Tautological classes on moduli spaces of curves with linear series and a
  push-forward formula when $\rho=0$
TAUTOLOGICAL CLASSES ON MODULI SPACES OF CURVES
WITH LINEAR SERIES AND A PUSH-FORWARD FORMULA
WHEN ρ = 0
DEEPAK KHOSLA
Abstract. We define tautological Chow classes on the moduli space Gr
triples consisting of a curve C, a line bundle L on C of degree d, and a linear
system V on L of dimension r. In the case where the forgetful morphism
to Mg has relative dimension zero, we describe the images of these classes
in A1(Mg). As an application, we compute the (virtual) slopes of several
different classes of divisors on Mg.
Contents
1. Introduction 1
2. Statement of Theorem 3
2-A. A limit linear series moduli stack 3
2-B. Tautological Classes 4
2-C. A Push-Forward Formula 5
3. Applications 5
3-A. The Gieseker-Petri Divisor 5
3-B. Hypersurface Divisors 6
3-C. Syzygy Divisors 7
3-D. Secant Plane Divisors 9
4. Special Families of Curves 10
4-A. Definitions of Families 10
4-B. Computations on the Special Families 11
4-C. Pull-Back Maps on Divisors 11
5. Proofs of Lemmas 13
6. Appendix 21
References 25
1. Introduction
The cone of effective divisors on a projective variety plays an important rôle in
the understanding of its birational geometry. In the case of the moduli space Mg
of genus-g curves, it has become apparent that the most interesting effective divisor
classes are those that arise from the extrinsic geometry of curves in projective space.
Indeed, in their pioneering work, Harris and Mumford [20] considered divisors of
curves admitting a degree-d branched cover of P1, where d = (g + 1)/2. More
Date: October 24, 2018.
http://arxiv.org/abs/0704.1340v1
2 D. KHOSLA
recently, work of Cukierman [2], Farkas-Popa [11], Khosla [21], and Farkas [10] [9]
has found effective divisor classes of smaller slope that those considered by Mumford
and Harris, and some of these classes have been used to improve on the best known
bounds on n for which Mg,n is of general type for fixed g [9].
Although all of these divisors are described by conditions on the space of em-
beddings of a curve in projective space, the techniques used to deal with them have
been varied. In this paper, we put all of the above calculations into a unified frame-
work and lay the ground for future work on the effective cone of Mg. Specifically
we consider the moduli stack Grd(Mg) of genus-g curves together with a g
d (linear
series). The set of C-valued points consists of triples (C,L, V ), where C is a genus-g
curve, L is a degree-d line bundle on C, and V ⊂ H0(L) is an (r + 1)-dimensional
subspace. The forgetful morphism
η : Grd(Mg) → Mg
is representable, proper, and generically smooth of relative dimension
ρ(g, r, d) = g − (r + 1)(g − d+ r)
[23], [16], [5], [6] [14], [7], [25], [13]. (When ρ < 0, then η is not dominant.)
If g, r, and d are chosen so that ρ = −1, then the image of η has a component of
codimension 1, and it is this divisor that Eisenbud and Harris use to show that Mg
is of general type when g ≥ 24 and g+1 is composite [8]. The closure of this divisor
in the moduli space Mirrg of irreducible nodal curves may also be interpreted as the
image of the virtual fundamental class of Grd(M
g ) under the proper push-forward
morphism
η∗ : A∗(G
g )) → A∗(M
where Grd(M
g ) is a partial compactification of G
d(Mg) using torsion-free sheaves.
If we now choose g, r, and d so that ρ = 0, we are led to the “second generation”
of effective divisors on Mg. In this case, the morphism η : G
d(Mg) → Mg is
generically finite. Since the work of Cukierman [2], every interesting effective divisor
class on Mg has been realized as the image under η of a divisor on G
d(Mg). For
example, the K3 locus in M10 [2], which was the first counterexample [11] to the
Harris-Morrison slope conjecture [19], can be interpreted as the image under η of
the divisor in G412(M10) of g
12’s which do not lie on a quadric. Again, the class of
its closure in Mirr10 may be realized as the proper push-forward of the corresponding
class in G412(M
10) under the morphism
η∗ : A∗(G
10)) → A∗(M
This latter class, in turn, is easily computed to be
2α− β − 6γ + η∗λ,
where α, β, and γ are certain tautological classes defined on Grd(M
g ). (See Sec-
tions 2-B and 3-C.)
In Section 2, we introduce a partial compactification Grd(M̃g) of G
d(Mg), which is
proper over an open substack M̃g of Mg that contains Mg and whose complement
in Mg has codimension 2. We define tautological virtual codimension-1 Chow
classes α, β, and γ on Grd(M̃g) and, in the case where ρ = 0, compute their images
under the proper push-forward morphism
η∗ : A3g−2(G
d(M̃g)) → A3g−2(M̃g) = A
1(Mg).
A PUSH-FORWARD FORMULA WHEN ρ = 0 3
This allows one to completely mechanically compute the slopes of all of the divisor
classes on Mg that have thus far been studied. As examples, in Section 3 we study
syzygy divisors, hypersurface divisors, Gieseker-Petri divisors, and secant plane
divisors.
In Section 4 we give the statements of a series of calculuations over special
families of stable curves. These calculations assemble to give the main result.
Finally Section 5 is devoted to the proofs of the lemmas stated in Section 4.
This work was carried out for my doctoral thesis under the supervision of Joe
Harris. I would like to thank Ethan Cotterill, Gavril Farkas, Johan de Jong, Martin
Olsson, Brian Osserman, and Jason Starr for helpful conversations.
2. Statement of Theorem
2-A. A limit linear series moduli stack.
Definition 2.1 ([24]). Let S be any scheme, and let g and n be non-negative
integers. An n-pointed stable curve of genus g over S is a proper flat morphism
π : X → S together with sections σ1 . . . , σn : S → X . Each geometric fiber Xs̄ must
be a reduced, connected, 1-dimensional scheme such that
(a) Xs̄ has only ordinary double points;
(b) Xs̄ intersects the sections σ1 . . . , σn at distinct points p1, . . . pn that lie on
the smooth locus of Xs̄;
(c) the line bundle ωXs̄(p1 + · · ·+ pn) is ample;
(d) dimH1(OXs̄) = g.
