NE(S
[2])), where
sloperat(NE(S
[2])) := inf
slope(X) | X is an irredu
ible rational 
urve in S[2]
(See � 6.1, 6.2 and 6.3 for further details.)
If now C ∈ |mH| is an irredu
ible 
urve of geometri
 genus pg(C) ≥ 2 and with hyperellipti
normalization, let g0(C) ≥ pg(C) be the arithmeti
 genus of the minimal partial desingularization of
C that 
arries the g12 (see � 2.1 and � 6.2). By the uni
ity of the g
2, C de�nes a unique irredu
ible
rational 
urve RC ⊂ S[2] with 
lass RC ∼alg mY − (g0(C)+12 )P
∆, 
f. (6.11). (This formula is also
valid if RC is asso
iated to a given g
2 on the normalization of an irredu
ible rational or ellipti
 
urve
C.) Thus, the higher g0(C) (or pg(C)) is, and the lower m is, the lower is the slope of RC . This
motivates the sear
h for 
urves on S with hyperellipti
 normalizations of high geometri
 genus, thus
�unexpe
ted� from Brill-Noether theory.
It is well-known that there exist �nitely many (nodal) rational 
urves, a one-parameter family
of (nodal) ellipti
 
urves, and a two-dimensional family of (nodal) 
urves of geometri
 genus 2 in
|H| (see � 5). Every su
h family yields in a natural way a two-dimensional family of irredu
ible
rational 
urves in S[2], 
f. � 2. Also note that, by a result of Ran [46℄, the expe
ted dimension of
a family of rational 
urves in a symple
ti
 fourfold, when
e a posteriori also of a family of 
urves
with hyperellipti
 normalizations lying on a K3, equals two (
f. Lemma 5.1). In [22, Examples
2.8 and 2.10℄ we found two-dimensional families of nodal 
urves of geometri
 genus 3 in |H| having
hyperellipti
 normalizations when pa(H) = 4 or 5. In this paper we generalize this:
Theorem 5.2. Let (S,H) be a general, smooth, primitively polarized K3 surfa
e of genus p =
pa(H) ≥ 4. Then the family of nodal 
urves in |H| of geometri
 genus 3 with hyperellipti
 normal-
izations is nonempty, and ea
h of its irredu
ible 
omponents is two-dimensional.
The proof takes the whole � 5 and relies on a general prin
iple of 
onstru
ting 
urves with hy-
perellipti
 normalizations on general K3s outlined in Proposition 5.11: �rst 
onstru
t a marked K3
surfa
e (S0,H0) of genus p su
h that |H0| 
ontains a family of dimension ≤ 2 of nodal (possibly
redu
ible) 
urves with the property that a desingularization of some δ > 0 of the nodes is a limit of a
hyperellipti
 
urve in the moduli spa
e Mp−δ of stable 
urves of genus p−δ and su
h that this family
is not 
ontained in a higher-dimensional su
h family. Then 
onsider the parameter spa
e Wp,δ of
pairs ((S,H), C), where (S,H) is a smooth, primitively marked K3 surfa
e of genus p and C ∈ |H|
is a nodal 
urve with at least δ nodes. Now map (the lo
al bran
hes of) Wp,δ into Mp−δ by partially
ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 3
normalizing the 
urves at δ of the nodes and mapping them to their respe
tive 
lasses. The existen
e
of the parti
ular family in |H0| ensures that the image of this map interse
ts the hyperellipti
 lo
us
Hp−δ ⊂ Mp−δ. A dimension 
ount then shows that the dimension of the parameter spa
e I ⊂ Wp,δ

onsisting of ((S,H), C) su
h that a desingularization of some δ > 0 of the nodes of C is a limit
of a hyperellipti
 
urve is at least 21. Now the dominan
e on the 19-dimensional moduli spa
e of
primitively marked K3 surfa
es of genus p follows as the dimension of the spe
ial family on S0 was
The te
hni
al di�
ulties in the proof of Proposition 5.11 mostly arise be
ause the 
urves in the
spe
ial family on S0 may be redu
ible (in fa
t, as in all arguments by degeneration, in pra
ti
al
appli
ations they will very often be). Therefore we need to partially desingularize families of nodal

urves, and this tool is provided in Appendix A by E. Sernesi. Moreover, we need a 
areful study of
the Severi varieties of redu
ible nodal 
urves on K3s, and here we use results of Tannenbaum [55℄.
Given Proposition 5.11, the proof of Theorem 5.2 is then a

omplished by 
onstru
ting a suitable
(S0,H0) in Proposition 5.19 with |H0| 
ontaining a desired two-dimensional family of spe
ial 
urves,
with δ = p− 3, and then showing that the 
urves in the spe
ial family on S0 in fa
t deform to 
urves
with pre
isely δ nodes on the general S in Lemma 5.20. As will be dis
ussed below, showing that
the spe
ial family on S0 is not 
ontained in a family of higher dimension of 
urves with the same
property, is quite deli
ate.
We also show that the asso
iated rational 
urves in S[2] 
over a threefold, 
f. Corollary 5.3, and
that g0 = pg = 3, 
f. Remark 5.23. Turning ba
k to the des
ription of NE(S
[2]), this shows that the

lass of the asso
iated rational 
urves in S[2] is Y − 3
P1∆, so that we obtain (
f. Corollary 6.27):
(6.28) sloperat(NE(S
[2])) ≤ 1
In Propositions 7.2 and 7.7 we present two in�nite series of examples of general primitively polarized
K3 surfa
es (S,H) of in�nitely many degrees su
h that S[2] 
ontains either a P2 (these examples
were shown to us by B. Hassett) or a threefold birational to a P1-bundle over a K3 and �nd the
two-dimensional families of 
urves with hyperellipti
 normalizations in |H| 
orresponding to the lines
and the �bres respe
tively. In parti
ular, these examples show that the bound (6.28) 
an be improved
for in�nitely many degrees of the polarization. Namely, for any n ≥ 6 and d ≥ 2, we get:
(7.4) sloperat(NE(S
[2])) ≤ 2
2n−9 if p = pa(H) = n
2 − 9n+ 20;
(7.9) sloperat(NE(S
[2])) ≤ 1
if p = pa(H) = d
Nevertheless, to our knowledge, (6.28) is the �rst non-trivial bound valid for any genus p of the
polarization.
The proofs of Propositions 7.2 and 7.7 are again by deformation, but unlike the proof of Proposition
5.11, we now deform S
0 of a spe
ial K3 surfa
e S0. The idea is to start with a spe
ial quarti
 surfa
e
S0 ⊂ P3 su
h that S[2]0 
ontains a P2 or a threefold birational to a P1-bundle over itself, perform
the standard involution on S
0 to produ
e a new su
h and then deform S
0 keeping the new one by
keeping a suitable polarization on the surfa
e that is di�erent from OS0(1). Here we use results on
deformations of symple
ti
 fourfolds by Hassett and Ts
hinkel [25℄ and Voisin [57℄.
By a result proved in [22℄, any irredu
ible 
urve C ∈ |H| with hyperellipti
 normalization must
satisfy g0(C) ≤ p+22 , where p = pa(H) (
f. Theorem 6.16 and (6.17)). It is then natural to ask
whether this inequality a
tually ensures the existen
e of su
h 
urves. We 
all this �The hyperellipti
4 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA
existen
e problem� and we see that a positive solution to this problem would yield a bound on the
slope of rational 
urves that is mu
h stronger than the ones obtained above, 
f. (6.25). In this sense,
Theorem 5.2 is hopefully only the �rst step towards stronger existen
e results.
The study of 
urves on S with hyperellipti
 normalizations is not the only way to obtain bounds
on the slope of the Mori 
one of S[2]. In fa
t, an irredu
ible 
urve C ∈ |mH| with a singular point x
of multipli
ity multx(C) yields an irredu
ible 
urve in S
with 
lass mY − (1/2)multx(C) (see the
proof of Theorem 6.18). In parti
ular, if p = pa(H), one has the bound (
f. Theorem 6.21)
(6.22) slope(NE(S[2])) ≤
p−1 ,
obtained by using well-known results on Seshadri 
onstants on S. This bound is stronger than
(6.28) but weaker than the bounds on the slope of the Mori 
one obtained from (7.4) and (7.9).
Moreover, one relatively easily sees that the best bound one 
an obtain by Seshadri 
onstants is in
any 
ase weaker than (7.4) and (7.9) and also weaker than the ones one 
ould obtain by solving �The
hyperellipti
 existen
e problem�, 
f. (6.25). In any 
ase, note that (6.22), (7.4) and (7.9) show that
the bounds tend to zero as the degree of the polarization tends to in�nity, that is,
(6.23) inf
slope(NE(S[2])) | S is a proje
tive K3 surfa
e
and likewise for sloperat(NE(S
[2])).
All the families of 
urves in |H| with hyperellipti
 normalizations we have seen above have in fa
t
dimension equal to two, the expe
ted one. Moreover, a 
ru
ial point in the proof of Theorem 5.2
is to bound the dimensions of families of irredu
ible 
urves with hyperellipti
 normalizations on the
spe
ial K3 surfa
e S0. This brings us over to the des
ription of the �rst part of this paper.
The problem of bounding the dimension of spe
ial families of 
urves on surfa
es, like in our 
ase of

urves with hyperellipti
 normalizations, is interesting in its own, may be studied for larger 
lasses of
surfa
es, and may lead to further appli
ations in other 
ontexts. Whereas methods from adjun
tion
theory have proved very useful for the study of smooth hyperellipti
 
urves on surfa
es [51, 53, 10℄,
these methods do not extend to the 
ase of singular 
urves, where in fa
t very little seems to be
known. Even in the relevant 
ase of nodal 
urves on smooth surfa
es, whose parameter spa
es (the
so-
alled Severi varieties) have re
eived mu
h attention over the years and have been studied also
in relation with moduli problems (see e.g. [49℄ for P2 and [21℄ for surfa
es of general type), the
dimension of their sublo
i 
onsisting of 
urves with hyperellipti
 normalizations is not determined.
The pre
ise question we address is whether there exists an upper bound on the dimension of families
of irredu
ible 
urves on a proje
tive surfa
e with hyperellipti
 normalizations. One easily sees that,
if the 
anoni
al system of the surfa
e is birational, then no 
urve with hyperellipti
 normalization

an move, 
f. e.g. [33℄. On the other hand, taking any surfa
e S admitting a (generi
ally) 2 : 1 map
onto a rational surfa
e R and pulling ba
k the families of rational 
urves on R, we obtain families of
arbitrarily high dimensions of 
urves on S having hyperellipti
 normalizations. Moreover, the in�nite
series of examples in Proposition 7.2 of general, primitively polarized K3 surfa
es (S,H) su
h that
S[2] 
ontains a P2 shows that one 
annot even hope, in general, to �nd a bound in the simplest 
ase
of Pi
ard number one: in fa
t, the (3m− 1)-dimensional family of rational 
urves in |OP2(m)| yields
a (3m− 1)-dimensional family of irredu
ible 
urves in |mH| having hyperellipti
 normalizations, 
f.
� 7.1. Nevertheless, for a large 
lass of surfa
es, it is possible to derive a geometri
 
onsequen
e on
the family V , when its dimension is greater than two:
ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 5
Theorem 4.6'. Let S be a smooth, proje
tive surfa
e with pg(S) > 0. Let V be a redu
ed and
irredu
ible s
heme parametrizing a �at family of irredu
ible 
urves on S with hyperellipti
 normal-
izations (of genus ≥ 2) su
h that dim(V ) ≥ 3. Then the algebrai
 equivalen
e 
lass [C] of the 
urves
parametrized by V has a de
omposition [C] = [D1] + [D2] into algebrai
ally moving 
lasses su
h
that [D1 + D2] ∈ V . Moreover the rational 
urves in S[2] 
orresponding to the irredu
ible 
urves
parametrized by V 
over only a (rational) surfa
e R ⊂ S[2].
In fa
t we prove a stronger result, 
f. Theorem 4.6, that in parti
ular relates the de
omposition
[C] = [D1] + [D2] to the g
2s on the normalizations of the 
urves parametrized by V . This additional
point will in fa
t be the 
ru
ial one in our appli
ation in the proof of Theorem 5.2. An immediate

orollary is that the �naïve� dimension bound one may hope for, thinking about the fa
t that rational

urves in S[2] arising from 
urves on S of geometri
 genera ≤ 2 move in dimension at most two, is
in fa
t true under additional hypotheses on V , 
f. Corollary 4.7. These are satis�ed if e.g. the
Néron-Severi group of S is of rank 1 and generated by the 
lass of a 
urve in V , and seem quite
natural, taking into a

ount the examples of large families mentioned above.
The idea of the proof of Theorem 4.6 is rather simple and geometri
 and illustrates well the ri
h
interplay between the properties of 
urves on S and those of subvarieties of S[2]. The proof relies on
the following two fundamental results:
The �rst is Mori's bend-and-break te
hnique (see Lemma 2.10 for the pre
ise version we need),
whi
h gives a breaking into redu
ible members of a family of rational 
urves of dimension ≥ 3 
overing
a surfa
e.
The se
ond is a suitable version of Mumford's well-known theorem on 0-
y
les on surfa
es with
pg > 0 (
f. Corollaries 3.2 and 3.4). The 
onsequen
e of parti
ular interest to us is that any threefold
in S[2] 
an only 
arry a two-dimensional 
overing family of rational 
urves when pg(S) > 0, 
f.
Proposition 3.6.
Combining those two ingredients, we see that any family satisfying the hypotheses of Theorem 4.6
yields a family of rational 
urves in S[2] of the same dimension ≥ 3, that 
an therefore only 
over
a surfa
e in S[2], on whi
h we 
an apply bend-and-break to produ
e a redu
ible member. Then we
have to show that we 
an also produ
e a de
omposition of the 
urves on S into algebrai
ally moving

lasses, and this is 
arried out in Proposition 4.3.
Beside the appli
ation in the proof of Theorem 5.2, we hope that Theorem 4.6 and the ideas behind
its proof will �nd more appli
ations. One is a Reider-like result for families of singular 
urves with
hyperellipti
 normalizations obtained in [33℄, where also more examples are given.
The paper is organized as follows. We go from the more general results to those pe
uliar to the

ase of K3 surfa
es. We start in � 2 with the 
orresponden
e between 
urves with hyperellipti
normalizations on any smooth surfa
e S and rational 
urves on S[2] and prove other preliminary
results, before turning to the bend-and-break lemma for families of rational 
urves 
overing a surfa
e
in S[2]. The version of Mumford's theorem we need for our purposes is proved in � 3, and then
rephrased in terms of rational quotients. Then we prove (a stronger version of) Theorem 4.6' in � 4.
We then turn to K3 surfa
es and prove Theorem 5.2 along the lines of the degeneration argument
sket
hed above. Se
tion 6, apart from some known fa
ts on the Hilbert s
heme of points on a K3
surfa
e, 
ontains the 
omputation of the 
lasses of rational 
urves in S[2] asso
iated to 
urves in S
with rational, ellipti
 or hyperellipti
 normalizations, as explained in � 2.1. The relation between the
existen
e of su
h a 
urve, its singular Brill-Noether number (an invariant introdu
ed in [22℄) and the
slope of the Mori 
one of S[2] is also dis
ussed, as well as the relation between the slope of the Mori

one and Seshadri 
onstants. We end the paper presenting the two series of examples of general K3
surfa
es whose Hilbert square 
ontains a P2 (respe
tively a threefold birational to a P1-bundle over
a K3) and dis
ussing the numeri
al properties of a line (respe
tively a �bre) in it, as well as those of
6 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA
the asso
iated singular 
urves in S with hyperellipti
 normalizations. In Appendix A by E. Sernesi
the reader will �nd a general result about partial desingularizations of families of nodal 
urves.
A
knowledgements. The authors thank L. Caporaso, O. Debarre, A. Iliev and A. Verra for useful
dis
ussions related to these problems. We are extremely grateful to: C. Ciliberto, for many valuable

