scholarly journals On the non-divisorial base locus of big and nef line bundles on K3[2]-type varieties

2020 ◽  
pp. 1-27
Author(s):  
Ulrike Rieß
Keyword(s):  
Moduli Spaces ◽  
Hilbert Scheme ◽  
Ample Line Bundle ◽  
Base Point ◽  
K3 Surfaces ◽  
Line Bundles ◽  
Codimension Two ◽  
Base Locus ◽  
Base Loci ◽  
Symplectic Varieties

Abstract We approach non-divisorial base loci of big and nef line bundles on irreducible symplectic varieties. While for K3 surfaces, only divisorial base loci can occur, nothing was known about the behaviour of non-divisorial base loci for more general irreducible symplectic varieties. We determine the base loci of all big and nef line bundles on the Hilbert scheme of two points on very general K3 surfaces of genus two and on their birational models. Remarkably, we find an ample line bundle with a non-trivial base locus in codimension two. We deduce that, generically in the moduli spaces of polarized K3[2]-type varieties, the polarization is base point free.

scholarly journals The standard conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K3 surfaces

2013 ◽  
Vol 149 (3) ◽  
pp. 481-494 ◽  
Author(s):  
François Charles ◽  
Eyal Markman
Keyword(s):  
Hilbert Scheme ◽  
K3 Surfaces ◽  
Cohomology Algebra ◽  
Hilbert Schemes ◽  
Projective Varieties ◽  
Hilbert Scheme Of Points ◽  
Complex Projective ◽  
Symplectic Varieties

AbstractWe prove the standard conjectures for complex projective varieties that are deformations of the Hilbert scheme of points on a K3 surface. The proof involves Verbitsky’s theory of hyperholomorphic sheaves and a study of the cohomology algebra of Hilbert schemes of K3 surfaces.

scholarly journals Refined Verlinde formulas for Hilbert schemes of points and moduli spaces of sheaves on K3 surfaces

2020 ◽  
Vol Volume 4 ◽  
Author(s):  
Lothar Göttsche
Keyword(s):  
Moduli Spaces ◽  
Generating Functions ◽  
K3 Surfaces ◽  
Line Bundles ◽  
Hilbert Schemes ◽  
Moduli Spaces Of Sheaves ◽  
Elliptic Genera ◽  
Determinant Line Bundles

We compute generating functions for elliptic genera with values in line bundles on Hilbert schemes of points on surfaces. As an application we also compute generating functions for elliptic genera with values in determinant line bundles on moduli spaces of sheaves on K3 surfaces.

scholarly journals Verra Four-Folds, Twisted Sheaves, and the Last Involution

2018 ◽  
Vol 2019 (21) ◽  
pp. 6661-6710 ◽  
Author(s):  
Chiara Camere ◽  
Grzegorz Kapustka ◽  
Michał Kapustka ◽  
Giovanni Mongardi
Keyword(s):  
Moduli Spaces ◽  
Hilbert Scheme ◽  
K3 Surfaces ◽  
Symplectic Manifolds ◽  
K3 Surface ◽  
Enriques Surfaces ◽  
Twisted Sheaves

Abstract We study the geometry of some moduli spaces of twisted sheaves on K3 surfaces. In particular we introduce induced automorphisms from a K3 surface on moduli spaces of twisted sheaves on this K3 surface. As an application we prove the unirationality of moduli spaces of irreducible holomorphic symplectic manifolds of K3[2]-type admitting non-symplectic involutions with invariant lattices U(2) ⊕ D4(−1) or U(2) ⊕ E8(−2). This complements the results obtained in [43], [13], and the results from [29] about the geometry of irreducible holomorphic symplectic (IHS) four-folds constructed using the Hilbert scheme of (1, 1) conics on Verra four-folds. As a byproduct we find that IHS four-folds of K3[2]-type with Picard lattice U(2) ⊕ E8(−2) naturally contain non-nodal Enriques surfaces.

scholarly journals Motives of moduli spaces on K3 surfaces and of special cubic fourfolds

2018 ◽  
Vol 161 (1-2) ◽  
pp. 109-124 ◽  
Author(s):  
Tim-Henrik Bülles
Keyword(s):  
Moduli Spaces ◽  
K3 Surfaces

scholarly journals On the Singular Sheaves in the Fine Simpson Moduli Spaces of 1-dimensional Sheaves