Theorem 2.2 ([3],[24]). Let g and n be non-negative integers such that 2g − 2 +
n > 0. The category Mg,n of families of n-pointed stable curves of genus g is an
irreducible Deligne-Mumford stack that is proper, smooth, and of finite type over
SpecZ.
Definition 2.3 ([3]). Let k be an algebraically closed field, and let X be an n-
pointed stable curve of over k. The dual graph ΓX of X is the following unoriented
graph:
(a) the set of vertices of ΓX is the set Γ
X of irreducible components of X ;
(b) the set of edges of ΓX is the set Γ
X which is the union of the singular and
marked points of X ;
(c) an edge x ∈ Γ1X has for extremities the irreducible components on which x
lies;
Definition 2.4. An n-pointed stable curve X → S is tree-like if the dual graph of
each geometric fiber is a tree.
Proposition 2.5 ([22]). The category M̃g,n of families of tree-like curves is an
open substack of Mg,n, whose complement has codimension 2.
Definition 2.6. Let π : X → S be a smooth genus-g curve. A grd on X is the data
of a line bundle L→ X of relative degree d together with a rank-r vector subbundle
V of π∗L [26, Definition 4.2].
Proposition 2.7. Let π : X → S be a smooth genus-g curve. The étale sheafifica-
tion of the functor
(Sch/S) → (Sets)
4 D. KHOSLA
given by
T 7→ {grd’s on XT → T}
is represented by a scheme Grd(X/S), proper over S. If Z is an irreducible compo-
nent of Grd(X/S), then
(1) dimZ ≥ dimS + ρ(g, r, d)
In this way, one can construct a Deligne-Mumford stack Grd , representable and
proper over Mg. In [22], Osserman and the author extend this construction to
families of tree-like stable curves. That is, for X a tree-like stable curve over S of
genus g, they construct an algebraic space Grd(X/S), proper over S and satisfying
the same dimension lower bound (1). In addition, they prove that if S = Spec k,
where k is an algebraically closed field, then there is an open substack of Grd(X/k)
isomorphic to the space of refined limit linear series onX . In this way they construct
a representable and proper Deligne-Mumford stack Grd over M̃g and hence over
M̃g,n by pulling back.
2-B. Tautological Classes.
Definition 2.8. Let g and n be as in Theorem 2.2 and let r and d be non-negative
integers for which ρ(g, r, d) = 0. Let U ⊂ M̃g,n be the open substack over which
η : Grd → M̃g,n is flat. Let [G
d ] ∈ A3g−3+n+ρ(G
d) be the class of the closure of
η−1(U). Similarly, let [Crd] ∈ A3g−2+n+ρ(C
d) be the class of the closure of (η ◦
π)−1(U) in the universal curve π : Crd → G
In the following we work over M̃g,1 in order to be able to consistently define the
universal line and vector bundles. There is a universal pointed quasi-stable curve
Yrd → G
d whose stabilization is the universal stable curve C
d → G
d . Let σ : G
d → Y
be the marked section. There is a universal line bundle L → Yrd of relative degree
d together with a trivialization σ∗L ∼= OGr
. On each geometric fiber, L has degree
1 on every exceptional curve, degree d on the pre-image of the component of stable
curve containing the marked point, and degree 0 on the pre-image of all other
components of the stable curve. There is a sub-bundle
V →֒ π∗L
which, over each point in Grd , is equal to the aspect of the limit linear g
d on the
component containing the marked point.
Remark 2.9. By a theorem of Harer [17], for g ≥ 3,
A3g−3(M̃g,1)Q = PicMg,1 ⊗Q = Qλ⊕Qδ0 ⊕Qδ1 ⊕ · · · ⊕Qδg−1 ⊕Qψ
where λ and ψ are the first Chern classes of the Hodge and tautological bundles
respectively, δ0 is the divisor of irreducible nodal curves, and δi is the divisor of
unions of curves of genus i and g− i, where the marked point lies on the component
of genus i.
Definition 2.10. We define “codimension-1” cycle classes in A3g−3+ρ(G
d) as fol-
lows.
α = π∗
c1(L)
2 ∩ [Yrd ]
β = π∗
c1(L) · c1(ω) ∩ [Y
γ = c1(V) ∩ [G
A PUSH-FORWARD FORMULA WHEN ρ = 0 5
2-C. A Push-Forward Formula. We now state our main result.
Theorem 2.11. Let g ≥ 1, and r, d ≥ 0, be integers for which
ρ = g − (r + 1)(g − d+ r) = 0,
and consider the map
η : Grd → M̃g,1
η∗ : A3g−3(G
d) → A3g−3(M̃g,1)
is the proper push-forward morphism on corresponding Chow groups, then
6(g − 1)(g − 2)
η∗α = 6(gd− 2g
2 + 8d− 8g + 4)λ
+ (2g2 − gd+ 3g − 4d− 2)δ0
(g − i)(gd+ 2ig − 2id− 2d)δi
− 6d(g − 2)ψ,
2(g − 1)
η∗β = 12λ− δ0 + 4
(g − i)(g − i− 1)δi − 2(g − 1)ψ,
2(g − 1)(g − 2)
η∗γ =
−(g + 3)ξ + 5r(r + 2)
λ− d(r + 1)(g − 2)ψ
(g + 1)ξ − 3r(r + 2)
(g − i)
iξ + (g − i− 2)r(r + 2)
where
1! · 2! · 3! · · · r! · g!
(g − d+ r)!(g − d+ r + 1)! · · · (g − d+ 2r)!
ξ = 3(g − 1) +
(r − 1)(g + r + 1)(3g − 2d+ r − 3)
g − d+ 2r + 1
3. Applications
In this section, we will apply Theorem 2.11 to various classes of divisors on
Grd(M
g ), where g, r, d are chosen so that ρ(g, r, d) = 0. We can parameterize such
choices using integers r, s ≥ 1 and setting g = (r + 1)(s+ 1) and d = r(s+ 2).