onversations and helpful 
omments on the subje
t and for having pointed out some mistakes in a
preliminary version of this paper; B. Hassett, for having pointed out the examples behind Proposition
7.2; E. Sernesi, for many helpful 
onversations and for his Appendix A. We �nally express our
gratitude to the Department of Mathemati
s, Università "Roma Tre" and to the Institut de Re
her
he
Mathématique Avan
ée, Université L. Pasteur et CNRS, where parts of this work have been done,
for the ni
e and warm atmosphere as well as for the kind hospitality.
2. Rational 
urves in S[2]
Let S be a smooth, proje
tive surfa
e. In this se
tion we gather some basi
 results that will be
needed in the rest of the paper. We �rst des
ribe the natural 
orresponden
e between rational 
urves
in S[2] and 
urves on S with rational, ellipti
 or hyperellipti
 normalizations. Then, in � 2.2, we
apply Mori's bend-and-break te
hnique to rational 
urves in Sym2(S) 
overing a surfa
e.
Re
all that we have the natural Hilbert-Chow morphism µ : S[2] → Sym2(S) that resolves
Sing(Sym2(S)) ≃ S. The µ-ex
eptional divisor ∆ ⊂ S[2] is a P1-bundle over S. The Hilbert-
Chow morphism gives an obvious one-to-one 
orresponden
e between irredu
ible 
urves in S[2] not

ontained in ∆ and irredu
ible 
urves in Sym2(S) not 
ontained in Sing(Sym2(S)). We will therefore
often swit
h ba
k and forth between working on S[2] and Sym2(S).
2.1. Irredu
ible rational 
urves in S[2] and 
urves on S. Let T ⊂ S × S[2] be the in
iden
e
variety, with proje
tions p2 : T → S[2] and pS : T → S. Then p2 is �nite of degree two, bran
hed
along ∆ ⊂ S[2]. (In parti
ular, T is smooth as ∆ is.)
Let X ⊂ S[2] be an irredu
ible rational 
urve not 
ontained in ∆. We will now see how X is
equivalent to one of three sets of data on S.
Let νX : X̃ ≃ P1 → X be the normalization and set X ′ := p−12 (X) ⊂ T . By the universal property
of blowing up, we obtain a 
ommutative square:
(2.1) C̃X
p2|X′
// X,
de�ning the 
urve C̃X , ν̃X and f . In parti
ular, ν̃X is birational and C̃X admits a g
2 (i.e., a 2 : 1
morphism onto P1, given by f ), but may be singular, or even redu
ible. Set ν̃ := pS |X′ ◦ν̃X : C̃X → S.
Assume �rst that C̃X is irredu
ible.
We set CX := ν̃(C̃X) ⊂ S. Sin
e X 6⊂ ∆, CX is a 
urve. As C̃X 
arries a g12, it is easily seen that
also the normalization of CX does, that is, CX has rational, ellipti
 or hyperellipti
 normalization.
Moreover, it is easily seen that ν̃ : C̃X → CX is generi
ally of degree one. Indeed, for general x ∈ CX ,
as x 6∈ pS(p−12 (∆)), we 
an write (pS |X′)−1(x) = {(x, x+ y1), . . . , (x, x+ yn)}, where n := deg ν̃. By
de�nition of p2, and sin
e X
′ = p−12 (X), we must have that ea
h (yi, x + yi) ∈ X ′, for i = 1, . . . , n,
and ea
h 
ouple ((x, x+yi), (yi, x+yi)) is the pushdown by ν̃X of an element of the g
2 on C̃X . Hen
e,
ea
h 
ouple (x, yi) is the pushdown by the normalization morphism of an element of the indu
ed g
on the normalization of CX . Sin
e x has been 
hosen general, x 6∈ Sing(CX), so that we must have
n = 1, as 
laimed.
ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 7
In parti
ular, by 
onstru
tion, ν̃ : C̃X → CX is a partial desingularization of CX , in fa
t, it is the
minimal partial desingularization of CX 
arrying the g
2 in question (whi
h is unique, if pg(CX) ≥ 2).
We have therefore obtained:
(I) the data of an irredu
ible 
urve CX ⊂ S together with a partial normalization ν̃ : C̃X → CX
with a g
2 on C̃X (unique, if pg(CX) ≥ 2), su
h that ν̃ is minimal with respe
t to the existen
e
of the g
Next we treat the 
ase where C̃X is redu
ible. In this 
ase, it must 
onsist of two irredu
ible
smooth rational 
omponents, C̃X = C̃X,1 ∪ C̃X,2, that are identi�ed by f .
If ν̃ does not 
ontra
t any of the 
omponents, set CX,i := ν̃(C̃X,i) ⊂ S and nX,i := deg ν̃| eCX,i , for
i = 1, 2. We therefore obtain:
(II) the data of a 
urve CX = nX,1CX,1 + nX,2CX,2 ⊂ S, with nX,i ∈ N, CX,i an irredu
ible,
rational 
urve, a morphism ν̃ : C̃X = C̃X,1 ∪ C̃X,2 → CX,1 ∪ CX,2 (resp. ν̃ : C̃X → CX,1 if
CX,1 = CX,2) that is nX,i : 1 on ea
h 
omponent and where C̃X,i is the normalization of CX,i,
and an identi�
ation morphism f : C̃X,1 ∪ C̃X,2 ≃ P1 ∪ P1 → P1.
If ν̃ 
ontra
ts one of the two 
omponents of C̃X , say C̃X,2, to a point xX ∈ S (it is easily seen
that it 
annot 
ontra
t both), then µ(X) ⊂ Sym2(S) is of the form {xX + CX}, for an irredu
ible

urve CX ⊂ S, whi
h is ne
essarily rational. It is easily seen that CX = ν̃(C̃X,1) and deg ν̃| eCX,1 = 1,
so that we obtain:
(III) the data of an irredu
ible rational 
urve CX ⊂ S together with a point xX ∈ S.
Note that in all 
ases (I)-(III), the support of the 
urve CX on S is simply
(2.2) Supp(CX) = one-dimensional part of {x ∈ S | x ∈ Supp(ξ) for some ξ ∈ X}
and the set is already purely one-dimensional pre
isely unless we are in 
ase (III) with xX 6∈ C.
Conversely from the data (I), (II) or (III) one re
overs an irredu
ible rational 
urve in S[2] not

ontained in ∆. Indeed, in 
ase (I) (resp. (II)), the g12 on C̃X (respe
tively, the identi�
ation f )
indu
es a P1 ⊂ Sym2(C̃X) and this is mapped to an irredu
ible rational 
urve in Sym2(S) by the
natural 
omposed morphism
Sym2(C̃X)
ν̃(2)
// Sym2(CX)
// Sym2(S).
The irredu
ible rational 
urve X ⊂ S[2] is the stri
t transform by µ of this 
urve. In 
ase (III),
X ⊂ S[2] is the stri
t transform by µ of {xX + CX} ⊂ Sym2(S).
We see that the data (III) 
orrespond pre
isely to rational 
urves of type {x0 + C} ⊂ Sym2(S),
where x0 ∈ S is a point and C ⊂ S is an irredu
ible rational 
urve. Moreover, it is easily seen that
the data (II) 
orrespond pre
isely to the images by
α : C̃1 × C̃2 ≃ P1 × P1 −→ C1 + C2 ⊂ Sym2(S),
resp.
α : Sym2(C̃) ≃ P2 −→ Sym2(C) ⊂ Sym2(S),
of irredu
ible rational 
urves in |n1F1 + n2F2| for n1, n2 ∈ N, resp. |nF | for an integer n ≥ 2, where
Pic(C̃1 × C̃2) ≃ Z[F1] ⊕ Z[F2], resp. Pic(Sym2(C̃)) ≃ Z[F ], and C1, C2, resp. C, are irredu
ible
rational 
urves on S and �˜� denotes normalizations. The data of type (II) will however not be
studied more in this paper, where we will fo
us on the other two, mostly on (I).
Note that an irredu
ible rational 
urve X ⊂ Sym2(S) arising from rational (resp. ellipti
) 
urves
C as in 
ase (I) moves in Sym2(C), whi
h is a surfa
e birational to P2 (resp. an ellipti
 ruled surfa
e),
8 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA
and a 
urve X ⊂ Sym2(S) of the form {xX +C} moves in the threefold {S +C}, whi
h is birational
to a P1-bundle over S, and 
ontains Sym2(C).
At the same time, it is well-known that if kod(S) ≥ 0, then rational 
urves on S do not move and
ellipti
 
urves move in at most one-dimensional families. This follows for instan
e from the following
general result (that we will later need in the 
ase pg = 2).
Lemma 2.3. Let S be a smooth, proje
tive surfa
e with kod(S) ≥ 0 
ontaining an n-dimensional
irredu
ible family of irredu
ible 
urves of geometri
 genus pg. Then n ≤ pg and if equality o

urs,
then either the family 
onsists of a single smooth rational 
urve; or kod(S) ≤ 1 and n ≤ 1; or
kod(S) = 0.
Proof. This is �folklore�. For a proof see [33℄. �
As a 
onsequen
e, if kod(S) ≥ 0, then rational 
urves in Sym2(S) arising from rational or ellipti

urves on S move in families of dimension at most two in Sym2(S).
On the other hand, irredu
ible rational 
urves X ⊂ Sym2(S) arising from 
urves on S with
hyperellipti
 normalizations of geometri
 genus pg ≥ 2 (ne
essarily of type (I)), move in a family
whose dimension equals that of the family of 
urves with hyperellipti
 normalizations in whi
h C ⊂ S
moves (by uni
ity of the g
2). Apart from some spe
ial 
ases, it is easy to see that the 
onverse is
also true:
Lemma 2.4. Let {Xb}b∈B be a one-dimensional irredu
ible family of irredu
ible rational 
urves in
Sym2(S) 
overing a (dense subset of a) proper, redu
ed and irredu
ible surfa
e Y ⊂ Sym2(S) that
does not 
oin
ide with Sing(Sym2(S)) ∼= S.
Then C = CXb in S for every b ∈ B (notation as above) if and only if either Y = Sym2(C0), with
either C0 ⊂ S an irredu
ible rational 
urve and C ≡ nC0 for n ≥ 1, or C0 = C ⊂ S an irredu
ible
ellipti
 
urve; or Y = C + C ′ := {p + p′ | p ∈ C, p′ ∈ C ′}, with C an irredu
ible rational 
urve
and C ′ ⊂ S any irredu
ible 
urve; or Y = C1 + C2, with C1, C2 ⊂ S irredu
ible rational 
urves and
C = n1C1 + n2C2 for n1, n2 ∈ N.
Proof. The "if" part is immediate. For the 
onverse, we treat the three 
ases (I)-(III) separately.
If C is as in (I), then 
learly Y ⊂ Sym2(C), so that Y = Sym2(C) and C must be either rational
or ellipti
, as Y is uniruled.
If C = n1C1 + n2C2 as in (II), then either C1 = C2 =: C0 and again Y = Sym
2(C0), or C1 6= C2
and Y = C1 + C2.
Finally, if C is as in (III), then, for every b ∈ B, we have {Xb}b∈B = {xb+C}b∈B for some xb ∈ S,
and the {xb}b∈B de�ne the desired 
urve C ′.
We note that by Lemma 2.3 also the rational 
urves in Sym2(S) arising from singular 
urves of
geometri
 genus 2 on S move in at most two-dimensional families. We will see below that this is a
general phenomenon, under some additional hypotheses. We will fo
us our attention on 
urves with
hyperellipti
 normalizations (of genus pg ≥ 2) in Se
tions 4-7.
2.2. Bend-and-break in Sym2(S). Let V ⊆ Hom(P1,Sym2(S)) be a redu
ed and irredu
ible sub-
s
heme (not ne
essarily 
omplete). We 
onsider the universal map
(2.5) PV := P
1 × V
// Sym2(S)
and assume that the following two 
onditions hold:
(2.6) For any v ∈ V, ΦV (P1 × v) 6⊆ Sing(Sym2(S)) ≃ S; and
(2.7) ΦV is generi
ally �nite
ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 9
(the latter just means that V indu
es a �at family of rational 
urves in Sym2(S) of dimension
dim(V )). Set
(2.8) RV := im(ΦV ),
the Zariski 
losure of im(ΦV ) in Sym
2(S). It is the (irredu
ible) uniruled subvariety of Sym2(S)

overed by the 
urves parametrized by V . In the language of [35, Def. 2.3℄, RV is the 
losure of the
lo
us of the family ΦV . Note that dim(RV ) ≥ 2 if dim(V ) ≥ 1 by (2.7). Moreover (
f. e.g. [24, Prop.
2.1℄),
(2.9) dim(RV ) ≤ 3 if kod(S) ≥ 0.
When RV is a surfa
e, using Mori's bend-and-break te
hnique we obtain the following result. In
the statement we underline the fa
t that the breaking 
an be made in su
h a way that, for general
ξ, η ∈ RV , two 
omponents of the redu
ible (or non-redu
ed) member at the border of the family
pass through ξ and η, respe
tively. This will be 
entral in our appli
ations (Proposition 4.3 and � 5,
where we prove Theorem 5.2). We give the proof be
ause we 
ould not �nd in literature pre
isely
the statement we will need.
Lemma 2.10. Assume that dim(V ) ≥ 3 and dim(RV ) = 2.
Let ξ and η be any two distin
t general points of RV . Then there is a 
urve Yξ,η in RV su
h that
Yξ,η is algebrai
ally equivalent to (ΦV )∗(P
v) and either
(a) there is an irredu
ible nonredu
ed 
omponent of Yξ,η 
ontaing ξ and η; or
(b) there are two distin
t, irredu
ible 
omponents of Yξ,η 
ontaing ξ and η, respe
tively.
Proof. Sin
e dim(V ) ≥ 3 by assumption, by (2.7) we 
an pi
k a one-dimensional smooth subs
heme
B = Bξ,η ⊂ V parametrizing 
urves in V su
h that (ΦV )∗(P1 × v) 
ontains both ξ and η, for every
v ∈ B. We therefore have a family of rational 
urves:
(2.11) ΦB := (ΦV )|B : P1 ×B −→ RV .
and two marked (distin
t) points x, y ∈ P1 su
h that ΦB(x×B) = ξ and ΦB(y ×B) = η, su
h that
ea
h ΦB(P
1 × v) is non
onstant, for any v ∈ B; in parti
ular ΦB(P1 ×B) is a surfa
e.
As in the proofs of [36, Lemma 1.9℄ and [35, Cor. II.5.5℄, let B be any smooth 
ompa
ti�
ation
of B. Consider the surfa
e P1 × B. Let 0 ∈ B denote a point at the boundary, P10 the �bre over 0
of the proje
tion onto the se
ond fa
tor and x0, y0 ∈ P10 ⊂ P1 ×B the 
orresponding marked points.
By the Rigidity Lemma [36, Lemma 1.6℄, ΦB 
annot be de�ned at the point x0, as in the proof of
[36, Cor. 1.7℄, and the same argument works for y0.
Therefore, to resolve the indetermina
ies of the rational map ΦB : P
1 × B − − → RV , we must
at least blow up P1 × B at the points x0 and y0. Now let W be the blow-up of P1 × B su
h that
ΦB : W −→ RV is an extension of ΦB , that is, we have a 
ommutative diagram
P1 ×B
//___ RV .
Let Ex0 := π
−1(x0) and Ey0 := π
−1(y0). Note that neither of these 
an be 
ontra
ted by ΦB , for
otherwise ΦB itself would be de�ned at x0 or y0.
Therefore the 
urve ΦB(Ex0) has an irredu
ible 
omponent Γξ 
ontaining ξ and the 
urve ΦB(Ey0)
has an irredu
ible 
omponent Γη 
ontaining η and by 
onstru
tion, Γξ+Γη ⊆ ΦB∗(π−1(P1× 0)) and
the latter is the desired 
urve Yξ,η. The two 
ases (a) and (b) o

ur as Γξ = Γη or Γξ 6= Γη,
respe
tively. �
10 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA
3. Rationally equivalent zero-
y
les on surfa
es with pg > 0
In this se
tion we extend to the singular 
ase a 
onsequen
e of Mumford's result on zero-
y
les
on surfa
es with pg > 0 (
f. [42, Corollary p. 203℄) and reformulate the results in terms of rational
quotients.
3.1. Mumford's Theorem. The main result of this subse
tion, whi
h we prove in detail for the
reader's 
onvenien
e, relies on the following generalization of Mumford's result (
f. [58, Chapitre 22℄
and referen
es therein, for a detailed a

