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Birational geometry of Beauville–Mukai systems III: Asymptotic behavior
Abstract
Suppose that a Hilbert scheme of points on a K3 surface of Picard rank one admits a rational Lagrangian fibration. We show that if the degree of the surface is sufficiently large compared to the number of points, then the Hilbert scheme is the unique hyperkähler manifold in its birational class. In particular, the Hilbert scheme is a Lagrangian fibration itself, which we realize as coming from a (twisted) Beauville–Mukai system on a Fourier–Mukai partner of . We also show that when the degree of the surface is small our method can be used to find all birational models of the Hilbert scheme.
... Other choices of m have been considered by [Marku06], [Saw07], and [BM14a]. In the third installment of this series [QS22c], we will use similar wall-crossing techniques to show that the Hilbert scheme itself admits a Lagrangian fibration when d is large relative to m (and d(m − 1) is a perfect square). ...
Via wall-crossing, we study the birational geometry of Beauville-Mukai systems on K3 surfaces with Picard rank one. We show that there is a class of walls which are always present in the movable cones of Beauville-Mukai systems. We give a complete description of the birational geometry of rank two Beauville-Mukai systems when the genus of the surface is small.
We study wall-crossing for the Beauville–Mukai system of rank three on a general genus two K3 surface. We show that such a system is related to the Hilbert scheme of ten points on the surface by a sequence of flops, whose exceptional loci can be described as Brill–Noether loci. We also obtain Brill–Noether type results for sheaves in the Beauville–Mukai system.
We show that for many moduli spaces M of torsion sheaves on K3 surfaces S,
the functor D(S) -> D(M) induced by the universal sheaf is a P-functor, hence
can be used to construct an autoequivalence of D(M), and that this
autoequivalence can be factored into geometrically meaningful equivalences
associated to abelian fibrations and Mukai flops. Along the way we produce a
derived equivalence between two compact hyperkaehler 2g-folds that are not
birational, for every g >= 2. We also speculate about an approach to Kawamata's
"K-equivalence implies D-equivalence" conjecture for moduli spaces of sheaves
on K3 surfaces.
Let be the Hilbert scheme of g points on a K3 surface S. Suppose that where C is a smooth curve with . We prove that is a Lagrangian fibration.
We study holomorphic symplectic manifolds which are fibred by abelian varieties. This structure is a higher dimensional analogue of an elliptic fibration on a K3 surface. We investigate when a holomorphic symplectic manifold is fibred in this way, and are led to several natural conjectures. We then study the geometry of these fibrations. The expectation is that this point of view will prove useful in understanding holomorphic symplectic manifolds, and possibly lead to a classification.
Using the techniques of Bayer–Macrì, we determine the walls in the movable cone of the Mukai system of rank two for a general K3 surface S of genus two. We study the (essentially unique) birational map to S^{[5]} and decompose it into a sequence of flops. We give an interpretation of the exceptional loci in terms of Brill–Noether loci.
Ricci-flat manifolds have been studied for many years and the interest in various aspects of their theory seems still to be growing. Their rich geometry has been explored with techniques from different branches of mathematics and the interplay of analysis, arithmetic, and geometry makes their theory highly attractive. At the same time, these manifolds play a prominent rôle in string theory and, in particular, in mirror symmetry. Physicists have come up with interesting and difficult mathematical questions and have suggested completely new directions that should be pursued.
We use wall-crossing with respect to Bridgeland stability conditions to
systematically study the birational geometry of a moduli space M of stable
sheaves on a K3 surface X:
1. We describe the nef cone, the movable cone, and the effective cone of M in
terms of the Mukai lattice of X.
2. We establish a long-standing conjecture that predicts the existence of a
birational Lagrangian fibration on M whenever M admits an integral divisor
class D of square zero (with respect to the Beauville-Bogomolov form).
These results are proved using a natural map from the space of Bridgeland
stability conditions Stab(X) to the cone Mov(X) of movable divisors on M; this
map relates wall-crossing in Stab(X) to birational transformations of M. In
particular, every minimal model of M appears as a moduli space of
Bridgeland-stable objects on X.
In this paper, we consider basic problems on moduli spaces of stable sheaves on abelian surfaces. Our main assumption is
the primitivity of the associated Mukai vector. We determine the deformation types, albanese maps, Bogomolov factors and their
weight 2 Hodge structures. We also discuss the deformation types of moduli spaces of stable sheaves on K3 surfaces.
A rational Lagrangian fibration f on an irreducible symplectic variety V is a rational map which is birationally equivalent to a regular surjective morphism with Lagrangian fibers. By analogy with K3 surfaces, it is natural to expect that a rational Lagrangian fibration exists if and only if V has a divisor D with Bogomolov–Beauville square 0. This conjecture is proved in the case when V is the Hilbert scheme of d points on a generic K3 surface S of genus g under the hypothesis that its degree 2g−2 is a square times 2d−2. The construction of f uses a twisted Fourier–Mukai transform which induces a birational isomorphism of V with a certain moduli space of twisted sheaves on another K3 surface M, obtained from S as its Fourier–Mukai partner.
We analyze the ample and moving cones of holomorphic symplectic manifolds, in light of recent advances in the minimal model
program. As an application, we establish a numerical criterion for ampleness of divisors on fourfolds deformation-equivalent
to punctual Hilbert schemes of K3 surfaces.
We construct a family of nef divisor classes on every moduli space of stable
complexes in the sense of Bridgeland. This divisor class varies naturally with
the Bridgeland stability condition.
For a generic stability condition on a K3 surface, we prove that this class
is ample, thereby generalizing a result of Minamide, Yanagida, and Yoshioka.
Our result also gives a systematic explanation of the relation between
wall-crossing for Bridgeland-stability and the minimal model program for the
moduli space.
We give three applications of our method for classical moduli spaces of
sheaves on a K3 surface:
1. We obtain a region in the ample cone in the moduli space of
Gieseker-stable sheaves only depending on the lattice of the K3.
2. We determine the nef cone of the Hilbert scheme of n points on a K3
surface of Picard rank one when n is large compared to the genus.
3. We verify the "Hassett-Tschinkel/Huybrechts/Sawon" conjecture on the
existence of a birational Lagrangian fibration for the Hilbert scheme in a new
family of cases.
This paper contains a description of one connected component of the space of stability conditions on the bounded derived category of coherent sheaves on a complex algebraic K3 surface.