3-A. The Gieseker-Petri Divisor. Petri’s theorem [14] states that if C is a gen-
eral curve, then for all line bundles L on C, the natural map
H0(L)⊗H0(KC ⊗ L
∨) → H0(KC)
is injective. This implies that if g, r, and d are chosen so that ρ = 0, and (C,L) is
a grd on a general curve C, then the natural map
V ⊗H0(KC ⊗ L
∨) → H0(KC)
6 D. KHOSLA
is an isomorphism. Away from a subset of codimension greater than 1, the sheaf
π∗(ω ⊗ L
∨) on Grd(M
g,1) is locally free, and the degeneracy locus of the map of
vector bundles
V ⊗ π∗(ω ⊗ L
∨) → π∗(ω)
defines the Gieseker-Petri divisor in Grd . We compute its class as follows. By
definition,
c1(π∗(ω)) = λ.
By Grothendieck-Riemann-Roch,
c1(π∗L)− c1(R
1π∗L) = π∗
ch(L) · tdCr
1 + c1(L) +
c1(L)
c1(ω)
c1(ω)
2 + κ
Thus,
c1π∗(ω ⊗ L
∨) = −c1(R
1π∗L) =
− γ + λ.
It follows that our degeneracy locus in Grd(M
g,1) has class
r + 1
(−α+ β) + (d+ 1− g)γ − rλ.
It is easy to see that the slope of the image divisor in Mirrg,1 will be symmetric
in r and s. Letting x = (r + 1) + (s + 1) and y = (r + 1)(s + 1) and applying
Theorem 2.11, we find that the slope of the Gieseker-Petri divisor in Mirrg is
6(2x+ 7y2 + 7xy + xy2 + 12y + y3)
y (4 + y) (y + 1 + x)
3-B. Hypersurface Divisors. Another natural substack in Grd is the locus of g
which lie on a hypersurface of degree k; that is, grd’s (L, V ) for which the restriction
Symk V → H0(L⊗k)
has a non-trivial kernel. If ρ = 0 and the above two vector spaces have the same
dimension, then this defines a virtual divisor in Grd . Namely, we look at the degen-
eracy locus of the map of vector bundles
Symk V → π∗(L
Note that if d > g − 1, then Lk is always non-special when k ≥ 2.
To compute the class of the degeneracy locus, observe first that
c1(Sym
k V) =
r + k
k − 1
A PUSH-FORWARD FORMULA WHEN ρ = 0 7
By Grothendieck-Riemann-Roch,
c1(π∗L
⊗k) = π∗
ch(L⊗k) · tdCr
1 + kc1(L) +
c1(L)
c1(ω)
c1(ω)
2 + κ
β + λ.
Applying Theorem 2.11, the image divisor in Mirrg has slope
f(k, r, s)
g(k, r, s)
where f and g are (rather large) polynomials in k, r, and s, which are, in turn,
related by the identity
r + k
= kr(s + 2)− (r + 1)(s+ 1) + 1.
3-C. Syzygy Divisors. Consider a basepoint-free grd (C,L, V ), so there is a map
f : C → PV ∨. On PV ∨, we have the tautological sequence
0 → OPV ∨(−1) → V
∨ ⊗OPV ∨ → Q→ 0.
For any i, consider the restriction map to C:
H1(∧iQ∨ ⊗OPV ∨(2)) → H
0(∧if∗Q∨ ⊗ L⊗2).
According to [10, Proposition 2.5], the map f fails Green’s property (Ni) if and
only if this restriction map degenerates to a certain rank. In the case where the
two vector spaces have the same dimension, f fails property (Ni) exactly when the
restriction map is not an isomorphism.
To globalize this, consider the tautological sequence on u : PV∨ → Grd :
0 → OPV∨(−1) → u
∗V∨ → Q→ 0.
We will remove from Grd the closed substack, isomorphic to C
d−1, of g
ds with
a basepoint, which has codimension greater than 1. Then there is a morphism
f : Yrd → PV
∨ commuting with the projection to Grd .
✲ PV∨
Our restriction map now globalizes to
i Q∨ ⊗OPV∨(2) → π∗ ∧
i f∗Q∨ ⊗ L⊗2.
Note also that all the higher direct images of these two bundles vanish [10, Propo-
sition 2.1].
Using the exact sequence
0 → ∧i+1Q∨ ⊗OPV∨(j) → ∧
i+1u∗V ⊗OPV∨(j) → ∧
iQ∨ ⊗OPV∨(j + 1) → 0
8 D. KHOSLA
for j = 0, 1, we have
ch(u∗(∧
iQ∨ ⊗OPV∨(2))) = ch(∧
i+1V ⊗ V)− ch∧i+2V
r + 1
(r + 1)−
r + 1
(r + 1) +
r + 1
γ + · · ·
= (i+ 1)
r + 2
+ (r + 2)
γ + · · ·
Applying Grothendieck-Riemann-Roch to π, we obtain that
chπ∗(∧
if∗Q∨ ⊗ L⊗2) = π∗
f∗ ch∧iQ∨ exp(2c1(L)) · tdYr
f∗ ch∧iQ∨ exp(2c1(L)) · tdCr
One computes
ch∧iQ∨ =
r − 1
(u∗γ − ζ)
r − 2
ζu∗γ +
r − 2
r − 2
ζ2 + · · · ,
where ζ = c1(OPV∨(1)). It follows that
chπ∗(∧
if∗Q∨ ⊗ L⊗2) = (2d+ 1− g)
r − 1
r − 1
r − 2
r − 2
r − 1
(2d+ 1− g)
r − 1
r − 2
In order for the two vector bundles to have the same dimension, therefore, we need
(i+ 1)
r + 2
= (2d+ 1− g)
r − 1
which is achieved by setting r = (i + 2)s + 2(i + 1). The class of our degeneracy
locus in Grd is
r − 1
r − 2
r − 2
r − 1
−(r + 2)
+ (2d+ 1− g)
r − 1
r − 2
A PUSH-FORWARD FORMULA WHEN ρ = 0 9
In Section 6 we prove that this locus is an actual effective divisor when i = 0 and
0 ≤ s ≤ 2. By Theorem 2.11 it follows that the slope of the image locus in Mirrg is
6f(i, t)
t(i− 2)g(i, t)
where
f(i, t) = (24i2 + i4 + 16 + 32i+ 8i3)t7 + (4i3 + i4 − 16i− 16)t6
+ (−13i2 − 7i3 + 12− i4)t5 + (−i2 − 14i− i4 − 24− 2i3)t4
+ (2i2 + 2i3 − 6i− 4)t3 + (17i2 + i3 + 50i+ 41)t2
+ (7i2 + 9 + 18i)t+ 2 + 2i,
g(i, t) = (12i+ i3 + 8 + 6i2)t6 + (−4i+ i3 − 8 + 2i2)t5
+ (−2− 11i− i3 − 7i2)t4 + (−i3 + 5i)t3
+ (5i+ 1 + 4i2)t2 + (7i+ 11 + i2)t+ 2 + 4i,
and t = s+ 1.