ount).
Theorem 3.1. (see [58, Prop. 22.24℄) Let T and Y be smooth proje
tive varieties. Let Z ⊂ Y × T
be a 
y
le of 
odimension equal to dim(T ). Suppose there exists a subvariety T ′ ⊂ T of dimension k0
su
h that, for all y ∈ Y , the zero-
y
le Zy is rationally equivalent in T to a 
y
le supported on T ′.
Then, for all k > k0 and for all η ∈ H0(T,ΩkT ), we have
[Z]∗η = 0 in H0(Y,ΩkY )
where, as 
ostumary, [Z]∗η denotes the di�erential form indu
ed on Y by the 
orresponden
e Z.
Mumford's original �symple
ti
� argument and the theorem above yield the following result (see
[42, Corollary p. 203℄).
Corollary 3.2. Let S be a smooth, irredu
ible proje
tive surfa
e with pg(S) > 0 and Σ ⊂ S[n] a
redu
ed, irredu
ible (possibly singular) 
omplete subs
heme su
h that µ(Σ) 6⊂ Sing(Symn(S)), where
µ : S[n] → Symn(S) is the Hilbert-Chow morphism.
If there exists a subvariety Γ ⊂ Symn(S) su
h that dim(Γ) ≤ 1, Γ 6⊂ Sing(Symn(S)) and all
the zero-
y
les parametrized by µ(Σ) are rationally equivalent to zero-
y
les supported on Γ, then
dim(Σ) ≤ n.
Proof. Let π : Σ̃ → Σ ⊂ S[n] be the desingularization morphism of Σ. Let Z = Λπ ⊂ Σ̃× S[n] be the
graph of π. Then Z ∼= Σ̃, so that codim(Z) = dim(S[n]), as in Theorem 3.1. By assumption, µ(Σ)
parametrizes zero-
y
les of length n on S that are all rationally equivalent to zero-
y
les supported on
Γ, with dim(Γ) ≤ 1. Sin
e µ(Σ) is not 
ontained in Sing(Symn(S)) by assumption, µ|Σ : Σ → µ(Σ)
is birational. If Γ′ denotes the stri
t transform of Γ under µ, we get that dim(Γ′) ≤ 1.
We 
an apply Theorem 3.1 with Z = Y = Σ̃, T = S[n] and T ′ = Γ′. Thus, for ea
h k > 1 and for
ea
h η ∈ H0(Ωk
), [Z]∗η = 0 in H0(Σ̃,Ωk
Let ω ∈ H0(S,KS) be a non-zero 2-form on S. As in [42, Corollary℄, we de�ne:
ω(n) :=
p∗i (ω) ∈ H0(Sn,Ω2Sn)
where Sn is the nth-
artesian produ
t and pi is the natural proje
tion onto the i
fa
tor, 1 ≤ i ≤ n.
The form ω(n) is Sym(n)-invariant and, sin
e we have that µ is surje
tive, this indu
es a 
anoni
al
2-form ω
µ ∈ H0(S[n],Ω2S[n]) (see [42, �1℄, where ω
µ = ηµ in the notation therein). From what we
observed above, [Z]∗(ω
µ ) = 0 as a form in H
0(Σ̃,Ω2
). Consider
(Symn(S))0 :=
xi | xi 6= xj , 1 ≤ i 6= j ≤ n and such that ω(xi) ∈ Ω2S,xi is not 0
Then (Symn(S))0 ⊂ Symn(S) is an open dense subs
heme that is isomorphi
 to its preimage via µ
in S[n]. For ea
h ξ ∈ (Symn(S))0, ξ is a smooth point and
πn : S
n → Symn(S)
ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 11
is étale over ξ. Thus, the 2-form ω(n) ∈ H0(Sn,Ω2Sn) is non-degenerate on the open subset (Sn)0 of
points in the preimage of (Symn(S))0, i.e. it de�nes a non-degenerate skew-symmetri
 form on the
tangent spa
e of (Sn)0.
Let π0n := πn|(Sn)0 ; sin
e π0n : (Sn)0 → (Sym
n(S))0 is étale, there exists a 2-form
0 ∈ H0((Symn(S))0,Ω2(Symn(S))0)
su
h that ω(n) = π∗n(ω
0 ) and ω
0 is also non-degenerate. Therefore, the maximal isotropi
 subspa
es
0 (ξ) are n-dimensional.
Now Σ ⊂ S[n] and Σ ∩ µ−1((Symn(S))0) 6= ∅, sin
e µ(Σ) 6⊂ Sing(Symn(S)) by assumption. Sin
e
Σ is redu
ed, let ξ ∈ Σ ∩ µ−1((Symn(S))0) be a smooth point. Then, sin
e Σsmooth = π−1(Σsmooth),
by abuse of notation we still denote by ξ ∈ Σ̃ the 
orresponding point. We know that [Z]∗ω[n]µ (ξ) = 0
in the tangent spa
e Tξ(Σ̃). Sin
e
ξ ∈ Σsmooth ∩ µ−1((Symn(S))0) ⊂ (Symn(S))0,
then [Z]∗(ω
µ ) = ω
0 |Σsmooth∩µ−1((Symn(S))0). This implies dim(Σ) ≤ n. �
3.2. The property RCC and rational quotients. Re
all that a variety T (not ne
essarily proper
or smooth) is said to be rationally 
hain 
onne
ted (RCC, for brevity), if for ea
h pair of very general
points t1, t2 ∈ T there exists a 
onne
ted 
urve Λ ⊂ T su
h that t1, t2 ∈ Λ and ea
h irredu
ible

omponent of Λ is rational (see [35℄). Furthermore, by [16, Remark 4.21(2)℄, if T is proper and RCC,
then ea
h pair of points 
an be joined by a 
onne
ted 
hain of rational 
urves.
Also re
all that, for any smooth variety T , there exists a variety Q, 
alled the rational quotient of
T , together with a rational map
(3.3) f : T −− → Q,
whose very general �bres are equivalen
e 
lasses under the RCC-equivalen
e relation (see, for in-
stan
e, [16, Theorem 5.13℄ or [35, IV, Thm. 5.4℄).
In this language, an equivalent statement of Corollary 3.2 is:
Corollary 3.4. Let S be a smooth, proje
tive surfa
e with pg(S) > 0. If Y ⊂ S[n] is a 
omplete
subvariety of dimension > n not 
ontained in Exc(µ), then any desingularization of Y has a rational
quotient of dimension at least two.
Proof. Let Ỹ be any desingularization of Y and Q its rational quotient. Up to resolving the indeter-
mina
ies of f : Ỹ −− → Q, we may assume that f is a proper morphism whose very general �bre is
a RCC-equivalen
e 
lass, so that in parti
ular ea
h �bre is RCC (see [35, Thm. 3.5.3℄).
If dim(Q) = 0, it follows that Ỹ (so also Y ) is RCC, 
ontradi
ting Corollary 3.2.
If dim(Q) = 1, then by 
utting Ỹ with dim(Y )− 1 general very ample divisors, we get a 
urve Γ′
that interse
ts every �bre of f . Every point of Ỹ is 
onne
ted by a 
hain of rational 
urves to some
point on Γ′. We thus obtain a 
ontradi
tion by Corollary 3.2 (with Γ the image of Γ′ in Sym2(S)). �
Let now RV be the variety 
overed by a family of rational 
urves in Sym
2(S) parametrized by V ,
as de�ned in (2.8), R̃V be any desingularization of RV and QV be the rational quotient of R̃V . Of

ourse dim(QV ) ≤ dim(RV )− 1, as RV is uniruled by 
onstru
tion.
Lemma 3.5. If dim(V ) ≥ dim(RV ), then dim(QV ) ≤ dim(RV )−2 (for any desingularization R̃V of
RV ). In parti
ular, if dim(V ) ≥ 2 and dim(RV ) = 2, then any desingularization of RV is a rational
surfa
e.
12 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA
Proof. With notation as in � 2.2, we have dim(PV ) ≥ dim(RV ) + 1, so that the general �bre of ΦV
is at least one-dimensional, 
f. (2.5). This means that, if ξ is a general point of RV , there exists a
family of rational 
urves in RV passing through ξ, of dimension ≥ 1. Of 
ourse the same is true for
a general point of R̃V . Thus, the very general �bre of f in (3.3) has dimension at least two, when
e
dim(QV ) ≤ dim(RV ) − 2. The last statement follows from the fa
t that any smooth surfa
e that is
RCC is rational (
f. [35, IV.3.3.5℄). �
Combining Corollary 3.4 and Lemma 3.5, we then get:
Proposition 3.6. If pg(S) > 0 and dim(V ) ≥ 2, then either
(i) RV is a surfa
e with rational desingularization; or
(ii) dim(V ) = 2, RV is a threefold and any desingularization of RV has a two-dimensional rational
quotient.
Proof. By (2.9), dim(RV ) = 2 or 3. If dim(RV ) = 2, then (i) holds by Lemma 3.5. If dim(RV ) = 3,
then dim(QV ) = 2 by Corollary 3.4. Hen
e dim(V ) = 2 by Lemma 3.5 and (ii) holds. �
Remark 3.7. Let S be a smooth, proje
tive surfa
e with pg(S) > 0 and let Y ⊂ S[2] be a uniruled
threefold di�erent from Exc(µ), where µ : S[2] → Sym2(S) is the Hilbert-Chow morphism.
Take a 
overing family {Cv}v∈V of rational 
urves on Y . By Corollary 3.4 the family must be
two-dimensional (see Lemma 3.5). Then the 
urves in the 
overing family yield, via the 
orrespon-
den
e des
ribed in � 2.1, 
urves on S with rational, ellipti
 or hyperellipti
 normalizations, and the

orresponden
e is one-to-one in the hyperellipti
 
ase. We therefore see that we must be in one of
the following 
ases:
(a) S 
ontains an irredu
ible rational 
urve Γ and
Y = {ξ ∈ S[2] | Supp(ξ) ∩ Γ 6= ∅};
(b) S 
ontains a one-dimensional irredu
ible family {E}v∈V of irredu
ible ellipti
 
urves and
Y = {ξ ∈ E[2]v }v∈V ;
(
) S 
ontains a two-dimensional, irredu
ible family of irredu
ible 
urves with hyperellipti
 nor-
malizations, not 
ontained in a higher dimensional irredu
ible family, and Y is the lo
us

overed by the 
orresponding rational 
urves in S[2].
(Note that in fa
t 
ase (b) 
an only o

ur for kod(S) ≤ 1 by Lemma 2.3 and 
ase (
) only when |KS |
is not birational. The latter fa
t is easy to see, 
f. e.g. [33℄.)
In the 
ase of K3 surfa
es, uniruled divisors play a parti
ularly important r�le [32, �5℄, 
f. � 7.
Now all 
ases (a)-(
) above o

ur on a general, proje
tive K3 surfa
e with a polarization of genus
≥ 6. In fa
t, 
ases (a) and (b) o

ur on any proje
tive K3 surfa
e sin
e it ne
essarily 
ontains a
one-dimensional family of irredu
ible, ellipti
 
urves and a zero-dimensional family of rational 
urves,
by a well-known theorem of Mumford (see the proof in [38, pp. 351-352℄ or [2, pp. 365-367℄). Case
(
) o

urs on a general primitively polarized K3 surfa
e of genus p ≥ 6 by Corollary 5.3 below with
a family of 
urves of geometri
 genus 3. In addition to this, in Proposition 7.7 we will see that there
is another threefold as in (
) arising from 
urves of geometri
 genus > 3 in the hyperplane linear
system on general proje
tive K3 surfa
es of in�nitely many degrees.
Moreover, there is not a one-to-one 
orresponden
e between families as in (a), (b) or (
) above
and uniruled threefolds in S[2]. In fa
t, in Proposition 7.2 we will see that there is a two-dimensional
family of 
urves with hyperellipti
 normalizations, as in (
), in the hyperplane linear systems on
general K3 surfa
es of in�nitely many degrees whose asso
iated rational 
urves 
over only a P2 in
S[2].
ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 13
4. Families of 
urves with hyperellipti
 normalizations
The purpose of this se
tion is to study the dimension of families of 
urves on a smooth proje
tive
surfa
e S with hyperellipti
 normalizations.
We �rst remark that it is not di�
ult to see that if |KS | is birational, then the dimension of su
h
a family is for
ed to be zero (see e.g. [33℄). At the same time it is easy to �nd obvious examples
of surfa
es, even with pg(S) > 0, with large families of 
urves with hyperellipti
 normalizations,
namely surfa
es admitting a �nite 2 : 1 map onto a rational surfa
e. (For examples of su
h 
ases,
see e.g. [26, 27, 28, 29, 48, 51, 53, 10℄ to mention a few.) In these 
ases one 
an pull ba
k the
families of rational 
urves on the rational surfa
e to obtain families of 
urves on S with hyperellipti
normalizations of arbitrarily high dimensions. Moreover, in Proposition 7.2 below we will see that
even a general, primitively polarized K3 surfa
e (S,H), for in�nitely many degrees, 
ontains a P2
in its Hilbert square, whi
h is not 
ontained in ∆ (but the surfa
e is not a double 
over of a P2,
by generality). Therefore, by the 
orresponden
e in � 2.1, S 
ontains large families of 
urves with
hyperellipti
 normalizations. One 
an see that in all these examples of large families the algebrai
equivalen
e 
lass of the members breaks into nontrivial e�e
tive de
ompositions. For example, in the
mentioned K3 
ase of Proposition 7.2, we will see that the 
urves in |OP2(n)| in P2 ⊂ S[2] 
orrespond
to 
urves in |nH|. In this se
tion we will see that this is a general phenomenon, with the help of
Lemma 2.10.
To this end, let V be a redu
ed and irredu
ible s
heme parametrizing a �at family of 
urves on S
all having 
onstant geometri
 genus pg ≥ 2 and hyperellipti
 normalizations. Let ϕ : C → V be the
universal family. Normalizing C we obtain, possibly restri
ting to an open dense subs
heme of V , a
�at family ϕ̃ : C̃ → V of smooth hyperellipti
 
urves of genus pg ≥ 2 (
f. [56, Thm. 1.3.2℄). Let ωeC/V
be the relative dualizing sheaf. As in [37, Thm. 5.5 (iv)℄, 
onsider the morphism γ : C̃ → P(ϕ̃∗(ωeC/V ))
over V . This morphism is �nite and of relative degree two onto its image, whi
h we denote by PV .
We thus obtain a universal family ψ : PV → V of rational 
urves mapping to Sym2(S), as in (2.5),
satisfying (2.6) and (2.7). (Stri
tly speaking, (2.5) denoted a universal family of maps, whereas it
now denotes a universal family of 
urves.) To summarize, re
alling (2.8), we have
(4.1) C̃
// PV
// RV
Also note that (4.1) is 
ompatible with the 
orresponden
e of 
ase (I) in � 2.1, in the sense that,
for general v ∈ V , we have (using the same notation as in � 2.1)
(4.2) π(ϕ̃−1(v)) = pS(p
2 Xv) = (pS)∗(p
2 Xv) = CXv , with Xv = µ
ΦV (ψ
−1(v))
⊂ S[2],
where µ is the Hilbert-Chow morphism (in parti
ular, pS and p2 are the �rst and se
ond proje
tions,
respe
tively, from the in
iden
e variety T ⊂ S × S[2]). Note that the se
ond equality in (4.2) follows
as pS is generi
ally one-to-one on the 
urves in question, as we saw in � 2.1. This will be 
entral in
the proof of the next result.
We now apply Lemma 2.10 to �break� the 
urves on S.
Proposition 4.3. Let S be a smooth, proje
tive surfa
e and V and RV as above. Assume that
dim(V ) ≥ 3 and dim(RV ) = 2 and let [C] be the algebrai
 equivalen
e 
lass of the members
parametrized by V .
Then there is a de
omposition into two e�e
tive, algebrai
ally moving 
lasses
[C] = [D1] + [D2]
14 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA
su
h that, for general ξ, η ∈ RV , there are e�e
tive divisors D′1 ∼alg D1 and D′2 ∼alg D2 su
h that
ξ ⊂ D′1 and η ⊂ D′2 and [D′1 +D′2] ∈ V , where V is the 
losure of V in the 
omponent of the Hilbert
s
heme of S 
ontaining V .
Proof. For general ξ, η ∈ RV , both being supported at two distin
t points on S, let B = Bξ,η ⊂ V be
as in the proof of Lemma 2.10 and B be any smooth 
ompa
ti�
ation of B. By abuse of notation,
we will 
onsider ξ and η as being points in S[2]. By (the proof of) Lemma 2.10, using the Hilbert-
Chow morphism, there is a �at family {Xb}b∈B of 
urves in the surfa
e µ−1∗ (RV ) ⊂ S[2] (where µ
is the Hilbert-Chow morphism as usual) parametrized by B, su
h that, for general b ∈ B, Xb is an
irredu
ible rational 
urve and
(4.4) CXb = (pS)∗(p
2 (Xb)) = π(ϕ̃
−1(b)),
with notation as in � 2.1 (
f. (4.2)). In parti
ular, {CXb}b∈B is a one-dimensional nontrivial subfamily
of the family {CXv}v∈V given by V . Moreover, for some b0 ∈ B \B, we have Xb0 ⊇ Yξ + Yη, where
Yξ and Yη are irredu
ible rational 
urves (possibly 
oin
iding) su
h that ξ ∈ Yξ and η ∈ Yη. Also
note that Yξ, Yη 6⊂ ∆ ⊂ S[2].
Pulling ba
k to the in
iden
e variety T ⊂ S × S[2], we obtain a �at family {X ′b := p
2 (Xb)}b∈B of