2017 ◽  
Vol 60 (3) ◽  
pp. 522-535 ◽  
Author(s):  
Oleksandr Iena ◽  
Alain Leytem
Keyword(s):  
Projective Plane ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Blow Up ◽  
Line Bundles ◽  
Hilbert Polynomial ◽  
Stable Sheaves ◽  
Closed Subvariety

AbstractIn the Simpson moduli space M of semi-stable sheaves with Hilbert polynomial dm − 1 on a projective plane we study the closed subvariety M' of sheaves that are not locally free on their support. We show that for d ≥4 , it is a singular subvariety of codimension 2 in M. The blow up of M along M' is interpreted as a (partial) modification of M \ M' by line bundles (on support).

scholarly journals K-MODULI OF CURVES ON A QUADRIC SURFACE AND K3 SURFACES

2021 ◽  
pp. 1-41
Author(s):  
KENNETH ASCHER ◽  
KRISTIN DEVLEMING ◽  
YUCHEN LIU
Keyword(s):  
Complete Intersection ◽  
Moduli Space ◽  
Moduli Spaces ◽  
K3 Surfaces ◽  
Moduli Of Curves ◽  
Quadric Surface ◽  
Intersection Curves

Abstract We show that the K-moduli spaces of log Fano pairs $\left(\mathbb {P}^1\times \mathbb {P}^1, cC\right)$ , where C is a $(4,4)$ curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ , complete intersection curves in $\mathbb {P}^3$ . This, together with recent results by Laza and O’Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of $(4,4)$ curves on $\mathbb {P}^1\times \mathbb {P}^1$ and the Baily–Borel compactification of moduli of quartic hyperelliptic K3 surfaces.

scholarly journals Moduli spaces of twisted K3 surfaces and cubic fourfolds

2020 ◽  
Vol 377 (3-4) ◽  
pp. 1453-1479
Author(s):  
Emma Brakkee
Keyword(s):  
Moduli Spaces ◽  
K3 Surfaces

scholarly journals Perverse Coherent Sheaves on Blow-ups at Codimension 2 Loci

2019 ◽  
Author(s):  
Naoki Koseki
Keyword(s):  
Moduli Space ◽  
Hilbert Scheme ◽  
Projective Variety ◽  
Blow Up ◽  
Smooth Projective Variety ◽  
Coherent Sheaves ◽  
Codimension Two ◽  
Higher Dimensional ◽  
Stable Sheaves ◽  
Closed Subvariety

Abstract Let $f \colon X \to Y$ be the blow-up of a smooth projective variety $Y$ along its codimension two smooth closed subvariety. In this paper, we show that the moduli space of stable sheaves on $X$ and $Y$ are connected by a sequence of flip-like diagrams. The result is a higher dimensional generalization of the result of Nakajima and Yoshioka, which is the case of $\dim Y=2$. As an application of our general result, we study the birational geometry of the Hilbert scheme of two points.

scholarly journals On Some Generalized Rapoport–Zink Spaces

2019 ◽  
Vol 72 (5) ◽  
pp. 1111-1187
Author(s):  
Xu Shen
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
K3 Surfaces ◽  
Connected Components ◽  
Shimura Varieties ◽  
Reductive Groups ◽  
Mixed Characteristic ◽  
Special Fibers ◽  
Type Spaces ◽  
Local Invariants

AbstractWe enlarge the class of Rapoport–Zink spaces of Hodge type by modifying the centers of the associated $p$-adic reductive groups. Such obtained Rapoport–Zink spaces are said to be of abelian type. The class of Rapoport–Zink spaces of abelian type is strictly larger than the class of Rapoport–Zink spaces of Hodge type, but the two type spaces are closely related as having isomorphic connected components. The rigid analytic generic fibers of Rapoport–Zink spaces of abelian type can be viewed as moduli spaces of local $G$-shtukas in mixed characteristic in the sense of Scholze.We prove that Shimura varieties of abelian type can be uniformized by the associated Rapoport–Zink spaces of abelian type. We construct and study the Ekedahl–Oort stratifications for the special fibers of Rapoport–Zink spaces of abelian type. As an application, we deduce a Rapoport–Zink type uniformization for the supersingular locus of the moduli space of polarized K3 surfaces in mixed characteristic. Moreover, we show that the Artin invariants of supersingular K3 surfaces are related to some purely local invariants.

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