3-D. Secant Plane Divisors. Given a curve C in Pr, we can ask whether there
is a k-plane meeting C in e points—that is, an e-secant k-plane. For example, an
m-secant 0-plane is just an m-fold point of C.
If (C,L, V ) is the associated grd, then this condition is described by saying that
there is an effective divisor E on C of degree e such that the restriction map
V → H0(LE)
has a kernel of dimension at least r − k.
To globalize this over Mirrg,1, we consider the relative Hilbert scheme of points on
a family of curves. For a stable curve X over S, the functor
(Sch/S) → (Sets)
T 7→ {subschemes Σ ⊂ XT , finite of degree e over T}
is represented by a scheme Hilbe(X/S), proper over S.
Now let H
d = Hilb
e(Crd/G
d), and consider the projection
p : H
d → G
Σ ⊂ H
d ×Grd C
be the universal subscheme. The is a natural map of vector bundles
p∗V → (π1)∗π
2L(Σ)
d , and we are looking for the rank-(k+1) locus of this map. The class of the
virtual degeneracy locus in Grd is, therefore, given by the Porteous formula as
(3) p∗∆e−k−1,r−k
c((π1)∗π
2L(Σ))/p
∗c(V)
In order to get a locus of expected codimension one in Grd , we need that
(e − k − 1)(r − k) = e+ 1
or ρ(e, r−k−1, r) = −1. Cotterill [1, Theorem 1] has proved that, in this case, one
obtains an actual effective divisor on Grd . The computation (3) can, in principle,
be carried out using the techniques in [28]. Cotterill [1] has made some progress
10 D. KHOSLA
towards an explicit calculation; there remains, however, some work to be done. The
final answer will have the form
Pαα+ Pββ + Pγγ + Pλλ+ Pδ0δ0,
where the coeffcients are rational functions in Q(r, s, e, k).
4. Special Families of Curves
Our strategy for proving Theorem 2.11 will be to pull back to various families of
stable curves over which the space of linear series is easier to analyze. In Section
4-A we define three families of pointed curves, and in Section 4-B we compute η∗
for these special families. In Section 4-C we compute the pull-backs of the standard
divisor classes on M̃g,1 to the base spaces of each of our families. Assembling the
results of these three sections, we compute η∗ over the whole moduli space.
4-A. Definitions of Families.
Definition 4.1. Let i : M0,g →֒ M̃g,1 be the family of marked stable curves defined
by sending a g-pointed stable curve
(C, p1, . . . , pg)
of genus 0 to the stable curve
Ei, p0
of genus g, where Ei are fixed non-isomorphic elliptic curves, attached to C at the
points pi, and p0 ∈ E1 is fixed as well.
E1 E2 E3 Eg· · ·
· · ·
Figure 1. i(C, p1, . . . , pg)
Definition 4.2. Let j : M̃2,1 →֒ M̃g,1 be the family of curves defined by sending
a marked curve (C, p) to the marked stable curve
(C ∪ C′, p0)
where (C′, p′, p0) is a fixed Brill-Noether-general curve in M̃g−2,2, attached nodally
to (C, p) at p′.
C C′•
Figure 2. j(C, p)
A PUSH-FORWARD FORMULA WHEN ρ = 0 11
Definition 4.3. Fix Brill-Noether general curves
(C1, p1) ∈ Mh,1
(C2, p2) ∈ Mg−h,1
and let C = C1 ∪C2 be their nodal union along the pi. Let kh : C1 →֒ M̃g,1 be the
map sending p ∈ C1 to the marked curve (C, p).
4-B. Computations on the Special Families.
Lemma 4.4. For the family
i : M0,g →֒ M̃g,1
we have
η∗α = η∗β = η∗γ = 0
Lemma 4.5. For the family
j : M̃2,1 →֒ M̃g,1
we have
η∗α =
2dN(d− 2g + 2)
3(g − 1)
(3ψ − λ− δ1) +
g − 1
(λ+ δ1 − 4ψ)
η∗β =
g − 1
(λ+ δ1 − 4ψ)
η∗γ =
3(g − 1)
(3ψ − λ− δ1),
where N and ξ are defined in the statement of Theorem 2.11.
Lemma 4.6. For the family
kh : C1 →֒ M̃g,1
we have
deg η∗α = −d
deg η∗β = −
2(g − h)− 1
deg η∗γ = −
r(r + 1)
4-C. Pull-Back Maps on Divisors.
Lemma 4.7. Let ǫi be the class of the closure of the locus on M0,g of stable curves
with two components, the component containing the first marked point having i
marked points.
(a) The classes ǫi are independent in H
2(M0,g;Q).
(b) For the family
i : M0,g →֒ M̃g,1
12 D. KHOSLA
we have the following pull-back map on divisor classes.
i∗λ = i∗ψ = i∗δ0 = 0
i∗δi = ǫi for i = 2, 3, . . . , g − 2
i∗δ1 = −
(g − i)(g − i− 1)
(g − 1)(g − 2)
i∗δg−1 = −
(g − i)(i− 1)
g − 2
Lemma 4.8. For the family
j : M̃2,1 →֒ M̃g,1
we have the following pull-back map on divisor classes.
j∗λ = λ j∗ψ = 0
j∗δ0 = δ0 j
∗δi = 0 i = 1, 2, . . . , g − 3
j∗δg−2 = −ψ j
∗δg−1 = δ1
Lemma 4.9. For the family
kh : C1 →֒ M̃g,1
we have the following pull-back map on divisor classes.
deg k∗hλ = 0 deg k
hψ = 2h− 1
deg k∗hδh = −1 deg k
hδg−h = 1
deg k∗hδi = 0 i 6= h, g − h
Proof of Theorem 2.11. Theorem 2.11 is now a consequence of the above lemmas.