urves in T , su
h that
(4.5) X ′b0 := p
2 (Xb) ⊇ p
2 (Yξ) + p
2 (Yη) =: Y
ξ + Y
Note that the family {X ′b}b∈B is in fa
t a family of 
urves in the in
iden
e variety T0 ⊂ S×µ−1∗ (RV ),
whi
h is a surfa
e 
ontained in T . Sin
e pS maps this family to a family of 
urves 
overing (an open
dense subset of) S, by (4.4), we see that (pS)|T0 is surje
tive, in parti
ular generi
ally �nite. Thus,

hoosing ξ and η general enough, we 
an make sure they lie outside of the images by p2 of the �nitely
many 
urves 
ontra
ted by (pS)|T0 . Hen
e q
−1(Yξ) and q
−1(Yη) are not 
ontra
ted by pS .
Therefore, re
alling (4.4) and (4.5) and letting b′ ∈ B be a general point, we get
C ∼alg (pS)∗X ′b′ ∼alg (pS)∗X ′b0 ⊇ (pS)∗Y
ξ + (pS)∗Y
η ⊇ Dξ +Dη,
where Dξ := p(q
−1Yξ) and Dη := p(q
−1Yη).
By 
onstru
tion we have Dξ ⊃ ξ and Dη ⊃ η, viewing ξ and η as length-two subs
hemes of S.
(Note that Dξ and Dη are not ne
essarily distin
t.) Possibly after adding additional 
omponents to
Dξ and Dη , we 
an in fa
t assume that
C ∼alg (pS)∗X ′b′ = Dξ +Dη,
with Dξ and Dη not ne
essarily redu
ed and irredu
ible. Sin
e this 
onstru
tion 
an be repeated
for general ξ, η ∈ RV and the set {x ∈ S | x ∈ Supp(ξ) for some ξ ∈ RV } is dense in S, as the

urves parametrized by V 
over the whole surfa
e S, the obtained 
urves Dξ and Dη must move in
an algebrai
 system of dimension at least one.
By 
onstru
tion, Dξ + Dη lies in the border of the family ϕ : C → V of 
urves on S, and as
su
h, [Dξ +Dη ] lies in the 
losure of V in the 
omponent of the Hilbert s
heme of S 
ontaining V .
Moreover, as the number of su
h de
ompositions is �nite (as S is proje
tive and the divisors are
e�e
tive), we 
an �nd one de
omposition [C] = [D1] + [D2] holding for general ξ, η ∈ RV . �
The next two results are immediate 
onsequen
es:
Theorem 4.6. Let S be a smooth, proje
tive surfa
e with pg(S) > 0. Then the following 
onditions
are equivalent:
(i) S[2] 
ontains an irredu
ible surfa
e R with rational desingularization, su
h that R 6= µ−1∗ (C1+
C2), µ
∗ (Sym
2(C)) for rational 
urves C,C1, C2 ⊂ S and R 6⊂ Exc(µ), where µ : S[2] →
Sym2(S) is the Hilbert-Chow morphism;
ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 15
(ii) S 
ontains a �at family of irredu
ible 
urves with hyperellipti
 normalizations of geometri
genus pg ≥ 3, parametrized by a redu
ed and irredu
ible s
heme V su
h that dim(V ) ≥ 3.
Furthermore, if any of the above 
onditions holds, then
(a) the rational 
urves in S[2] that 
orrespond to the irredu
ible 
urves parametrized by V , 
over
only the surfa
e R in S[2]; and
(b) the algebrai
 equivalen
e 
lass [C] of the 
urves parametrized by V has an e�e
tive de
om-
position [C] = [D1] + [D2] into algebrai
ally moving 
lasses su
h that, for general ξ, η ∈ R,
there are e�e
tive divisors D′1 ∼alg D1 and D′2 ∼alg D2 su
h that ξ ⊂ D′1, η ⊂ D′2 and
[D′1 + D
2] ∈ V , where V is the 
losure of V in the 
omponent of the Hilbert s
heme of S

ontaining V .
Proof. Assume (ii) holds. By Proposition 3.6 we have that RV ⊂ Sym2(S) is a surfa
e with rational
desingularization, so that (i) holds.
Assume now that (i) holds. Then R 
arries a family of rational 
urves of dimension n ≥ 3. By
Lemma 2.4 and the assumptions in (i), this yields an n-dimensional family of 
urves on S that have
rational, ellipti
 or hyperellipti
 normalizations. From Lemma 2.3, we get (ii).
Finally, assume that these 
onditions hold. Then (a) follows from Proposition 3.6 again, where R
is the proper transform via µ of the surfa
e RV therein; �nally, (b) follows from Proposition 4.3. �
Corollary 4.7. Let S be a smooth, proje
tive surfa
e with pg(S) > 0 and V be a redu
ed, irredu
ible
s
heme parametrizing a �at family of irredu
ible 
urves with hyperellipti
 normalizations (of geometri
genus ≥ 2). Denote by [C] the algebrai
 equivalen
e 
lass of the members of V .
If [C] has no de
omposition into e�e
tive, algebrai
ally moving 
lasses, then dim(V ) ≤ 2.
In parti
ular, Corollary 4.7 holds when e.g. NS(S) = Z[C].
The examples with the double 
overs of smooth rational surfa
es and the result in Proposition 7.2
mentioned above, show that the results above are natural.
The statement in Theorem 4.6(b) shows that in fa
t the length-two zero-dimensional s
hemes on
the 
urves in the family 
orresponding to the elements of the g
2s on their normalization, are in fa
t
�generi
ally 
ut out� by moving divisors in a �xed algebrai
 de
omposition of the 
lass of the members
in the family. This reminds of the nowadays well-known results of Reider and their generalizations
[47, 8, 9℄. In fa
t, Theorem 4.6(b) 
an be used to prove a Reider-like result involving the arithmeti
and geometri
 genera of the 
urves in the family, 
f. [33℄. Moreover, the pre
ise statement in Theorem
4.6(b) will be 
ru
ial in the next se
tion, where we will prove existen
e of 
urves with hyperellipti
normalizations by degeneration methods.
5. Nodal 
urves of geometri
 genus 3 with hyperellipti
 normalizations on K3
surfa
es
In the rest of the paper we will fo
us on the existen
e of 
urves with �Brill-Noether spe
ial�
hyperellipti
 normalizations (i.e. of geometri
 genera > 2) and in this se
tion we will see that
Theorem 4.6(b) is parti
ularly suitable to prove existen
e results by degeneration arguments.
To do this and to dis
uss some 
onsequen
es on S[2], we will in the rest of the paper fo
us on K3
surfa
es, whi
h in fa
t were one of our original motivations for this work.
We start with the following observation 
ombining a result of Ran, already mentioned in the
Introdu
tion, with the results from the previous se
tion.
Lemma 5.1. Let S be a smooth, proje
tive K3 surfa
e and L be a globally generated line bundle of
se
tional genus p ≥ 2 on S. Let |L|hyper ⊆ |L| be the subs
heme parametrizing irredu
ible 
urves in
|L| with hyperellipti
 normalizations.
16 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA
Then, any irredu
ible 
omponent of |L|hyper has dimension ≥ 2, with equality holding if L has no
de
omposition into moving 
lasses.
Proof. Any n-dimensional 
omponent of |L|hyper yields an n-dimensional family of irredu
ible rational

urves in S[2]. By [46, Cor. 5.1℄, we have n ≥ 2. The last statement follows from Corollary 4.7. �
The main aim of this se
tion is to apply Theorem 4.6(b) to prove:
Theorem 5.2. Let (S,H) be a general, smooth, primitively polarized K3 surfa
e of genus p =
pa(H) ≥ 4. Then the family of nodal 
urves in |H| of geometri
 genus 3 with hyperellipti
 normal-
izations is nonempty, and ea
h of its irredu
ible 
omponents is two-dimensional.
In [22℄ we studied whi
h linear series may appear on normalizations of irredu
ible 
urves on K3
surfa
es. To do so, we introdu
ed a singular Brill-Noether number ρsing(pa, r, d, pg) whose negativity,
when Pic(S) ≃ Z[H], ensures non-existen
e of 
urves in |H|, with pa = pa(H) and of geometri
 genus
pg, having normalizations admitting a g
d (we will return to this in � 6.3 below). Moreover, in [22,
Examples 2.8 and 2.10℄, we already gave examples of nodal 
urves with hyperellipti
 normalizations
with geometri
 genus 3 and arithmeti
 genus 4 or 5. Theorem 5.2 shows that this is a general
phenomenon. The proof will be given in the remainders of this se
tion. Moreover, we will also
determine the dimension of the lo
us 
overed in S[2] by the rational 
urves asso
iated to 
urves in a

omponent of the family:
Corollary 5.3. Let (S,H) be a general, smooth, primitively polarized K3 surfa
e of genus p =
pa(H) ≥ 6. Then the subs
heme of |H| parametrizing nodal 
urves of geometri
 genus 3 with hyper-
ellipti
 normalizations 
ontains a two-dimensional 
omponent V su
h that dim(RV ) = 3.
This 
orollary in parti
ular shows that all three 
ases in Remark 3.7 o

ur on a general K3 surfa
e.
In � 6.2-6.3 we will both 
ompute the 
lasses of the 
orresponding rational 
urves in S[2] (see (6.26))
and dis
uss some of the 
onsequen
es of Theorem 5.2 on the Mori 
one of S[2].
Before starting on the proof of Theorem 5.2, we re
all that, for any smooth surfa
e S and any
line bundle L on S, su
h that |L| 
ontains smooth, irredu
ible 
urves of genus p := pa(L), and any
positive integer δ ≤ p, one denotes by V|L|,δ the lo
ally 
losed and fun
torially de�ned subs
heme
of |L| parametrizing the universal family of irredu
ible 
urves in |L| having δ nodes as the only
singularities and, 
onsequently, geometri
 genus pg := p − δ. These are 
lassi
ally 
alled Severi
varieties of irredu
ible, δ-nodal 
urves on S in |L|.
It is nowadays well-known, as a dire
t 
onsequen
e of Mumford's theorem on the existen
e of nodal
rational 
urves on K3 surfa
es (see the proof in [38, pp. 351-352℄ or [2, pp. 365-367℄) and standard
results on Severi varieties, that if (S,H) is a general, primitively polarized K3 surfa
e of genus p ≥ 3,
then the Severi variety V|H|,δ is nonempty and regular, i.e. it is smooth and of the expe
ted dimension
p− δ, for ea
h δ ≤ p (
f. [55, Lemma 2.4 and Theorem 2.6℄; see also e.g. [15, 20℄).
The regularity property follows from the fa
t that, sin
e by de�nition V|L|,δ parametrizes irredu
ible

urves, the nodes of these 
urves impose independent 
onditions on |L| (
f. [15, 20℄ and [55, Remark
2.7℄). From equisingular deformation theory, this implies that suitable obstru
tions to some lo
ally
trivial deformations are zero. In other words, it implies �rst that, for any δ′ > δ, V|L|,δ′ ⊂ V |L|,δ (see
[52, Anhang F℄, [59℄ and [50, Thm. 4.7.18℄ for P2 and [55, � 3℄ forK3s). Furthermore, if [C] ∈ V|L|,δ+k,
k > 0, is a general point of an irredu
ible 
omponent, the fa
t that the nodes impose independent

onditions allows to 
learly des
ribe what V |L|,δ looks like lo
ally around the point [C]: it is the
union of
smooth bran
hes through [C], ea
h bran
h 
orresponding to a 
hoi
e of δ "marked"
(or "assigned") nodes among the δ+ k nodes of C, and these bran
hes interse
t transversally at [C];
moreover, the other k "unassigned" nodes of C disappear when one deforms [C] in the 
orresponding
bran
h of V |L|,δ (see [52, Anhang F℄, [59℄ and [49, � 1℄ for P
and [55, � 3℄ for K3s).
ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 17
The situation is slightly di�erent for redu
ible, nodal 
urves in |L|. Sin
e they appear in the proof
of Theorem 5.2, we also have to take 
are of this 
ase. To this end, we de�ne the �degenerated�
version of V|L|,δ by
W|L|,δ :=
C ∈ |L| | C, not ne
essarily irredu
ible, has only nodes(5.4)
as singularities and at least δ nodes
For the same reasons as above, W|L|,δ is a lo
ally 
losed subs
heme of |L|. Note that
(5.5) W|L|,δ = ∪δ′≥δV|L|,δ′ if all the 
urves in |L| are irredu
ible,
whi
h is a partial 
ompa
ti�
ation of V|L|,δ.
Let [C] ∈ W|L|,δ. Choosing any subset {p1, . . . , pδ} of δ of its nodes, one obtains a pointed 
urve
(C; p1, . . . , pδ), where p1, . . . , pδ are also 
alled themarked (or assigned) nodes of C (
f. [55, De�nitions
3.1-(ii) and 3.6-(i)℄).
Re
all that there exists an algebrai
 s
heme, whi
h we denote by
(5.6) B(C; p1, p2, . . . , pδ),
lo
ally 
losed in |L|, representing the fun
tor of in�nitesimal deformations of C in |L| that preserve
the marked nodes, i.e. the fun
tor of lo
ally trivial in�nitesimal deformations of the pointed 
urve
(C; p1, . . . , pδ) (
f. [55, Proposition 3.3℄, where we have identi�ed the s
hemes therein with their
proje
tions into the linear system |L|). In other words, B(C; p1, p2, . . . , pδ) is the lo
al bran
h of
W|L|,δ around [C] ∈W|L|,δ, 
orresponding to the 
hoi
e of the δ marked nodes. We have:
Theorem 5.7. (
f. [55, Theorem 3.8℄) Let (C; p1, . . . , pδ) be as above. Assume that the general
element of |L| is a smooth, irredu
ible 
urve and that the partial normalization of C at the δ marked
nodes p1, . . . , pδ is a 
onne
ted 
urve.
Then B(C; p1, p2, . . . , pδ) is smooth at the point [(C; p1, p2, . . . , pδ)] of dimension dim(|L|)− δ.
Proof. This follows from [55, Theorem 3.8℄ sin
e, by our assumptions, the pointed 
urve (C; p1, . . . , pδ)
is virtually 
onne
ted in the language of [55, De�nition 3.6℄. �
For the proof of Theorem 5.2 we need to re
all other fundamental fa
ts. We �rst de�ne, for any
globally generated line bundle L of se
tional genus p := pa(L) ≥ 2, on a K3 surfa
e S, and any
integer δ su
h that 0 < δ ≤ p− 2, the lo
us in the Severi variety V|L|,δ,
(5.8) V
hyper
|L|,δ :=
C ∈ V|L|,δ | its normalization is hyperellipti
Observe that in parti
ular, for any p ≥ 3, one always has V hyper|L|,p−2 = V|L|,p−2 6= ∅ and, by regularity
of V|L|,p−2, this is smooth and of dimension two.
Let Mg be the moduli spa
e of smooth 
urves of genus g, whi
h is quasi-proje
tive of dimension
3g−3 for g ≥ 2. Denote by Mg its Deligne-Mumford 
ompa
ti�
ation. Then Mg is the moduli spa
e
of stable, genus g 
urves. Let Hg ⊂ Mg denote the lo
us of hyperellipti
 
urves, whi
h is known to
be an irredu
ible variety of dimension 2g − 1 (see e.g. [1℄) and Hg ⊂ Mg be its 
ompa
ti�
ation.
Moreover, re
all from [23, Def.(3.158)℄ that a nodal 
urve C (not ne
essarily irredu
ible) is stably
equivalent to a stable 
urve C ′ if C ′ is obtained from C by 
ontra
ting to a point all smooth rational

