Given a strict simple degeneration @math the first three authors previously constructed a degeneration @math of the relative degree @math Hilbert scheme of @math -dimensional subschemes. In this paper we investigate the geometry of this degeneration, in particular when the fibre dimension of @math is at most @math . In this case we show that @math is a dlt model. This is even a good dlt model if @math has this property. We compute the dual complex of the central fibre @math and relate this to the Kontsevich-Soibelman skeleton. For a type II degeneration of @math surfaces we show that the stack @math carries a nowhere degenerate relative logarithmic @math -form. Finally we discuss the relationship of our degeneration with the constructions of Nagai.
In @cite_10 , Koll 'ar study (among other things) minimal dlt models of @math -dimensional Hyperk "ahler manifolds. Most relevant to our results, is that they establish a strong result linking properties of the dual complex @math of the special fibre @math of a minimal dlt model @math and the monodromy action on the second cohomology @math of a general fibre of @math . More precisely, in [Theorem 0.10] KLSV17 they prove that the dimension of the geometric realization of @math is @math , @math and @math , if and only if the index of nilpotency of the log monodromy operator on @math is @math , @math or @math respectively. Starting with a type II degeneration @math of K3 surfaces, our degeneration @math yields a dual complex of dimension @math , and it is indeed well known that the nilpotency index is @math in this case.
@cite_10
2613622891 1893146140 1645016350 2550342065
Using the Minimal Model Program, any degeneration of K-trivial varieties can be arranged to be in a Kulikov type form, i.e. with trivial relative canonical divisor and mild singularities. In the hyper-K "ahler setting, we can then deduce a finiteness statement for monodromy acting on @math , once one knows that one component of the central fiber is not uniruled. Independently of this, using deep results from the geometry of hyper-K "ahler manifolds, we prove that a finite monodromy projective degeneration of hyper-K "ahler manifolds has a smooth filling (after base change and birational modifications). As a consequence of these two results, we prove a generalization of Huybrechts' theorem about birational versus deformation equivalence, allowing singular central fibers. As an application, we give simple proofs for the deformation type of certain geometric constructions of hyper-K "ahler manifolds (e.g. Debarre--Voisin or Laza--Sacc a--Voisin). In a slightly different direction, we establish some basic properties (dimension and rational homology type) for the dual complex of a Kulikov type degeneration of hyper-K "ahler manifolds. This is the second in a series of papers in which we derive a @math -expansion for the two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In this model, two phases appear, which interact via a nonlocal Coulomb type energy. Here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction, thus creating small "droplets" of the minority phase in a "sea" of the majority phase. In our previous paper, we computed the @math -limit of the leading order energy, which yields the averaged behavior for almost minimizers, namely that the density of droplets should be uniform. Here we go to the next order and derive a next order @math -limit energy, which is exactly the Coulombian renormalized energy obtained by Sandier and Serfaty as a limiting interaction energy for vortices in the magnetic Ginzburg-Landau model. The derivation is based on the abstract scheme of Sandier-Serfaty that serves to obtain lower bounds for 2-scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem. Without thus appealing to the Euler-Lagrange equation, we establish for all configurations which have "almost minimal energy" the asymptotic roundness and radius of the droplets, and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense. Via a kind of @math -equivalence, the obtained results also yield an expansion of the minimal energy for the original Ohta-Kawasaki energy. This leads to expecting to see triangular lattices of droplets as energy minimizers. Let f be a polynomial of degree n in ZZ[x_1,..,x_n], typically reducible but squarefree. From the hypersurface f=0 one may construct a number of other subschemes Y by extracting prime components, taking intersections, taking unions, and iterating this procedure. We prove that if the number of solutions to f=0 in ^n is not a multiple of p, then all these intersections in ^n_ just described are reduced. (If this holds for infinitely many p, then it holds over as well.) More specifically, there is a_Frobenius splitting_ on ^n_ compatibly splitting all these subschemes Y . We determine when a Gr "obner degeneration f_0=0 of such a hypersurface f=0 is again such a hypersurface. Under this condition, we prove that compatibly split subschemes degenerate to compatibly split subschemes, and stay reduced. Our results are strongest in the case that f's lexicographically first term is i=1 ^n x_i. Then for all large p, there is a Frobenius splitting that compatibly splits f's hypersurface and all the associated Y . The Gr "obner degeneration Y' of each such Y is a reduced union of coordinate spaces (a Stanley-Reisner scheme), and we give a result to help compute its Gr "obner basis. We exhibit an f whose associated Y include Fulton's matrix Schubert varieties, and recover much more easily the Gr "obner basis theorem of [Knutson-Miller '05]. We show that in Bott-Samelson coordinates on an opposite Bruhat cell X^v_ in G B, the f defining the complement of the big cell also has initial term i=1 ^n x_i, and hence the Kazhdan-Lusztig subvarieties X^v_ w degenerate to Stanley-Reisner schemes. This recovers, in a weak form, the main result of [Knutson '08]. Author(s): Johnson, William Andrew | Advisor(s): Scanlon, Thomas | Abstract: This dissertation is a collection of results in model theory, related in one way or another to fields, NIP theories, and elimination of imaginaries. The most important result is a classification of dp-minimal fields, presented in Chapter 9. We construct in a canonical fashion a non-trivial Hausdorff definable field topology on any unstable dp-minimal field. Usingthis we classify the dp-minimal pure fields and valued fields up to elementary equivalence. Furthermore we prove that every VC-minimal field is real closed or algebraically closed.In Chapter 11, we analyze the theories of existentially closed fields with several valuations and orderings, as studied by van den Dries. We show that these model complete theories are NTP2, and analyze forking, dividing, and burden in these theories. The theory of algebraically closed fields with n independent valuation rings turns out to be an example of such a theory. This provides a new and natural example of an NTP2 theory which is neithersimple nor NIP, nor even a conceptual hybrid of something simple and something NIP.In Chapter 8, we exhibit a bad failure of elimination of imaginaries in a dense o-minimal structure. We produce an exotic interpretable set which cannot be put in definable bijection with a definable set, after naming any amount of parameters. However, we show that these exotic interpretable sets are still amenable to some of the tools of tame topology: they mustadmit nice definable topologies locally homeomorphic to definable sets.Chapter 12 proves the existence of Z nZ-valued definable strong Euler characteristics on pseudofinite fields, which measure the non-standard “size” of definable sets, mod n. The non-trivial result is that these “sizes” are definable in families of definable sets. This could probably be proven e cohomology, but we give a more elementary proof relying heavily on the theory of abelian varieties.We also present simplified and new proofs of several model-theoretic facts, including the definability of irreducibility and Zariski closure in ACF (Chapter 10), and elimination of imaginaries in ACVF (Chapter 6). This latter fact was originally proven by Haskell, Hrushovski, and Macpherson. We give a proof that is drastically simpler, inspired by Poizat’s proofs of elimination of imaginaries in ACF and DCF.