The main point is that the pull-backs of the classes η∗α, η∗β, and η∗γ to our special
families coincide with the classes computed in Section 4-B. For example, to see this
for j∗η∗γ, form the fiber the fiber square
j∗Grd
✲ Grd
M̃2,1
✲ M̃g,1.
Notice that although j is a regular embedding, j′ need not be. Nonetheless, ac-
cording to Fulton [12, Chapter 6], there is a refined Gysin homomorphism
j! : Ak(G
d) → Ak−l(j
∗Grd),
where l is the codimension of j, which commutes with push-forward:
! = j∗η∗.
We need to check that
j!c1(V) ∩ [G
d ] = c1(j
′∗V) ∩ [j∗Grd ].
A PUSH-FORWARD FORMULA WHEN ρ = 0 13
Since
j!c1(V) ∩ [G
d ] = c1(j
′∗V) ∩ j![Grd ]
[12, Proposition 6.3], it is enough to check that
j![Grd ] = [j
∗Grd ].
Generalizing the dimension upper bound in [27, Corollary 5.9] to the multi-component
case [22], we obtain
dim j∗Grd = dimM̃2,1.
This implies that the codimension of j′ is equal to that of j, so the normal cone of
j′ is equal to the pull-back of the normal bundle of j, and the result follows.
Now, for example, to compute η∗γ, write
η∗γ = aλ−
biδi + cψ
Our goal is to solve for a, b0, b1, . . . , bg−1, c. Using Lemmas 4.6 and 4.9, we may solve
for c and write bg−i in terms of bi. From Lemmas 4.4 and 4.7, we may further solve
for b1, b2, . . . , bg−2 in terms of bg−1. It remains to determine a, b0, and bg−1. This
is done by pulling back to M̃2,1, which has Picard number 3, and using Lemmas
4.5 and 4.8. The other push-forwards are computed similarly. �
5. Proofs of Lemmas
In this section, we give proofs of the lemmas stated in Sections 4-B and 4-C.
Proof of Lemma 4.4. If C0,g → M0,g is the universal stable curve, then i
∗C̃g,1 is
formed by attaching M0,g ×Ei to C0,g along the marked sections σi : M0,g → C0,g.
We have the following fiber square.
i∗Crd
✲ i∗C̃g,1
i∗Grd
✲ M0,g
By the Plücker formula for P1, given [C] ∈ M0,g, a limit linear series on i(C)
must have the aspect
(d− r − 1)pi + |(r + 1)pi|
on each Ei. The line bundle L → i
∗Crd is, therefore, the pull-back from i
∗C̃g,1 of the
bundle which is given by
π∗2OE1dp
on M0,g ×E1 and is trivial on all other components. Thus α = β = 0. The vector
bundle V ⊂ π∗L is trivial with fiber isomorphic to
H0(OE1(r + 1)p) ⊂ H
0(OE1dp)
so γ = 0 as well. �
Before proving Lemma 4.5 we state an elementary result in Schubert calculus.
14 D. KHOSLA
Lemma 5.1. [16, p. 266] For integers r and d with 0 ≤ r ≤ d, let
X = G(r,Pd)
be the Grassmannian of r-planes in Pd. For integers
0 ≤ b0 ≤ b1 ≤ · · · ≤ br ≤ d− r,
let σb = σbr ,...,b0 be the corresponding Schubert cycle of codimension
bi. Let
ζ = σ1,1,...,1,0 be the special Schubert cycle of codimension r. If k is an integer for
which
bi = dimX = (r + 1)(d− r),
then ∫
ζk · σb =
i=0(k − d+ r + ai)!
0≤i<j≤r
(aj − ai),
where ai = bi + i.
Proof of Lemma 4.5. Since M2,1 is a smooth Deligne-Mumford stack, it is enough,
by the moving lemma, to prove Lemma 4.5 for a family over a complete curve
B →֒ M̃2,1
which intersects the boundary and Weierstrass divisors transversally. If π : C → B
is the universal stable genus-2 curve and σ : B → C is the marked section, then
j∗C̃g,1 is formed by attaching C to B×C
′ along the marked section Σ = σ(B) ⊂ C.
We begin by assuming that B is disjoint from the closure of the Weierstrass locus
W . In this case we claim that
j∗Grd → B
is a trivial N -sheeted cover of the form B × X , where is X a zero-dimensional
scheme of length N . Indeed, for any curve (C, p) in M̃2,1 \W there are two (limit
linear) grds on C with maximum ramification at p; the vanishing sequences are
a1 = (d− r − 2, d− r − 1, . . . , d− 4, d− 3, d),
a2 = (d− r − 2, d− r − 1, . . . , d− 4, d− 2, d− 1).
If C is smooth, the two linear series are
(d− r − 2)p+ |(r + 2)p|
(d− r − 2)p+ |rp+KC |.
There are analogous series on nodal curves outside the closure of the Weierstrass
locus. In the case of irreducible nodal curves, the sheaves are locally free.
For each of the two grds on C with maximum ramification at p, there are finitely
many grds on C
′ with compatible ramification. Specifically, there are
(2g − 2− d)N
2(g − 1)
of type a1 and
2(g − 1)
of type a2, for total of N limit linear series counted with multiplicity. Since C
fixed, the cover j∗Grd → B is a trivial N -sheeted cover.
A PUSH-FORWARD FORMULA WHEN ρ = 0 15
Consider a reduced sheet B1 ≃ B of type a1. (We assume for simplicity that
the sheet is reduced—the computation is the same in the general case.) Then the
universal line bundle L on j∗Crd is given as
OC on C
π∗2L1 on B1 × C
for some line bundle L1 on C
′ of degree d. It follows that α = β = γ = 0 on B1.