omponents of C meeting the other 
omponents in only one or two points.
18 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA
As above, we de�ne the degenerated version of V
hyper
|L|,δ by
hyper
|L|,δ :=
C ∈W|L|,δ | there exists a desingularization C̃ of δ of the(5.9)
nodes of C, su
h that C̃ is stably equivalent to a
(stable) 
urve C ′ with [C ′] ∈ Hpa(L)−δ
Note that, by de�nition, any su
h C̃ is 
onne
ted. Similarly as in (5.5), we have:
(5.10) W
hyper
|L|,δ = ∪δ′≥δV
hyper
|L|,δ if all the 
urves in |L| are irredu
ible.
Theorem 5.2 will be a dire
t 
onsequen
e of the next three results, Propositions 5.11 and 5.19 and
Lemma 5.20. The 
entral degeneration argument is given by the following:
Proposition 5.11. Let p ≥ 3 and δ ≤ p− 2 be positive integers. Assume there exists a smooth K3
surfa
e S0 with a globally generated, primitive line bundle H0 on S0 with pa(H0) = p and su
h that
hyper
|H0|,δ (S0) 6= ∅ and dim(W
hyper
|H0|,δ (S0)) ≤ 2.
Then, on the general, primitively marked K3 surfa
e (S,H) of genus p, W
hyper
|H|,δ (S) is nonempty
and equidimensional of dimension two.
Proof. Let Bp be the moduli spa
e of primitively marked K3 surfa
es of genus p. It is well-known
that Bp is smooth and irredu
ible of dimension 19, 
f. e.g. [2, Thm.VIII 7.3 and p. 366℄. We let
b0 = [(S0,H0)] ∈ Bp. Similarly as in [5℄, 
onsider the s
heme of pairs
(5.12) Wp,δ :=
(S,C) | [(S,H)] ∈ Bp and [C] ∈W|H|,δ(S)
and the natural proje
tion
(5.13) π : Wp,δ −→ Bp.
(The fa
t that Wp,δ is a s
heme, in fa
t a lo
ally 
losed s
heme, follows from the already mentioned
proof of Mumford's theorem on the existen
e of nodal rational 
urves as in [38, pp. 351-352℄ or [2,
pp. 365-367℄.)
Note that for general [(Sb,Hb)] = b ∈ Bp we have
π−1(b) = ∪δ′≥δV|Hb|,δ′(Sb)
by (5.5) (as Pic(Sb) ≃ Z[Hb]), so that π−1(b) is nonempty, equidimensional and of dimension g :=
p− δ, by the regularity property re
alled above. In parti
ular, π is dominant. Observe that Wp,δ is
singular in 
odimension one, so in parti
ular it is not normal.
For brevity, let W := Wp,δ and let C
f→ W be the universal 
urve. As in Theorem A.1, (i) and (ii),
in Appendix A, there exists a 
ommutative diagram
// W,
where α is a �nite, unrami�ed morphism de�ning a marking of all the δ-tuples of nodes of the �bres
of f (
f. Theorem A.1, with V = W, E(δ) = W(δ)). Pre
isely, by using notation as in Theorem A.1,
if for w ∈ W the 
urve C(w) has δ + τ nodes, τ ∈ Z+, α−1(w) 
onsists of
elements, sin
e any
ηw ∈ α−1(w) parametrizes an unordered, marked δ-tuple of the δ + τ nodes of C(w).
Let ηw ∈ W(δ). Then ηw is represented by a pointed 
urve (C; p1, p2, . . . , pδ), where (S,C) ∈ W
and where p1, p2, . . . , pδ are δ marked nodes on C.
ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 19
Let W(S,H) (resp. W(δ)(S,H)) be the �bre of π (resp. of α ◦ π) over [(S,H)] ∈ Bp, and let
α(S,H) : W(δ)(S,H) −→ W(S,H)
be the indu
ed morphism. For ηw ∈ W(δ)(S,H) as above, we have
(5.14) T[ηw](W(δ)(S,H))
∼= T[(C;p1,p2,...,pδ)](B(C; p1, p2, . . . , pδ)),
where B(C; p1, p2, . . . , pδ) is as in (5.6). Indeed, sin
e α is �nite and unrami�ed, then also α(S,H)
is. Therefore, it su�
es to 
onsider the image of the di�erential dα(S,H)[ηw ]. The latter is given by
�rst-order deformations of C in S (equivalently in |H|) that are lo
ally trivial at the δ marked nodes;
these are pre
isely given by T[(C;p1,p2,...,pδ)](B(C; p1, p2, . . . , pδ)) (
f. [55, Remark 3.5℄).
Let W̃(δ) be the smooth lo
us of W(δ). By Theorem 5.7 and by (5.14), together with the fa
t that
Bp is smooth, W̃(δ) 
ontains all the pairs (S,C) with δ marked nodes on C, su
h that |C| is globally
generated (i.e. its general element is a smooth, irredu
ible 
urve) and the partial normalization of
C at these marked nodes is a 
onne
ted 
urve. More pre
isely, by the proof of Mumford's theorem
on the existen
e of nodal rational 
urves on K3 surfa
es, as in [38, pp. 351-352℄ or [2, pp. 365-367℄),
any irredu
ible 
omponent of W(δ) has dimension ≥ 19 + p − δ = 19 + g; furthermore, by (5.14),
dim(T[ηw ](W(δ)(S,H))) = g, where ηw represents (S,C) with C with the δ marked nodes. It also
follows that W(δ) is smooth, of dimension 19 + g at these points.
If we restri
t C
to W̃(δ), from Theorem A.1, (iv) and (v), we have a 
ommutative diagram
W̃(δ)
// W,
where α̃ = α|fW(δ) and where f̃ is the �at family of partial normalizations at δ nodes of the 
urves
parametrized by α(W̃(δ)) (in the notation of Theorem A.1 in Appendix A, f̃ = f in (v) and C̃ = C
in (iii) and (iv)).
There is an obvious rational map
W̃(δ)
//___ Mg,
de�ned on the open dense subs
heme W̃
⊂ W̃(δ) su
h that, for ηw ∈ W̃0(δ), C̃(ηw) is stably equivalent
to a stable 
urve of genus g.
Set ψ := c|fW0
. By de�nition, for any ηw ∈ W̃0(δ), the map ψ 
ontra
ts all possible smooth rational

omponents of C̃(ηw) meeting the other 
omponents in only one or two points and maps the resulting
stable 
urve into its equivalen
e 
lass in Mg.
Pi
k any C0 ∈ W hyper|H0|,δ (S0) and let w0 = [(S0, C0)] ∈ W be the 
orresponding point. Now |H0| is
globally generated and the normalization of C0 at some δ nodes satisfying the 
onditions in (5.9) is
a 
onne
ted 
urve. Therefore, letting ηw0 ∈ α−1(w0) be the point 
orresponding to marking these δ
nodes, we have that ηw0 ∈ W̃0(δ) and the map c is de�ned at ηw0 .
Let Ṽ ⊆ W̃0
be the irredu
ible 
omponent 
ontaining ηw0 ; then, as proved above, dim(Ṽ) = 19+g.
By assumption, ψ(Ṽ) ∩Hg 6= ∅. Hen
e, for any irredu
ible 
omponent K ⊆ ψ(Ṽ) ∩Hg, we have
(5.15) dim(K) ≥ dim(ψ(Ṽ)) + dim(Hg)− dim(Mg) = dim(ψ(Ṽ)) + 2− g.
20 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA
Pi
k any K 
ontaining ψ(ηw0) and let I ⊆ ψ−1|eV (K) be any irredu
ible 
omponent 
ontaining ηw0 .
Sin
e the general �bre of ψ|eV
has dimension dim(Ṽ)− dim(ψ(Ṽ)) = 19 + g − dim(ψ(Ṽ)), from (5.15)
we have
dim(I) = dim(K) + 19 + g − dim(ψ(Ṽ))(5.16)
≥ dim(ψ(Ṽ)) + 2− g + 19 + g − dim(ψ(Ṽ)) = 21.
Consider now
(5.17) π ◦ (α̃|I) : I −→ Bp.
Sin
e, by assumption, the �bre over b0 = [(S0,H0)] is at most two-dimensional, we 
on
lude from
(5.16) that π◦(α̃|I) is dominant, that all the �bres are pre
isely two-dimensional and that dim(I) = 21.
This shows that W
hyper
|H|,δ 6= ∅ for general [(S,H)] ∈ Bp and Lemma 5.1 implies that in fa
t any
irredu
ible 
omponent of W
hyper
|H|,δ (S) has dimension two. �
Remark 5.18. In parti
ular, Lemma 5.1, Proposition 5.11 and [22, Examples 2.8 and 2.10℄ prove
Theorem 5.2 for p = 4 and 5.
We next 
onstru
t the desired spe
ial primitively marked K3 surfa
e:
Proposition 5.19. Let d ≥ 2 and k ≥ 1 be integers. There exists a K3 surfa
e S0 with
Pic(S0) = Z[E]⊕ Z[F ]⊕ Z[R]
and interse
tion matrix 
E2 E.F E.R
F.E F 2 F.R
R.E R.F R2
0 d k
d 0 k
k k −2
and su
h that the following 
onditions are satis�ed:
(a) |E| and |F | are ellipti
 pen
ils;
(b) R is a smooth, irredu
ible rational 
urve.
(
) H0 := E +F +R is globally generated, in parti
ular the general member of |H0| is a smooth,
irredu
ible 
urve of arithmeti
 genus p := 2k + d;
(d) the only e�e
tive de
ompositions of H0 are
H0 ∼ E + F +R ∼ (E + F ) +R ∼ (E +R) + F ∼ (F +R) + E.
Proof. Sin
e the latti
e has signature (1, 2), then, by a result of Nikulin [43℄ (see also [39, Cor.
2.9(i)℄), there is a K3 surfa
e S0 with that as Pi
ard latti
e. Performing Pi
ard-Lefs
hetz re�e
tions
on the latti
e, we 
an assume that H0 is nef, by [2, VIII, Prop. 3.9℄. Straightforward 
al
ulations on
the Pi
ard latti
e rules out the existen
e of e�e
tive divisors Γ satisfying Γ2 = −2 and Γ.E < 0 or
Γ.F < 0, or Γ2 = 0 and Γ.H0 = 1. Hen
e (a) and (
) follow from [48, Prop. 2.6 and (2.7)℄. Similarly
one 
omputes that if Γ > 0, Γ2 = −2 and Γ.R < 0, then Γ = R, proving (b).
Similarly, (d) is proved by dire
t 
al
ulations using the nefness of E, F and H0 and re
alling that
by Riemann-Ro
h and Serre duality a divisor D on a K3 surfa
e is e�e
tive and irredu
ible only if
D2 ≥ −2 and D.N > 0 for some nef divisor N . �
The following result, together with (5.10) and Proposition 5.11, now 
on
ludes the proof of Theo-
rem 5.2 and Corollary 5.3. From Remark 5.18, we need only 
onsider p ≥ 6.
Lemma 5.20. Let p ≥ 6 be an integer. There exists a smooth K3 surfa
e S0 with a globally generated,
primitive line bundle H0 on S0 with p = pa(H0) su
h that
(a) W
hyper
|H0|,p−3(S0) 6= ∅;
ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 21
(b) dim(W
hyper
|H0|,p−3(S0)) = 2;
(
) there exists a 
omponent of W
hyper
|H0|,p−3(S0) whose general member deforms to a 
urve [Ct] ∈
hyper
|Ht|,p−3(St), for general [(St,Ht)] ∈ Bp;
(d) for general [(St,Ht)] ∈ Bp, the two-dimensional irredu
ible 
omponent Vt ⊆ V hyper|Ht|,p−3(St)
given by (
), satis�es dim(RVt) = 3 (with notation as in � 2.2).
Proof. Set k = 1 if p is even and k = 2 if p is odd and let d := p− 2k ≥ 2. Consider the marked K3
surfa
e (S0,H0) in Proposition 5.19.
We will 
onsider two general smooth ellipti
 
urves E0 ∈ |E| and F0 ∈ |F | and 
urves of the form
C0 := E0 ∪ F0 ∪R,
with transversal interse
tions and a desingularization
(5.21) C̃0 = Ẽ0 ∪ F̃0 ∪ R̃→ C0
of the δ := p − 3 = d + 2k − 3 nodes marked in Figure 1 below, that is, all but one of ea
h of the
interse
tion points E0 ∩ F0, E0 ∩R and F0 ∩R.
E 0 F 0
−−−−−−−−−−−−−−−−>
E 0 F 0
partial
normalization
k points k points
k=1,2 k=1,2 
d points
Figure 1. The 
urves C0 and C̃0
Then [C0] ∈W hyper|H0|,p−3, as C̃0 is stably equivalent to a union of two smooth ellipti
 
urves interse
ting
in two points (
f. [23, Exer
ise (3.162)℄), proving (a). Clearly the 
losure of the family we have
22 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA

onstru
ted is isomorphi
 to |E| × |F | ≃ P1 × P1, and is therefore two-dimensional. Denote by
W0 ⊂W hyper|H0|,p−3 this two-dimensional subs
heme.
We will now show that any irredu
ible 
omponent W of W
hyper
|H0|,p−3 has dimension ≤ 2.
A 
entral observation, whi
h will be used together with Theorem 4.6(b), will be that, with the
above 
hoi
es of k, we have
(5.22) E.H0 = F.H0 = d+ k = p− k is odd.
We start by 
onsidering families of redu
ible 
urves. These are all 
lassi�ed in Proposition 5.19(d).
If the general element in W is of the form D ∪R, for D ∈ |E +F |, then in order to have a partial
desingularization D̃ ∪ R̃ to be (degenerated) hyperellipti
, we must have deg(D̃ ∩ R̃) = 2, so that we
must desingularize 2(k − 1) of the interse
tion points of D ∩R. Finally, as pa(D̃ ∪ R̃) = 3, we must
have pa(D̃) = 2. Therefore W ⊆ WD × {R} ≃ WD, where WD ⊂ |D| is a subfamily of irredu
ible

urves of geometri
 genus ≤ 2. It follows that dim(W ) ≤ dim(WD) ≤ 2, by Lemma 2.3.
If the general element in W is of the form D ∪ E, for D ∈ |F + R|, then in order to have a
partial desingularization D̃ ∪ R̃ that is (degenerated) hyperellipti
, we must have deg(D̃ ∩ Ẽ) = 2.
If the proje
tion W → |E| is dominant, this means that g12(D̃) ⊆ |f∗E|| eD, where f : S̃ → S
denotes the 
omposition of blow-ups of S that indu
es the partial desingularization D̃∪ R̃→ D∪R.
But this would mean that |f∗E|| eD, whi
h is base point free on D̃, is 
omposed with the g
2(D̃), a

ontradi
tion, as deg(O eD(f
∗E)) = E.D = E.H0 is odd by (5.22). Therefore, the proje
tion W → |E|
is not dominant, when
e dim(W ) ≤ dim(|D|) = 1
D2+1 = k ≤ 2, as desired. By symmetry, the 
ase
where the general element in W is of the form D ∪ F , for D ∈ |E +R| is treated in the same way.
Finally, we have to 
onsider the 
ase of a family W ⊆ |H0| of irredu
ible 
urves.
In this 
ase assume dim(W ) ≥ 3, and let C be a general 
urve parametrized by W . Then by
Theorem 4.6 (b), there exists an e�e
tive de
omposition into moving 
lasses H0 ∼M +N su
h that
2(C̃) ⊆ |f∗M || eC , |f
∗N || eC ,
where f : S̃ → S denotes the su