Next consider a sheet B2 ≃ B of type a2. Over B2×C
′ the universal line bundle
L is isomorphic to π∗2L2 for some L2 of degree d on C
′. It remains to determine
L over C. Now ωC(−2p) gives the correct line bundle for all [C] ∈ B2; however,
it has the wrong degrees on the components of the singular fibers. As our first
approximation to L on C we take
ωC/B2(−2Σ)
Let ∆ ⊂ C be the pull-back of the divisor on C of curves of the form C1 ∪C2, where
the Ci have genus one, and the marked points lie on different components. Then
ωC/B2(−2Σ +∆)
has the correct degree on the irreducible components on each fiber. It remains
only to normalize our line bundle by pull-backs from the base B2. In this case,
L|C is required to be trivial along Σ since L is a pull-back from C
′ on the other
component. If σ : B2 → C is the marked section, we let
Ψ = σ∗ωC/B2
be the tautological line bundle on B2. Then
σ∗OC∆ ∼= OB2
σ∗OCΣ ∼= Ψ
It follows that on C,
ωC/B2(−2Σ +∆)⊗ π
∗Ψ⊗−3 on C
π∗2L2 on B2 × C
Thus, if we let
ω = c1(ωC/B2)
σ = c1(OCΣ)
δ = c1(OC∆)
on C and let
ψ = c1(Ψ)
on B2, then
c1(L) =
ω − 2σ + δ − 3π∗ψ on C
dπ∗2p on B2 × C
For the relative dualizing sheaf ω
j∗ eCg,1/B2
, we have
c1(ωj∗C/B2) =
ω + σ on C
(2(g − 2)− 1)π∗2p on B2 × C
16 D. KHOSLA
Figure 3. The morphism C → B2.
To compute the products of these classes on j∗Crd , recall the following formulas on
π : C2,1 → M2,1
π∗ω = 2 π∗δ = 0 π∗σ = 1
2 = 12λ− δ0 − δ1 π∗σ
2 = −ψ π∗δ
2 = −δ1
π∗(δ.σ) = 0 π∗(ω.δ) = δ1 π∗(σ.ω) = ψ.
Then we compute
α = π∗
c1(L)
= 12λ− δ0 − 8ψ
β = π∗
c1(L) · c1(ω)
= 12λ− δ0 − 8ψ
on B2. Since the marked point lies on C
′, V is trivial on B2, so γ = 0 on B2.
Finally we consider the case where B (transversally) intersects the Weierstrass
locus. In this case
η : j∗Grd → B
is the union of a trivial N -sheeted cover of B and a 1-dimensional scheme lying
over each point of the divisor W . It will suffice to compute α and γ on
Grd(j[C, p])
where C is a smooth genus-2 curve, and p ∈ C is a Weierstrass point. (Note that
β is automatically zero.)
There is a single grd on C with maximal ramification at p, namely
(d− r − 2)p+ |(r + 2)p|,
which has vanishing sequence
(d− r − 2, d− r − 1, . . . , d− 4, d− 2, d).
We claim that we only need to consider components of Grd(C ∪C
′) with this aspect
on C. Indeed, any grd on C with ramification 1 less at p is still a subseries of
|dp|. There will be finitely many corresponding aspects on C′ so that, as before,
α = β = γ = 0 on these components.
A PUSH-FORWARD FORMULA WHEN ρ = 0 17
It remains to consider the components of Grd(C ∪ C
′) where the aspect on C′
has ramification (0, 1, 2, 2, . . . , 2) or more at p′. We are reduced to studying the
one-dimensional scheme
S = Grd(C
′; p′, (0, 1, 2, 2, . . . , 2)).
To simplify computations we specialize C′ to a curve which is the union of P1
with g− 2 elliptic curves E1, . . . , Eg−2 attached at general points p1, . . . , pg−2, and
where the marked point p0 lies on E1, and the point of attachment p
′ lies on the
P1. There will be two types of components of S: those on which the aspects on the
E1 E2 Eg−2
· · ·
· · ·
Figure 4. The curve C′.
Ei are maximally ramified at pi, and those on which the aspect on one Ei varies.
Again, as in the proof of Lemma 4.4, we need only consider the latter case.
Assume that for some i, the ramification at pi of the g
d on Ei is one less than
maximal. There are two possibilities: either the series is of the form
(d− r − 1)pi + |rpi + q| for q ∈ Ei,
which for q 6= pi imposes on the P
1 the ramification condition
(1, 1, . . . , 1),
or the grd is a subseries of
(d− r − 2)pi + |(r + 2)pi|
containing
(d− r)pi + |rpi|,
which generically imposes on the P1 the ramification condition
(0, 1, 1, . . . , 1, 2).
In the first case the components are parameterized by Ei, and we compute that
α = −2 on each such irreducible component, irrespective of whether i = 1 or not.
By Grothendieck-Riemann-Roch, γ = −1 when i = 1 and is zero otherwise. In the
second case the grds are parameterized by a P
1. Because the line bundle is constant,
α = 0. On each such P1, the vector bundle V may be viewed as the tautological
bundle of rank r+1 on the Grassmannian of vector subspaces of a fixed vector space
of dimension r+2 containing a subspace of dimension r. It follows that γ = −1 on
each P1.
Let X = G(r,Pd) be the Grassmannian of r-planes in Pd. Let
ζ = σ1,1,...,1,0
18 D. KHOSLA
be the special Schubert cycle of codimension r. Collecting our calculations, we have
that on Grd(C ∪ C
α = −2(g − 2)
σ2,2,...,2,1,0 · σ1,1,...,1 · ζ
= −2(g − 2)
σ3,3,...,3,2,1 · ζ
γ = −
σ2,2,...,2,1,0 · (σ1,1,...,1 + σ2,1,1,...,1,0) · ζ
σ2,2,...,2,1,0 · (σ1,0,0,...,0 · ζ) · ζ
(σ3,2,2,...,2,1,0 + σ2,2,...,2,0 + σ2,2,...,2,1,1) · ζ
(σ3,2,2,...,2,1,0 + ζ
2) · ζg−2
σ3,2,2,...,2,1,0 · ζ
g−2 −
From Lemma 5.1 we compute,
−2d(2g − 2− d)N
3(g − 1)
3(g − 1)
Since the class of the Weierstrass locus in M̃2,1 is 3ψ−λ−δ1, the lemma follows. �
Proof of Lemma 4.6. Because the curves (Ci, pi) are Brill-Noether general, k
a trivial N -sheeted cover of C1 of the form C1 ×X , where X is a zero-dimensional
scheme of length N . Fix a sheet G ∼= C1 in k
d ; this choice corresponds to aspects
Vi ⊂ H
0(Ci, Li)
where Li are degree-d line bundles on Ci. If (a0, a1, . . . , ar) is the vanishing sequence
of V1 at p1, then we know that
0 = ρ(h, r, d)−
(ai − i)
= (r + 1)(d− r)− hr −
r(r + 1),
ai = (r + 1)d−
r(r + 1)− hr.
Let C1 be the blow-up of C1 × C1 at (p1, p1), E the exceptional divisor, and e its
first Chern class. We may construct the universal curve k∗hC̃g,1 → C1 by attaching
C1 × C2 to C1 along C1 × {p2} and the proper transform of C1 × {p1}. Over the
sheet G, the universal line bundle L on k∗hC
π∗2L1 ⊗OC1−dE ⊗ π
1 (dp1)
A PUSH-FORWARD FORMULA WHEN ρ = 0 19
on C1 and
π∗2L2(−dp2)⊗ π
on C1 × C2. Thus
c1(L) =
dπ∗2p− de on C1
−dπ∗1p on C1 × C2.
The relative dualizing sheaf ω
eCg,1/C1
is isomorphic to
π∗2ωC1 ⊗OCE ⊗ π
1OC1−p1
on C1 and
π∗2ωC2(p2)
on C1 × C2. We have
c1(ω) =
−π∗1p+ (2h− 2)π
2p+ e on C1
(2(g − h)− 1)π∗2p on C1 × C2.
Thus, on G,
degα = c1(L)
2 = −d2
deg β = c1(L) · c1(ω) = −d
2(g − j)− 1
The formulas for η∗α and η∗β now follow.
To calculate γ on G, notice that it suffices to compute c1(V
′), where
V ′ = V ⊗ L1(−dp1)
is a sub-bundle of
π1∗(π
2L1(−dE)).
We claim there is a bundle isomorphism
OC1(aj − d)p1
−→ V ′
To describe the map, pick a basis (σ0, σ1, . . . , σr) of V1 ⊂ H
0(L1) with σi vanish-
ing to order ai at p1. Given local sections τi of OC1(ai − d)p1, let the image of
(τ0, τ1, . . . , τr) be the section
of V ′. This is clearly an isomorphism away from p1 and is checked to be an isomor-
phism over p1 as well. Using (4), we have that on G,
deg γ = deg V ′ =
(ai − d)
r(r + 1)− rh
which finishes the proof of the lemma. �
20 D. KHOSLA
Proof of Lemma 4.7. To prove the independence of the ǫi, consider the curves
Bj →֒ M0,g
for j = 2, 3, . . . , g−3 given by taking a fixed stable curve in ǫj and moving a marked
point on the component with g − j marked points. Let B1 →֒ M0,g be the curve
given by moving the first marked point along a fixed smooth curve.The intersection
matrix
(ǫi · Bj)

g − 1 0 0 0 · · · 0 0 0
−1 1 0 0 · · · 0 0 g − 3
0 −1 1 0 · · · 0 0 g − 4
. . .
0 0 0 0 · · · −1 1 3
0 0 0 0 · · · 0 −1 2

where the rows correspond to the Bj for j = 1, 2, . . . , g − 3, and the columns to
the ǫi for i = 2, 3, . . . , g − 2. Since this matrix is non-singular, the first part of the
lemma follows.
To derive the formula for the pull-back, we follow Harris and Morrison [19,
Section 6.F]. Let B be a smooth projective curve, π : C → B a 1-parameter family
of curves in M0,g transverse to the boundary strata. Then π has smooth total
space, and the fibers of π have at most two irreducible components. Let
σi : B → C
be the marked sections. Denote by Σi the image curve σi(B) in C. Then on B,
δ1 = Σ
δg−1 =
Σ2j ,
where we are using D2 to denote π∗(D
2) for a divisor D on C. We now contract the
component of each reducible fiber which meets the section Σ1. If Σj is the image
of Σj under this contraction, then we have
Σ21 = Σ
Σ2j =
(i − 1)ǫi
The Σj are sections of a P
1-bundle, so
0 = (Σj − Σk)
2 = Σ
k − 2Σj · Σk
A PUSH-FORWARD FORMULA WHEN ρ = 0 21
(g − 2)
2≤j,k≤g
k) = 2
2≤j,k≤g
Σj · Σk
It follows that
δg−1 =
(i − 1)(i− 2)
g − 2
− (i− 1)
(i− 1)(i− g)
g − 2
Similarly, we can show that
δ1 + δg−1 =
i(i− g)
g − 1
so the formula for δ1 follows as well. �
The proofs of Lemmas 4.8 and 4.9 are straightforward, so we omit them.
6. Appendix
In this section, we prove that the locus in Section 3-C defined for i = 0 and s = 2
is, in fact, a divisor in G624(M21). The case s = 1 is similar, and s = 0 is clear.
We first establish the following result.
Lemma 6.1. The space G624(M21) is irreducible.
Proof. Note that a g624 is residual to a g
16; that is
L ∈W 624(C) ⇐⇒ KC ⊗ L
∗ ∈W 216(C)
for any smooth curve C of genus 21. Thus there is a dominant rational map
V16,21 −→ G
from the Severi variety of irreducible plane curves of degree 16 and genus 21. Since
V16,21 is irreducible [18] and maps dominantly to G
24, so G
24 is irreducible. �
Proposition 6.2. The substack Ẽ of G624(M21) defined in Section 3-C has codi-
mension 1.