ession of blow ups of S that indu
es the normalization C̃ → C.
From Proposition 5.19(d) we see that we must have
2(C̃) ⊆ |f∗E|| eC , or |f
∗F || eC ,
whi
h means that either |f∗E|| eC or |f
∗F || eC is 
omposed with the g
2(C̃), again a 
ontradi
tion, as
both have odd degree by (5.22). We have therefore proved (b).
To prove (
) we will show that any [C0] ∈W hyper|H0|,p−3 in the two-dimensional, irredu
ible 
omponent
W0 
onsidered above in fa
t deforms to a 
urve [Ct] ∈W hyper|Ht|,p−3(St), for general [(St,Ht)] ∈ Bp, that
has pre
isely δ = p− 3 nodes (
f. (5.10)).
To this end, denote by S → Bp the universal family of K3 surfa
es, f̃ : C̃ → W̃(δ) and I ⊂ W̃(δ) as
in the proof of Proposition 5.11, and let ϕ : C̃I → I be the restri
tion of f̃ .
Sin
e the �ber over [(S0,H0)] of I → Bp as in (5.17) 
ontains an open, dense subset of P1 × P1,
we 
an �nd a smooth, irredu
ible 
urve B ⊂ I satisfying: for x ∈ B general, ϕ−1(x) is a (partial)
desingularization of δ = p−3 of the nodes of a 
urve in W|Ht|,δ(St) (
f. (5.4)), for general [(St,Ht)] ∈
Bp, and ϕ
−1(x) ∈ H3 ⊂ M3; moreover B 
ontains a point x0 ∈ I su
h that ϕ−1(x0) is C̃0 as in
(5.21), for C0 general in W0.
Let ϕB : C̃B → B be the indu
ed universal 
urve. Sin
e the dualizing sheaf of ϕ−1B (x0) = C̃0 is
globally generated (as ea
h 
omponent interse
ts the others in two points), we in fa
t have, possibly
after substituting B with an open neighbourhood of x0, a morphism γB : C̃B → P(ϕ̃∗(ωeC/B)) over
ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 23
B that is 2 : 1 on the general �bre ϕ−1B (x) and 
ontra
ts the rational 
omponent R̃ of ϕ
B (x0) and
maps the two ellipti
 
urves Ẽ0 and F̃0 ea
h 2 : 1 onto (di�erent) P
s (
f. (5.21) and Figure 1).
Let ν : C̃′B → C̃B be the normalization and
// P(ϕ̃∗(ωeCB/B
the Stein fa
torization of γB ◦ ν. In parti
ular, γ2 is �nite of degree two onto its image. Moreover,
ν◦ϕB : C̃′B → B is a �at family whose general �ber (ν◦ϕB)−1(x) is a desingularization of ϕ
B (x) ∈ C̃B.
Let pg be the geometri
 genus of this general �bre.
Let D ⊂ C̃′B be the stri
t transform via γ1 of the 
losure of the bran
h divisor of γ2 on the
smooth lo
us of C̃
B . By Riemann-Hurwitz, for general x ∈ B, we have D.ϕ
B (x) = 2pg + 2,
whereas D.ϕ−1B (x0) ≥ 8, as the 
urve γ1(ϕ
B (x0)) 
ontains two smooth ellipti
 
urves, ea
h being
mapped 2 : 1 by γ2 onto (di�erent) P
s. This implies pg = 3. Sin
e, for general x ∈ B, we have
pg ≤ pa(ϕ−1B (x)) = p − δ = 3, we �nd that ϕ
B (x) is smooth. This means that the general 
urve in
W|Ht|,δ(St), for (St,Ht) ∈ Bp general, has pre
isely δ = p− 3 nodes, proving (
).
To prove (d), again we 
onsider the morphism (up to possibly restri
ting I as above)
γI : CI → P(ϕ∗(ωCI/I))
over I whi
h, apart some possible 
ontra
tions of rational 
omponents in spe
ial �bres over I, is
relatively 2 : 1 onto its image. We have a natural morphism h : CI → S, indu
ing a natural map
Φ : im(γI)−− → Sym2(S),
whose domain has nonempty interse
tion with every �bre over Bp.
Let R := im(Φ). Then R ∩ Sym2(St) = RVt , for general [(St,Ht)] ∈ Bp. One easily sees that
{Sym2(E′)}E′∈|E| ∪ {Sym2(F ′)}F ′∈|F | ⊆ R ∩ Sym2(S0).
Sin
e the two varieties on the left are threefolds, we have dim(Φ−1(ξ0)) = 0 for general ξ0 ∈ R ∩
Sym2(S0) ⊂ R. Therefore, for general ξ ∈ R, we have dim(Φ−1(ξ)) = 0, so that dim(R) = dim(CI) =
dim(I) + 1 = 22, when
e dim(RVt) = 22− dim(Bp) = 3. �
Remark 5.23. For general [(St,Ht)] ∈ Bp the obtained 
urves in the last proof have in fa
t δ = p−3
non-neutral nodes (
f. [22, �3℄). In fa
t a desingularization of less than p− 3 nodes of Ct admits no
2s, as 
learly a desingularization of less than p − 3 nodes of C0 is not stably equivalent to a 
urve
in the hyperellipti
 lo
us H3 ⊂ M3.
6. On the Mori 
one of the Hilbert square of a K3 surfa
e
In this se
tion we �rst summarize 
entral results on the Hilbert square of a K3 surfa
e and show
how to 
ompute the 
lass of a rational 
urve in S[2]. Then we dis
uss the relations between the
existen
e of 
urves on S and the slope of the Mori 
one of S[2], that is, the 
one of e�e
tive 
lasses
in N1(S
[2])R. In parti
ular, we show how to dedu
e the bound (6.28) from Theorem 5.2 and (6.22)
from known results about Seshadri 
onstants. Finally, we dis
uss the relation between the existen
e
of a 
urve on S with given singular Brill-Noether number and the slope of the Mori 
one of S[2].
6.1. Preliminaries on S[2] for a K3 surfa
e. Re
all that for any smooth surfa
e S we have
(6.1) H2(S[2],Z) ≃ H2(S,Z)⊕ Ze,
where ∆ := 2e is the 
lass of the divisor parametrizing 0-dimensional subs
hemes supported on a
single point (see [7℄). So we may identify a 
lass in H2(S,Z) with its image in H2(S[2],Z). When
S is a K3 surfa
e the 
ohomology group H2(S[2],Z) is endowed with a quadrati
 form q, 
alled
24 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA
the Beauville-Bogomolov form, su
h that its restri
tion to H2(S,Z) is simply the 
up produ
t on
S, the two fa
tors H2(S,Z) and Ze are orthogonal with respe
t to this form and q(e) = −2. The
de
omposition (6.1) indu
es an isomorphism
(6.2) Pic(S[2]) ≃ Pic(S)⊕ Z[e],
and ea
h divisor D on S 
orresponds to the divisor on S[2], by abuse of notation also denoted by D,

onsisting of length-two subs
hemes with some support on D.
Given a primitive 
lass α ∈ H2(S[2],Z), there exists a unique 
lass wα ∈ H2(S[2],Q) su
h that
α.v = q(wα, v), for all v ∈ H2(S[2],Z), and one sets
(6.3) q(α) := q(wα).
We denote also by ρα ∈ H2(S[2],Z) the 
orresponding primitive (1, 1)-
lass su
h that ρα = cwα, for
some c > 0 (for further details, we refer the reader to [25℄).
If now Pic(S) = Z[H], then the Néron-Severi group of S[2] has rank two. We may take as generators
of N1(S
[2])R the 
lass P
∆ of a rational 
urve in the ruling of the ex
eptional divisor ∆ ⊂ S[2], and
the 
lass of the 
urve in S[2] de�ned as follows
{ξ ∈ S[2]|Supp(ξ) = {p0, y} | y ∈ Y },
where Y is a 
urve in |H| and p0 is a �xed point on S. By abuse of notation, we still denote the 
lass
of the 
urve in S[2] by Y . Note that we always have that
(6.4) P
∆ lies on the boundary of the Mori 
one.
Indeed, the 
urve P1∆ is 
ontra
ted by the Hilbert-Chow morphism S
[2] → Sym2(S), so that the
pull-ba
k of an ample divisor on Sym2(S) is nef, but zero along P1∆.
Therefore, des
ribing the Mori 
one NE(S[2]) amounts, by (6.4), to 
omputing
(6.5) slope(NE(S[2])) := inf
| aY − bP1∆ ∈ N1(S[2]) is e�e
tive, a, b ∈ Q+
We will also 
all the (possibly in�nite) number a/b asso
iated to an irredu
ible 
urve X ∼alg aY −bP1∆
with a > 0 and b ≥ 0, the slope of the 
urve X and denote it by slope(X). Thus, the smaller slope(X)
is, the nearer is X to the boundary of NE(S[2]).
By a general result due to Huybre
hts [32, Prop. 3.2℄ and Bou
ksom [11℄, a divisor D on S[2] is
ample if and only if q(D) > 0 and D.R > 0 for any (possibly singular) rational 
urve R ⊂ S[2]. As
a 
onsequen
e, if the Mori 
one is 
losed then the boundary (whi
h remains to be determined) is
generated by the 
lass of a rational 
urve (the other boundary is generated by P1∆, by (6.4)). This
means that one would have slope(NE(S[2])) = sloperat(NE(S
[2])), where
(6.6)
sloperat(NE(S
[2])) := inf
| aY − bP1∆ ∈ N1(S[2]) is the 
lass of a rational 
urve, a, b ∈ Q+
(A priori, one only has slope(NE(S[2])) ≤ sloperat(NE(S[2])).)
Hassett and Ts
hinkel [25℄ make a pre
ise predi
tion on the geometri
 and numeri
al properties of
su
h extremal rational 
urves in S[2]. Indeed, a

ording to their 
onje
tures [25, p. 1206 and Conj.
3.6℄, the extremal ray R has to be generated either by the 
lass of a line inside a P2, su
h that
q(R) = −5
as in (6.3), or by the 
lass of a rational 
urve that is a �bre of a P1-bundle over a K3
surfa
e and su
h that q(R) = −2 or −1
ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 25
6.2. The 
lasses of rational 
urves in S[2]. Assume that Pic(S) = Z[H] with pa(H) = pa ≥ 2.
Let X ⊂ S[2] be an irredu
ible rational 
urve. Let CX ⊂ S be the 
orresponding 
urve as in � 2.1
and assume that CX ∈ |mH| with m ≥ 1. (In parti
ular, m ≥ 2 if we are in 
ase (II)). We 
an write
X ∼alg a1Y + a2P1∆.
Sin
e X.H = m(2pa − 2), Y.H = 2pa − 2 and P1∆.H = 0 by the very de�nition of H as a divisor in
S[2], and Y.e = 0 and P1∆.e = −2, we obtain, de�ning g0(X) := X.e − 1,
(6.7) X ∼alg mY −
(g0(X) + 1
To 
ompute g0(X), 
onsider the diagram (2.1). Sin
e ν
XOX(∆) ≃ (ν∗XOX(e))⊗2, the double 
over
f is de�ned by ν∗XOX(∆). By Riemann-Hurwitz we therefore get
(6.8) g0(X) = pa(C̃X).
Note that in the 
ases (II) and (III) in the 
orresponden
e in � 2.1, X.e = g0(X) + 1 is pre
isely
the length of the interse
tion s
heme C̃X,1 ∩ C̃X,2, where C̃X = C̃X,1 ∪ C̃X,2. In 
ase (III), sin
e
ν̃ : C̃X → S 
ontra
ts one of the two 
omponents of C̃X to a point xX ∈ S, we obtain that
(6.9) g0(X) = multxX (CX)− 1 (if CX is of type (III)).
One 
an 
he
k that for all divisors D in S[2] one has X.D = q(wX ,D) with
(6.10) wX := mH −
(g0(X) + 1
e ∈ H2(S[2],Q).
In parti
ular, 2wX ∈ H2(S[2],Z).
From (6.5) and (6.7) we see that sear
hing for irredu
ible rational 
urves in (or at least �near�) the
boundary of the Mori 
one of S[2], or with negative square q(X), amounts to sear
hing for irredu
ible

urves in |mH| with (partial) hyperellipti
 normalizations of high genus (
ase (I)), or to irredu
ible
rational 
urves in |mH| with high multipli
ity at a point (
ase (III)), or to irredu
ible rational 
urves
on S with some 
orresponden
e between some 
overings of their normalizations (
ase (II)). Moreover,
we should sear
h for 
urves with as low m as possible. Now m ≥ 2 in 
ase (II), as remarked above.
Moreover, any rational 
urve in |H| on a general S is nodal, by a result of Chen [13, Thm. 1.1℄ (the
same is also 
onje
tured for rational 
urves in |mH| for m > 1, see [14, Conj. 1.2℄), so that g0(X) ≤ 1
if CX is of type (III) in these 
ases, by (6.9). Hen
e, we see that the most natural 
andidates are
irredu
ible 
urves in |H| with hyperellipti
 normalizations.
By the above, an irredu
ible 
urve C ∈ |mH| with hyperellipti
 normalization de�nes, by the
uni
ity of the g
2, a unique irredu
ible rational 
urve X = RC ⊂ S[2] with 
lass
(6.11) RC ∼alg mY −
(g0(C) + 1
where g0(C) := g0(RC) is well-de�ned as
(6.12) g0(C) := the arithmeti
 genus of a minimal partial desingularization of C admitting a g
(For example, if C is nodal, then we simply take the desingularization of the non-neutral nodes of C,

f. [22, �3℄). From (6.5) we then get
(6.13) slope(NE(S[2])) ≤ 2m
g0(C) + 1
pg(C) + 1
, if there exists a C ∈ |mH| with hyp. norm.
and, by (6.3) and (6.10),
(6.14) q(RC) = 2m
2(pa − 1)−
(g0(C) + 1)
≤ 2m2(pa − 1)−
(pg(C) + 1)
26 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA
In parti
ular, the higher g0(C) (or pg(C)) is - thus the more �unexpe
ted� the 
urve on S is from a
Brill-Noether theory point of view - the lower is the bound on the slope of NE(S[2]) and the more
negative is the square q(RC) in S
6.3. The invariant ρsing, Seshadri 
onstants, the �hyperellipti
 existen
e problem� and
the slope of the Mori 
one. In [22℄ we introdu
ed a singular Brill-Noether invariant
(6.15) ρsing(pa, r, d, g) := ρ(g, r, d) + pa − g,
in order to study linear series on the normalization of singular 
urves. Pre
isely, we proved
Theorem 6.16. Let S be a K3 surfa
e su
h that Pic(S) ≃ Z[H] with pa := pa(H) ≥ 2. Let C ∈ |H|
and C̃ → C be a partial normalization of C, su
h that g := pa(C̃).
If ρsing(pa, r, d, g) < 0, then C̃ 
arries no g
Proof. One easily sees that the proof of [22, Thm. 1℄ also holds for a partial normalization of C. �
For r = 1 and d = 2, we have
(6.17) ρsing(pa, 1, 2, g) < 0 ⇔ g >
pa + 2
In parti
ular, a 
onsequen
e of Theorem 6.16 is the following:
Theorem 6.18. Let S be a smooth, proje
tive K3 surfa
e with Pic(S) ≃ Z[H] and pa := pa(H) ≥ 2.
Let Y and P1∆ be the generators of N1(S
[2])R with notation as in � 6.1.
If X ∈ N1(S[2])Z with X ∼alg Y − kP1∆, then k ≤
Proof. We 
an assume thatX is an irredu
ible 
urve. Then, pre
isely as in the 
ase of a rational 
urve,
X 
orresponds either to the data of an irredu
ible 
urve C ∈ |H| on S, with a partial normalization
C̃ admitting a 2 : 1 morphism onto the normalization X̃ of X, or to the data of an irredu
ible 
urve
C ∈ |H| on S together with a point x0 := xX ∈ S. (The 
ase 
orresponding to 
ase (II) in � 2.1 does
not o

ur, sin
e the 
oe�
ient of Y is one, pre
isely as in the 
ase of a rational X explained above.)
In the latter 
ase µ(X) = {x0 + C} ⊂ Sym2(S), where µ : S[2] → Sym2(S) is the Hilbert-Chow
morphism as usual, and one easily 
omputes k = (1/2)multx0(C) as in the rational 
ase above. Sin
e