Proof. Since G624(M21) is irreducible, it suffices to exhibit a smooth curve with a
24 not lying on a quadric. Let
S = Bl21 P
be the blow-up of P2 at 21 general points, and consider the linear system
∣∣∣13H − 2
Ej − 3
22 D. KHOSLA
on S, where H is the hyperplane class and Ei are the exceptional divisors. A
calculation using Macaulay 2 (see Proposition 6.3) shows that a general member C
of ν is irreducible and smooth of genus 21. The series
∣∣∣6H −
embeds S in P6 as the rank-2 locus of general 3×6 matrix of linear forms [4, Section
20.4]. The ideal of S is therefore generated by cubics, so S does not lie on quadric.
It follows that C, which embeds in P6 in degree 24, does not lie on a quadric.
Proposition 6.3. Let Σ be a set of 21 general points in P2 and let S = BlΣP
be the blow-up of P2 at Σ. If H is the line class on S and E1, . . . , E21 are the
exceptional divisors, then the linear system
∣∣∣13H − 2
Ej − 3
on S contains a smooth connected curve.
Proof. We begin by showing that it is enough to exhibit a single set of 21 points
over a finite field for which the above statement is true.
Let Hilb
2 be the Hilbert scheme of k points in P2, and let
Σk ⊂ Hilb
2 ×P2
be the universal subscheme. Let
B ⊂ Hilb9Z P
2 ×Hilb12Z P
be the irreducible open subset over which the composition
Σ = π−11 Σ9 ∪ π
2 Σ12
✲ Hilb9ZP
2 ×Hilb12Z P
2 ×P2
Hilb9ZP
2 ×Hilb12Z P
is étale, where π1 and π2 are the obvious projections. Let
π : S = BlΣ P
B → B
be the smooth surface over B whose fibers are blow-ups of P2 at 21 distinct points.
If E9 and E12 are the exceptional divisors, let
L = OS13H − 2E9 − 3E12.
We may further restrict B to an open over which π∗L is locally free of rank at least
C ⊂ Pπ∗L ×B S
is the universal section, then the projection
C → Pπ∗L
is flat, so it suffices to find a single smooth fiber in order to conclude that the
general fiber is smooth. To this end we use Macaulay 2 [15] and work over a finite
field.
A PUSH-FORWARD FORMULA WHEN ρ = 0 23
i1 : S = ZZ/137[x,y,z];
Following Shreyer and Tonoli [29], we realize our points in P2 as a determinental
subscheme.
i2 : randomPlanePoints = (delta,R) -> (
k:=ceiling((-3+sqrt(9.0+8*delta))/2);
eps:=delta-binomial(k+1,2);
if k-2*eps>=0
then minors(k-eps,
random(R^(k+1-eps),R^{k-2*eps:-1,eps:-2}))
else minors(eps,
random(R^{k+1-eps:0,2*eps-k:-1},R^{eps:-2})));
i3 : distinctPoints = (J) -> (
singJ = minors(2, jacobian J) + J;
codim singJ == 3);
Let Σ9 and Σ12 be our subsets of 9 and 12 points, respectively.
i4 : Sigma9 = randomPlanePoints(9,S);
o4 : Ideal of S
i5 : Sigma12 = randomPlanePoints(12,S);
o5 : Ideal of S
i6 : (distinctPoints Sigma9, distinctPoints Sigma12)
o6 = (true, true)
o6 : Sequence
Their union is Σ.
i7 : Sigma = intersect(Sigma9, Sigma12);
o7 : Ideal of S
i8 : degree Sigma
o8 = 21
24 D. KHOSLA
Next we construct the 0-dimensional subscheme Γ whose ideal consists of curves
double through points of Σ9 and triple through points of Σ12.
i9 : Gamma = saturate intersect(Sigma9^2, Sigma12^3);
o9 : Ideal of S
Let us check that Γ imposes the expected number of conditions (9 · 3+ 12 · 6 = 99)
on curves of degree 13.
i10 : hilbertFunction (13, Gamma)
o10 = 99
Pick a random curve C of degree 13 in the ideal of Γ.
i11 : C = ideal (gens Gamma
* random(source gens Gamma, S^{-13}));
o11 : Ideal of S
We check that C is irreducible.
i12 : # decompose C
o12 = 1
To check smoothness, let Csing be the singular locus of C.
i13 : Csing = (ideal jacobian C) + C;
o13 : Ideal of S
i14 : codim Csing
o14 = 2
A double point will contribute 1 to the degree of Csing if it is transverse and
more otherwise. Similarly, a triple point will contribute 4 to the degree of Csing if it
is transverse and more otherwise. So for C to be smooth in the blow-up, we must
have that
degCsing = 9 + 4 · 12 = 57
A PUSH-FORWARD FORMULA WHEN ρ = 0 25
i15 : degree Csing
o15 = 57
Definition 6.4. Let E be the effective codimension-1 Chow cycle which is the
image of Ẽ under the map
η : G624 → M
Proposition 6.5. The class of E ⊂ Mirr21 is given as
[E] = 2459λ− 377δ0.
Proof. Applying Equation (2) from Section 3-C,
[E] = η∗[Ẽ] = η∗(2α− β + λ− 8γ)
2459N
where N is the degree of η. �
Corollary 6.6. The slope conjecture is false in genus 21.
Proof. Since
< 6 +
this is an immediate consequence of [11, Corollary 1.2]. �
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Department of Mathematics, University of Texas at Austin, 1 University Station
C1200, Austin, Texas 78712, USA
E-mail address: dkhosla@math.utexas.edu
http://www.math.uiuc.edu/Macaulay2/
	1. Introduction
	2. Statement of Theorem
	2-A. A limit linear series moduli stack
	2-B. Tautological Classes
	2-C. A Push-Forward Formula
	3. Applications
	3-A. The Gieseker-Petri Divisor
	3-B. Hypersurface Divisors
	3-C. Syzygy Divisors
	3-D. Secant Plane Divisors
	4. Special Families of Curves
	4-A. Definitions of Families
	4-B. Computations on the Special Families
	4-C. Pull-Back Maps on Divisors
	5. Proofs of Lemmas
	6. Appendix
	References