learly multx0(C) ≤ 2 if pa = 2 and multx0(C) ≤ 3 if pa = 3, we have k ≤
in these two 
ases.
If pa ≥ 4, then from dim |H| − 3 − (pa − 4) = 1 and the fa
t that being singular at a given point
imposes at most three independent 
onditions on |H|, we 
an �nd an irredu
ible 
urve C ′ ∈ |H|,
di�erent from C, singular at x0, and passing through at least pa − 4 points of C. Therefore
2pa − 2 = H2 = C ′.C ≥ multx0(C ′) ·multx0(C) + pa − 4 ≥ 2multx0(C) + pa − 4,
when
e multx0(C) ≤ (pa + 2)/2, so that k ≤ (pa + 2)/4.
In the �rst 
ase, then, pre
isely as in the rational 
ase above,
(6.19) k =
pa(C̃) + 1
− pg(X)
from Riemann-Hurwitz. By Brill-Noether theory on X̃ , it follows that C̃ 
arries a g1d, with
d ≤ 2⌊pg(X) + 3
By Theorem 6.16 we have ρsing(pa(C), 1, d, pa(C̃)) ≥ 0, when
e pa(C̃) ≤ d−1+pa(C)/2. The desired
result now follows. �
ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 27
By the proof of Theorem 6.18 we see that if C ∈ |mH| is an irredu
ible 
urve and x0 ∈ C, then
the 
lass of the 
orresponding 
urve µ−1∗ {x0+C} ⊂ S[2] is given by mY − (1/2)multx0(C)P1∆. Hen
e
slope(NE(S[2])) ≤ inf
C∈|mH|
multx(C)
= inf
C∈|mH|
multx(C)
It follows that
(6.20) slope(NE(S[2])) ≤ ε(H)
pa − 1
where
ε(H) := inf
multx(C)
(and the in�mum is taken over all irredu
ible 
urves C ⊂ S passing through x) is the (global) Seshadri

onstant of H (
f. [17, � 6℄, [18℄ or [4℄). These 
onstants are very di�
ult to 
ompute. The only 
ase
where they have been 
omputed on general K3 surfa
es is the 
ase of quarti
 surfa
es, where one
has ε(H) = 2 by [3℄, yielding the bound slope(NE(S[2])) ≤ 1. As a 
omparison, the bound one gets
from (6.13) using the singular 
urves of genus two in |H| is slope(NE(S[2])) ≤ 2/3. However, it is
well-known that ε(H) ≤
H2 on any surfa
e, see e.g. [54, Rem. 1℄. Hen
e, by (6.20) we obtain
Theorem 6.21. Let (S,H) be a primitively polarized K3 surfa
e of genus pa := pa(H) ≥ 2 su
h
that Pic(S) ≃ Z[H]. Then (
f. (6.5))
(6.22) slope(NE(S[2])) ≤ ε(H)
pa − 1
pa − 1
In parti
ular, (6.22) shows that there is no lower bound on the slope of the Mori 
one of S[2] of
K3 surfa
es, as the degree of the polarization tends to in�nity, that is,
(6.23) inf
slope(NE(S[2])) | S is a proje
tive K3 surfa
e
The same fa
t about sloperat(NE(S
[2])) will follow from (7.4) and (7.9) below.
Note that one always has ε(H) > ⌊
H2⌋ − 1 under the hypotheses of Theorem 6.21. Indeed, if
ε(H) <
H2, then there is an x ∈ S and an irredu
ible 
urve C su
h that ε(H) = C.H
multx(C)
, see e.g.
[44, Cor. 2℄. Sin
e one easily 
omputes dim |H ⊗ I(⌊
H2⌋−1)
x | ≥ 2, we 
an �nd a D ∈ |L| su
h that
D 6⊇ C, multx(D) ≥ ⌊
H2⌋ − 1 and D passes through at least one additional point of C. Thus
ε(H) =
multx(C)
multx(C)
≥ multx(C) ·multx(D) + 1
multx(C)
> multx(D) ≥ ⌊
H2⌋ − 1,
as desired. It follows that
(6.24)
pa − 1
2pa − 2⌋ − 1
pa − 1
, for (S,H) as in Theorem 6.21,
showing that there is a natural limit to how good a bound one 
an get on slope(NE(S[2])) by using
Seshadri 
onstants.
The bound in (6.22) is not (ne
essarily) obtained by rational 
urves in S[2]. However, the presen
e
of pg(X) in (6.19) above tends to indi
ate that the better bounds will be obtained by rational 
urves
in S[2]. (Of 
ourse, if the Mori 
one is 
losed, then the bound will indeed be obtained by rational

urves, as explained at the end of � 6.1.) In fa
t, the bound (6.22) above will be improved, for
in�nitely many values of H2, in Propositions 7.2 and 7.7 below by rational 
urves.
We now return to the study of irredu
ible rational 
urves in S[2] and to sloperat(NE(S
[2])).
Given Theorem 6.16 and (6.17), a natural question to ask is the following:
28 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA
Hyperellipti
 existen
e problem (HEP). For 3 ≤ pg ≤ pa+22 , does there exist a singular 
urve
in |H| with hyperellipti
 normalization of geometri
 genus pg?
By (6.13) we have that
a positive solution to (HEP) for �maximal� pg = ⌊
pa + 2
⌋ =⇒(6.25)
sloperat(NE(S
[2])) ≤
if pa is even;
if pa is odd
and, by (6.14), the q-square of the asso
iated rational 
urves would be mu
h less than what predi
ted
by Hassett and Ts
hinkel [25, Conj. 3.1℄. Moreover, the bounds in (6.25) would be mu
h stronger
than the bound given by the right hand inequality in (6.22), and even stronger than the best bounds
one 
ould obtain from Seshadri 
onstants (
ompare the left hand side inequality in (6.22) with (6.24)).
It is natural to try to solve (HEP) using nodal 
urves, as one has better 
ontrol of their deformations
and their parameter spa
es (the Severi varieties 
onsidered in � 5). After the positive answer to the
hyperellipti
 existen
e problem for the spe
i�
 values pg = 3 and pa = 4, 5 in [22, Examples 2.8 and
2.10℄, Theorem 5.2 gives the �rst examples, at least as far as we know, of positive answers to the
hyperellipti
 existen
e problem for primitively polarized K3 surfa
es of any degree.
In Remark 5.23 we showed that pg(C) = g0(C) = 3 for these 
onstru
ted 
urves C ∈ |H| (
f.
(6.12)), so that the 
lasses of the asso
iated rational 
urves RC ⊂ S[2] are, using (6.10),
(6.26) wRC = H − 2e,
q(wRC ) = q(RC) = 2p− 10 ≥ −2.
Moreover, using (6.13), Theorem 5.2 yields (
f. (6.6)):
Corollary 6.27. Let (S,H) be a general, primitively polarized K3 surfa
e of genus pa(H) ≥ 4. Then
(6.28) sloperat(NE(S
[2])) ≤ 1
Note that the existen
e of nodal 
urves of geometri
 genus 2 in |H|, whi
h was already known and
followed from the nonemptiness of the Severi varieties on general K3 surfa
es, as explained in the
beginning of � 5, leads to the less good bound of
. Therefore, again as far as we know, (6.28) is the
�rst �nontrivial� bound on the slope of rational 
urves holding for all degrees of the polarization. As
already mentioned, for in�nitely many degrees of the polarization we will in fa
t improve this bound
in Propositions 7.2 and 7.7 below.
Remark 6.29. One may also look for irredu
ible singular 
urves with hyperellipti
 normalizations
in |mH|, m ≥ 2. In [22, Corollary 4℄, we also proved that, apart from some spe
ial numer-
i
al 
ases (where we were not able to 
on
lude), the negativity of ρsing(pa(mH), 1, 2, g) implies
the non-existen
e of irredu
ible nodal 
urves in |mH| with hyperellipti
 normalizations. A posi-
tive solution to the hyperellipti
 existen
e problem for singular 
urves in |mH| would then pro-
vide an even better bound on the slope of the Mori 
one. Namely, one would for instan
e get
slope(NE(S[2])) ≤ 4/[m(pa(H) + 4)] for even pa. Whereas we tend to believe that the nonnegativity
of ρsing should imply existen
e of 
urves with hyperellipti
 normalizations for the spe
i�
 values of
pa and g in a primitive linear system |H| on a general K3, we are not sure what to expe
t for 
urves
in |mH| when m > 1. For instan
e, the degeneration methods to prove existen
e as in the proof of
Theorem 5.2 will 
ertainly get more di�
ult, be
ause the irredu
ibility of the obtained 
urves after
deformation is not automati
ally ensured.
ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 29
Remark 6.30. We do not know whether there will always be 
omponents in |H|hyper (whenever
nonempty) of singular 
urves with hyperellipti
 normalizations su
h that the singularities of the
general member are as ni
e as possible, that is, all nodes and all non-neutral [22, �3℄.
7. P2s and threefolds birational to P1-bundles in the Hilbert square of a general
K3 surfa
e
We now give an in�nite series of examples of general, primitively polarized K3 surfa
es (S,H), of
in�nitely many degrees su
h that S[2] 
ontains either a P2 or a threefold birational to a P1-bundle,
thus showing both possibilities o

urring in Proposition 3.6.
Both series of examples are similar to Voisin's 
onstru
tions in [57, � 3℄. The idea is to start with
a smooth quarti
 surfa
e S0 su
h that S
0 
ontains an �obvious� P
or threefold birational to a P1-
bundle over S0, use the involution on the quarti
 to produ
e another su
h P
or uniruled threefold,
and then deform S0 keeping the latter one and loosing the �rst one in the Hilbert square.
We remark that the question of existen
e of P2s in S[2] when S is K3 is a very interesting problem
be
ause of the following fa
t: a P2 in S[2] gives rise to a birational map from S[2] onto another
hyperkähler fourfold, and 
onversely any birational transformation X −− → X ′ between proje
tive,
symple
ti
 fourfolds 
an be fa
torized into a �nite sequen
e of Mukai �ops (
f. [41, Thm. 0.7℄), by
[60, Thm. 2℄, see also [12, 30, 62℄. Therefore, in the 
ase of a K3 surfa
e, if S[2] 
ontains no P2s,
then S[2] admits no other birational model than itself.
Also uniruled divisors have an in�uen
e on the birational geometry of a hyperkähler manifold X.
Indeed, Huybre
hts proved in [32, Prop. 4.2℄ that a 
lass α in the 
losure of the positive 
one CX lies
in the 
losure of the birational Kähler 
one BKX if and only if q(α,D) ≥ 0, for all uniruled divisors
D ⊂ X. (Re
all that the positive 
one CX is the 
onne
ted 
omponent of {α ∈ H1,1(X,R) : q(α) ≥ 0}

ontaining the 
one KX of all Kähler 
lasses of X, and the birational Kähler 
one BKX equals by
de�nition ∪f :X−−→X′f∗KX′ , where f is a bimeromorphi
 map onto another hyperkähler manifold
X ′).
7.1. P2s in S[2]. The �rst nontrivial 
ase, the 
ase of degree 10, is parti
ularly easy, so we begin
with that one.
Example 7.1. (Hassett) Let S ⊂ P6 be a general K3 surfa
e of degree 10. By [40℄ the surfa
e S is
a 
omplete interse
tion S = G ∩ T ∩ Q, where G := Grass(2, 5) is the Grassmannian of lines in P4
embedded in P9 by its Plü
ker embedding, T is a general 6-dimensional linear subspa
e of P9, and Q
is a hyperquadri
 in P9. Set Y := G∩T . Then Y is a Fano 3-fold of index 2. Let F (Y ) be its variety
of lines. It is 
lassi
ally known (see e.g. [19℄ for a modern proof) that F (Y ) ∼= P2. Then we may
embed this plane in S[2] by mapping the point 
orresponding to a line [ℓ] to ℓ ∩Q. By generality, S
does not 
ontain any line, so that this map is a morphism.
The 
onstru
tion behind the following result, generalizing the previous example, was shown to us
by B. Hassett.
Proposition 7.2. Let (S,H) be a general primitively polarized K3 surfa
e of degree H2 = 2(n2 −
9n+ 19), for n ≥ 6. Then S[2] 
ontains a P2.
The 
lass wℓ ∈ H2(S[2],Q) 
orresponding to a line ℓ ⊂ P2 is
(7.3) wℓ = H −
2n − 9
In parti
ular
(7.4) sloperat(NE(S
[2])) ≤ 2
2n− 9 .
30 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA
Moreover the 
urves C ⊂ S with hyperellipti
 normalizations asso
iated to the lines ℓ ⊂ P2 ⊂ S[2]
lie in |H|, have geometri
 genus pg = 2n− 10, and ρsing(pa(C), 1, 2, pg) = n(n− 13) + 42 ≥ 0.
Proof. Consider the latti
e ZF ⊕ ZG with interse
tion matrix
F 2 F.G
G.F G2
, n ≥ 6.
Sin
e it has signature (1, 1), then, by a result of Nikulin [43℄ (see also [39, Cor. 2.9(i)℄), there is an
algebrai
 K3 surfa
e S0 with the given Pi
ard latti
e. Performing Pi
ard-Lefs
hetz re�e
tions on the
latti
e, we 
an assume that G is nef, by [2, VIII, Prop. 3.9℄. By Riemann-Ro
h and Serre duality, we
have G > 0 and F > 0. Straightforward 
omputations on the Pi
ard latti
e rules out the existen
e
of divisors Γ satisfying Γ2 = −2 and Γ.F ≤ 0 or Γ.G ≤ 1; or Γ2 = 0 and Γ.F = 1 or Γ.G = 1, 2.
By [48℄ it follows that both |F | and |G| are base point free, ϕ|F | : S0 → P2 is a double 
over and
ϕ|G| : S0 → P3 is an embedding onto a smooth quarti
 not 
ontaining lines. As explained in � 4, S

ontains a P2 arising from the double 
over.
If ℓ0 is a line on the P
, the 
orresponding 
lass in H2(S
0 ,Q) is wℓ0 = 2F − 3e, whi
h 
oin
ides
with the 
orresponding integral 
lass ρℓ0 (
f. [25, Example 5.1℄).
As S0 is a quarti
 surfa
e not 
ontaing lines, S
0 admits an involution
ι : S
0 → S
0 ; ξ 7→ (ℓξ ∩ S0) \ ξ,
by [6, Prop. 11℄, where ℓξ is the line determined by ξ, and the sign \ means that we take the residual
subs
heme. The 
orresponding involution on 
ohomology is given by (
f. e.g. [45, (4.1.6)-(4.1.7)℄)
v 7→ q(G− e, v) · (G− e)− v.
The involution sends the P2 into another P2, and the 
orresponding 
lass asso
iated to a line on it is
(7.5) q(G− e, 2F − 3e) · (G− e)− (2F − 3e) = 2((n − 3)G− F )− (2n− 9)e.
In order to obtain a generalK3 with the desired property we now deform S
0 . Pre
isely, we 
onsider a
general deformation of S
0 su
h that (i) e remains algebrai
 and (ii) ι(P
2) is preserved. Deformations
satisfying (i) form a 
ountable union of hyperplanes in the deformation spa
e of S
0 , whi
h is smooth
and of dimension 21, and may be 
hara
terized as those of the form S[2], where S is a K3 surfa
e (see
[7, Thm. 6 and Rem. 2℄). Deformations preserving ι(P2) 
an be 
hara
terized as those preserving
the image in H2(S[2],Z) of the 
lass of the line in ι(P2) as an algebrai
 
lass (see [25, Thm. 4.1 and
Cor. 4.2℄ or [57℄), that is, using (7.5), those deformations keeping H := (n− 3)G−F ∈ Pic(S[2]0 ), or,
equivalently, H ∈ Pic(S), by (6.2). As H2 = [(n− 3)G−F ]2 = 2(n2 − 9n+19) ≥ 2 for n ≥ 6 and H
is primitive, those deformations form a divisor in the 20-dimensional spa
e of deformations keeping
e algebrai
, by [34, Thm. 14℄.
We therefore obtain a 19-dimensional spa
e of deformations of S
0 , whose general member is S
where (S,H) is a general primitively polarized (algebrai
) K3 surfa
e of degree H2 = 2(n2−9n+19),
n ≥ 6, and S[2] 
ontains a plane.
The 
lass wℓ ∈ H2(S[2],Q) 
orresponding to the line ℓ is as in (7.3), yielding (7.4).
As S is general, it does not 
ontain smooth rational 
urves, so that the P2 is not of the form
C [2], for a smooth rational 
urve C on S. By Lemma 2.4, the lines in the P2 in S[2] give rise to
a two-dimensional family V of 
urves on S with hyperellipti
 normalizations, so that RV = µ(P
where µ : S[2] → Sym2(S) is the Hilbert-Chow morphism. By (7.3) we have ℓ.H = H2, so that, by
the very de�nition of the divisor H in H2(S[2],Z), the lines in the P2 
orrespond to 
urves C ∈ |H|.
Comparing (6.10) and (7.3), we see that g0(C) = 2n − 10, 
f. (6.12). Now we note that the general
line in the P2 is not tangent to ∆ = 2e. (Indeed, this follows by deformation sin
e in S
0 we have
ON FAMILIES OF RATIONAL CURVES IN THE HILBERT SQUARE OF A SURFACE 31
that ι(P2) ∩ ∆ is a smooth plane sexti
, sin
e we have a 
omposite map S0 → P2 → ι(P2) that is
�nite of degree two, when
e rami�ed along a smooth sexti
, as S0 is a smooth K3.) Therefore we
have pg(C) = 2n− 10. We 
ompute ρsing = n(n− 13) + 42 ≥ 0 (re
all that n ≥ 6). �
The examples 
ontained in the above proposition is interesting in several regards.
Noti
e �rst that q(ℓ) = −5/2, 
f. (6.3), in a

ordan
e with the predi
tion in [25, Conj. 3.6℄.
The proposition shows in parti
ular that the 
orresponden
e in Remark 3.7 is not one-to-one and
also shows that the 
ase dim(V ) = dim(RV ) = 2 of Proposition 3.6 a
tually o

urs.
The result also gives nontrivial examples of 
urves in |H| with hyperellipti
 normalizations and
positively answers the hyperellipti
 existen
e problem for pa = n
2−9n+20 and pg = 2n−10, n ≥ 6.
Moreover (7.4) shows that there is no lower bound on sloperat(NE(S
[2])) as the degree of the
polarization tends to in�nity. The same follows from (7.9) in Proposition 7.7 below. Both the
bounds (7.4) and (7.9) below in fa
t yield better bounds on slope(NE(S[2])) than (6.22).
Finally, the 
oni
s on the P2 give a �ve-dimensional family V (2) of irredu
ible 
urves with hyperel-
lipti
 normalizations on S. Of 
ourse this family has obvious non-integral members, 
orresponding to
non-integral 
oni
s. More generally, for any m ≥ 3, the (3m−1)-dimensional family of nodal rational

urves in |OP2(m)| (
f. [15, Thm. 1.1℄) yields 
orresponding families V (m) of 
urves in |mH| with
hyperellipti
 normalizations with dimV (m) = 3m − 1 ≥ 5 and dim(RV ) = 2, showing in parti
ular
that the 
ase dim(V ) > dim(RV ) = 2 of Proposition 3.6 a
tually o

urs.
In the 
ase of the 
oni
s, we 
ompute pg = 4n− 19 as above and as pa(2H) = 4n2 − 36n+ 77, we
get ρsing = 4n(n− 11) + 117 ≥ −3 in these 
ases. This does not 
ontradi
t [22, Thm. 1℄.
7.2. Threefolds birational to P1-bundles in S[2]. We start with an expli
it example in the spe
ial

ase of a quarti
 surfa
e.
Example 7.6. In the 
ase of a general quarti
 S in P3 we 
an �nd a P1-bundle over S in S[2], arising
from the two-dimensional family of hyperplane se
tions of geometri
 genus two. In fa
t, taking the
tangent plane through the general point of S we get a nodal 
urve of geometri
 genus 2. We obtain in
this way a family V of nodal 
urves with hyperellipti
 normalizations in the hyperplane linear system.
This family is parametrized by an open subset of S, and the lo
us in S[2] 
overed by the asso
iated
rational 
urves is birational to a P1-bundle over this open subset. To see this, set Cp := (S ∩ TpS),
and let C̃p be the normalization of Cp. Note that the g
2 on C̃p, viewed on Cp, is given by the pen
il
of lines in TpS through the node p. If, for two distin
t points p, q ∈ S, the g12s on C̃p and C̃q had two

ommon points, say x and y (so that the map ΦV in (2.5) sends (p, x+ y) and (q, x+ y) to the same
point x+ y in Sym2(S)), then the line TpS ∩ TqS, whi
h is bitangent to S, would also pass through
x and y. This is absurd, as deg(S) = 4.
By (6.10), the 
lass w ∈ H2(S[2],Q) 
orresponding to the 
urves of geometri
 genus 2 is w = H− 3
when
e q(w) = −1/2, as predi
ted by [25, Conj. 3.6℄. Moreover, performing the usual involution on
the quarti
, we send the 
onstru
ted uniruled threefold to another one, with 
orresponding �bre 
lass
given by e, so that it simply is the P1-bundle ∆ over S. This shows that also our original threefold
was smooth, so in fa
t a P1-bundle over S.
We now give an in�nite series of examples of general K3s whose Hilbert squares 
ontain threefolds
birational to P1-bundles.
Proposition 7.7. Let (S,H) be a general primitively polarized K3 surfa
e of degree H2 = 2(d2−1),
for d ≥ 2. Then S[2] 
ontains a threefold birational to a P1-bundle over a K3 surfa
e.
The 
lass wf ∈ H2(S[2],Q) 
orresponding to a �bre is
(7.8) wf = H − de ∈ H2(S[2],Z).
32 F. FLAMINI, A. L. KNUTSEN, G. PACIENZA
In parti
ular
(7.9) sloperat(NE(S
[2])) ≤ 1
Moreover the 
urves C ⊂ S with hyperellipti
 normalizations asso
iated to the �bres of the threefold
lie in |H|, have geometri
 genus pg = 2d− 1, and ρsing(pa(C), 1, 2, pg) = d(d − 4) + 4 ≥ 0.
Proof. This time we start with the latti
e ZF ⊕ ZG with interse
tion matrix
F 2 F.G
G.F G2
, d ≥ 2.
As in Proposition 7.2 one easily shows that there is an algebrai
 K3 surfa
e S0 with Pic(S0) =
ZF ⊕ ZG and that ϕ|G| : S0 → P3 is an embedding onto a smooth quarti
 not 
ontaining lines and
F is a smooth, irredu
ible rational 
urve. (Note that F [2] = P2 and performing the same pro
edure
on this plane as in the proof of Proposition 7.2, one gets pre
isely the same series of examples as
above.)
We now 
onsider the divisor F ⊂ S[2]0 , de�ned as the length-two s
hemes with some support along
F . One easily sees that this is a threefold birational to a P1-bundle over S0 and that the 
lass in
0 ,Z) 
orresponding to the �bres f is ρf = F , 
f. [25, Example 4.6℄.
The involution on the quarti
 sends this threefold to another threefold birational to a P1-bundle
over S0 and the 
orresponding 
lass of the �bres is
(7.10) q(G− e, F ) · (G− e)− F = dG− F − de.
Note that this threefold satis�es the 
onditions in [25, Thm. 4.1℄ by [25, Example 4.6℄, so that,
as in the previous example, we 
an deform S
0 , keeping e algebrai
 and H := dG − F . We thus
obtain a 19-dimensional spa
e of deformations of S
0 , whose general member is S
, where (S,H) is
a general, primitively polarized (algebrai
) K3 surfa
e of degree H2 = 2(d2−1) ≥ 6 and S[2] 
ontains
a threefold birational to a P1-bundle, again over a K3 surfa
e (see also [25, Thm. 4.3℄).
The unique 
lass wf ∈ H2(S[2],Q) 
orresponding to a �bre f is as in (7.8) and yields (7.9).
By (7.8) we have f.H = H2, so that, by the very de�nition of the divisorH inH2(S[2],Z), the �bres
f of Y 
orrespond to 
urves C ∈ |H|. Comparing (6.10) and (7.8), we see that g0(C) = 2d − 1 ≥ 3,

f. (6.12). As in the proof of Proposition 7.2, one 
an see that the general �bre of Y is not tangent
to ∆ = 2e, so that in fa
t we have pg(C) = 2d−1. In parti
ular, Y is not one of the obvious uniruled
threefolds arising from the rational 
urves on S, or the one-dimensional families of ellipti
 
urves on
S. A 
omputation shows that ρsing = d(d − 4) + 4 ≥ 0. �
Again, a few 
omments are in order.
The square of the 
lass of the �bres of the uniruled threefolds 
onstru
ted above is q(f) = −2, as
predi
ted in [25, Conj. 3.6℄.
The obtained family V of 
urves on S with hyperellipti
 normalizations has dim(V ) = 2 and
dim(RV ) = 3, showing that also this 
ase of Proposition 3.6 a
tually o

urs. This family gives
nontrivial examples of 
urves in |H| with hyperellipti
 normalizations and positively answers the
hyperellipti
 existen
e problem for pa = 2(d
2 − 1) and pg = 2d − 1 for every d ≥ 2. Note that the

ase d = 2 is the 
ase des
ribed in [22, Example 2.8℄.
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EDOARDO SERNESI: PARTIAL DESINGULARIZATIONS OF FAMILIES OF NODAL CURVES 35
Appendix A.
PARTIAL DESINGULARIZATIONS OF FAMILIES OF NODAL CURVES
EDOARDO SERNESI
In this Appendix we show how to 
onstru
t simultaneous partial desingularizations of families
of nodal 
urves, generalizing a well known pro
edure of simultaneous total desingularization, as
des
ribed in [4℄.
We work over an algebrai
ally 
losed �eld k of 
hara
teristi
 0. For every morphism X → Y , and
for every y ∈ Y , we denote by X(y) the s
heme-theoreti
 �bre of y.
Theorem A.1. Let
f : C // V
be a �at proje
tive family of 
urves, with C and V algebrai
 s
hemes, su
h that all �bres have at most
ordinary double points (nodes) as singularities. Let δ ≥ 1 be an integer. Then there is a 
ommutative
diagram:
with the following properties:
(i) α is �nite and unrami�ed, the square is 
artesian, and q is an étale 
over of degree δ.
(ii) The left triangle de�nes a marking of all δ-tuples of nodes of �bres of f . In parti
ular f ′
parametrizes all 
urves of the family f having ≥ δ nodes and, for ea
h η ∈ E(δ), Dδ(η) ⊂ C′(η)
is a set of δ nodes of the 
urve C′(η).
(iii) The diagram is universal with respe
t to properties (i) and (ii). Pre
isely, if
Ẽ ×V C
is a diagram having the properties analogous to (i) and (ii), then there is a unique fa
torization
// E(δ)
su
h that q̃ and f̃ are obtained by pulling ba
k q and f ′ by ϕ.
If moreover E(δ) is normal, then the above diagram 
an be enlarged as follows:
Work done during a visit to the Institut Mittag-Le�er (Djursholm, Sweden), whose support is gratefully a
knowl-
edged. I am grateful to F. Flamini, A. L. Knutsen and G. Pa
ienza for a

epting this note as an Appendix to their
paper, and to F. Flamini for some useful remarks.
36 EDOARDO SERNESI: PARTIAL DESINGULARIZATIONS OF FAMILIES OF NODAL CURVES
where:
(iv) β is a birational morphism su
h that, for ea
h η ∈ E(δ), the restri
tion:
β(η) : C(η) // C′(η)
is the partial normalization at the nodes Dδ(η).
(v) The 
omposition f̄ := f ′ ◦ β is �at.
Proof. Consider the �rst relative 
otangent sheaf T
. Sin
e all �bres of f are nodal, T1

ommutes
with base 
hange ([3, Lemma 4.7.5℄ or [5℄), thus on every �bre C(v), v ∈ V , it restri
ts to T1
, whi
h
is the stru
ture sheaf of the s
heme of nodes of C(v). It follows that we have
C/V = OE
for a 
losed subs
heme E ⊂ C supported on the nodes of the �bres of f . Consider the 
omposition
fE : E ⊂ C
By 
onstru
tion it follows that fE is �nite and unrami�ed. Now �x δ ≥ 1 and 
onsider the �bre
produ
t:
E ×V · · · ×V E︸ ︷︷ ︸
Sin
e fE is �nite and unrami�ed, it follows from [1, Exp.1, Prop. 3.1℄, and by indu
tion on δ (see [3,
Lemma 4.7.11(i)℄), that we have a disjoint union de
omposition:
E ×V · · · ×V E = ∆
where ∆ is the union of all the diagonals, and Eδ 
onsists of all the ordered δ-tuples of distin
t points
of E mapping to the same point of V ; moreover the natural proje
tion morphism
Eδ // V
is �nite and unrami�ed.
There is a natural a
tion of the symmetri
 group Σδ on Eδ that 
ommutes with the proje
tion to
V . We denote the quotient Eδ/Σδ by E(δ). Sin
e the 
omposition
Eδ // E(δ) // V
is �nite and unrami�ed and the �rst morphism is an étale 
over, the morphism α : E(δ) → V is �nite
and unrami�ed. Note that if, for a 
losed point v ∈ V , C(v) has δ+ t nodes as the only singularities,
with t > 0, then α−1(v) has degree
. Now let
Dδ = {(η, e) : e ∈ Supp(η)} ⊂ E(δ) ×V E
EDOARDO SERNESI: PARTIAL DESINGULARIZATIONS OF FAMILIES OF NODAL CURVES 37
Then the �rst proje
tion de�nes the tautologi
al family:
(A.2) Dδ
⊂ E(δ) ×V E ⊂ E(δ) ×V C
whi
h is an étale 
over of degree δ. The �bre Dδ(η) is the δ-tuple parametrized by η, for ea
h
η ∈ E(δ)2. We therefore have the following diagram:
where we have denoted by C
′ = E(δ) ×V C. The �bres of f ′ are all the 
urves of the family f having
≥ δ nodes. For ea
h η ∈ E(δ) the divisor Dδ(η) ⊂ C′(η) marks the set of δ nodes parametrized by η.
This proves (i) and (ii).
(iii) follows from the fa
t that α : E(δ) → V is the relative Hilbert s
heme of degree δ of fE : E → V ,
and (A.2) is the universal family.
Assume that E(δ) is normal. Then we 
an normalize C
lo
ally around Dδ as in [4, Theorem 1.3.2℄,
to obtain a birational morphism β having the required properties (iv) and (v). �
A typi
al example of the situation 
onsidered in the theorem is when V parametrizes a 
omplete
linear system of 
urves on an algebrai
 surfa
e. If the morphism fE is self-transverse of 
odimension
1 (see [3, De�nition 4.7.13℄) then the Severi variety of irredu
ible δ-nodal 
urves is nonsingular and
of 
odimension δ, and E(δ) is nonsingular (see [3, Lemma 4.7.14℄), so that the theorem applies and
the simultaneous partial desingularization exists. This happens for example for the linear systems of
plane 
urves [3, Proposition 4.7.17℄.
Referen
es
[1℄ Revêtements étales et groupe fondamentale, Séminaire de Géométrie Algébrique du Bois Marie 1960-61
(SGA1), Le
ture Notes in Math. 224. Springer, Berlin, 1971.
[2℄ R. Hartshorne, Algebrai
 Geometry, Graduate Texts in Mathemati
s 52. Springer-Verlag, New York-
Heidelberg, 1977.
[3℄ E. Sernesi, Deformations of Algebrai
 S
hemes, Grundlehren der Mathematis
hen Wissens
haften 334.
Springer-Verlag, Berlin, 2006.
[4℄ B. Tessier, Résolution simultanée I, II. Le
ture Notes in Math. 777, 71�146. Springer, Berlin, 1980.
[5℄ J. Wahl, Deformations of plane 
urves with nodes and 
usps, Amer. J. of Math. 96 (1974), 529�577.
Edoardo Sernesi, Dipartimento di Matemati
a, Università di Roma Tre, Largo San Leonardo Murialdo 1,
00146, Roma, Italy. e-mail sernesi�mat.uniroma3.it.
If δ = 1 then E(1) = E and D1 ⊂ E ×V E is the diagonal.
	1. Introduction
	2. Rational curves in S[2]
	2.1. Irreducible rational curves in S[2] and curves on S
	2.2. Bend-and-break in Sym2(S)
	3. Rationally equivalent zero-cycles on surfaces with pg>0
	3.1. Mumford's Theorem
	3.2. The property RCC and rational quotients
	4. Families of curves with hyperelliptic normalizations
	5. Nodal curves of geometric genus 3 with hyperelliptic normalizations on K3 surfaces
	6. On the Mori cone of the Hilbert square of a K3 surface
	6.1. Preliminaries on S[2] for a K3 surface
	6.2. The classes of rational curves in S[2]
	6.3. The invariant sing, Seshadri constants, the ``hyperelliptic existence problem'' and the slope of the Mori cone
	7. P2s and threefolds birational to P1-bundles in the Hilbert square of a general K3 surface
	7.1. P2s in S[2]
	7.2. Threefolds birational to P1-bundles in S[2]
	References
	Appendix A. 
	References
