scholarly journals The Automorphism Group of the Hilbert Scheme of Two Points on a Generic Projective K3 Surface

2016 ◽  
pp. 1-15 ◽  
Author(s):  
Samuel Boissiére ◽  
Andrea Cattaneo ◽  
Marc Nieper-Wisskirchen ◽  
Alessandra Sarti
Keyword(s):  
Automorphism Group ◽  
Hilbert Scheme ◽  
K3 Surface

scholarly journals INTEGRAL CONSTRAINTS ON THE MONODROMY GROUP OF THE HYPERKÄHLER RESOLUTION OF A SYMMETRIC PRODUCT OF A K3 SURFACE

2010 ◽  
Vol 21 (02) ◽  
pp. 169-223 ◽  
Author(s):  
EYAL MARKMAN
Keyword(s):  
Hilbert Scheme ◽  
Prime Power ◽  
Monodromy Group ◽  
Euler Number ◽  
Hodge Structure ◽  
K3 Surface ◽  
Monodromy Operator ◽  
First Case ◽  
Hyperkähler Varieties ◽  
Restriction Homomorphism

Let S[n]be the Hilbert scheme of length n subschemes of a K3 surface S. H2(S[n],ℤ) is endowed with the Beauville–Bogomolov bilinear form. Denote by Mon the subgroup of GL [H*(S[n],ℤ)] generated by monodromy operators, and let Mon2be its image in OH2(S[n],ℤ). We prove that Mon2is the subgroup generated by reflections with respect to +2 and -2 classes (Theorem 1.2). Thus Mon2does not surject onto OH2(S[n],ℤ)/(±1), when n - 1 is not a prime power.As a consequence, we get counterexamples to a version of the weight 2 Torelli question for hyperKähler varieties X deformation equivalent to S[n]. The weight 2 Hodge structure on H2(X,ℤ) does not determine the bimeromorphic class of X, whenever n - 1 is not a prime power (the first case being n = 7). There are at least 2ρ(n - 1) - 1distinct bimeromorphic classes of X with a given generic weight 2 Hodge structure, where ρ(n - 1) is the Euler number of n - 1.The second main result states, that if a monodromy operator acts as the identity on H2(S[n],ℤ), then it acts as the identity on Hk(S[n],ℤ), 0 ≤ k ≤ n + 2 (Theorem 1.5). We conclude the injectivity of the restriction homomorphism Mon → Mon2, if n ≡ 0 or n ≡ 1 modulo 4 (Corollary 1.6).

scholarly journals A CLASSIFICATION OF LAGRANGIAN PLANES IN HOLOMORPHIC SYMPLECTIC VARIETIES

2015 ◽  
Vol 16 (4) ◽  
pp. 859-877 ◽  
Author(s):  
Benjamin Bakker
Keyword(s):  
Hilbert Scheme ◽  
Rational Curve ◽  
K3 Surface ◽  
Higher Dimensional ◽  
Extremal Curve ◽  
Symplectic Varieties

Classically, an indecomposable class $R$ in the cone of effective curves on a K3 surface $X$ is representable by a smooth rational curve if and only if $R^{2}=-2$. We prove a higher-dimensional generalization conjectured by Hassett and Tschinkel: for a holomorphic symplectic variety $M$ deformation equivalent to a Hilbert scheme of $n$ points on a K3 surface, an extremal curve class $R\in H_{2}(M,\mathbb{Z})$ in the Mori cone is the line in a Lagrangian $n$-plane $\mathbb{P}^{n}\subset M$ if and only if certain intersection-theoretic criteria are met. In particular, any such class satisfies $(R,R)=-\frac{n+3}{2}$, and the primitive such classes are all contained in a single monodromy orbit.

scholarly journals EPW cubes

2019 ◽  
Vol 2019 (748) ◽  
pp. 241-268 ◽  
Author(s):  
Atanas Iliev ◽  
Grzegorz Kapustka ◽  
Michał Kapustka ◽  
Kristian Ranestad
Keyword(s):  
Moduli Space ◽  
Hilbert Scheme ◽  
Symplectic Manifolds ◽  
K3 Surface ◽  
Degeneracy Loci ◽  
Double Covers ◽  
Formula Type

Abstract We construct a new 20-dimensional family of projective six-dimensional irreducible holomorphic symplectic manifolds. The elements of this family are deformation equivalent with the Hilbert scheme of three points on a K3 surface and are constructed as natural double covers of special codimension-three subvarieties of the Grassmannian G(3,6) . These codimension-three subvarieties are defined as Lagrangian degeneracy loci and their construction is parallel to that of EPW sextics, we call them the EPW cubes. As a consequence we prove that the moduli space of polarized IHS sixfolds of K3 -type, Beauville–Bogomolov degree 4 and divisibility 2 is unirational.

scholarly journals Computing cup products in integral cohomology of Hilbert schemes of points on K3 surfaces

2016 ◽  
Vol 19 (1) ◽  
pp. 78-97
Author(s):  
Simon Kapfer
Keyword(s):  
Computer Program ◽  
Hilbert Scheme ◽  
K3 Surfaces ◽  
K3 Surface ◽  
Hilbert Schemes ◽  
Integral Cohomology ◽  
Cup Products ◽  
By Products

We study cup products in the integral cohomology of the Hilbert scheme of $n$ points on a K3 surface and present a computer program for this purpose. In particular, we deal with the question of which classes can be represented by products of lower degrees.Supplementary materials are available with this article.

scholarly journals THE ABELIAN FIBRATION ON THE HILBERT CUBE OF A K3 SURFACE OF GENUS 9

2007 ◽  
Vol 18 (01) ◽  
pp. 1-26 ◽  
Author(s):  
ATANAS ILIEV ◽  
KRISTIAN RANESTAD
Keyword(s):  
General Result ◽  
Hilbert Scheme ◽  
Alternative Proof ◽  
Hilbert Cube ◽  
K3 Surface ◽  
Cubic Curves ◽  
General Fiber ◽  
Twisted Cubic

In this paper we construct an abelian fibration over P3on the Hilbert cube of the primitive K3 surface of genus 9. After the abelian fibration constructed by Hassett and Tschinkel on the Hilbert square on the primitive K3 surface of genus 5, this is the second example where the abelian fibration is constructed directly on [Formula: see text]. The recent more general result of Sawon proves the existence of an abelian fibration on the Hilbert scheme [Formula: see text] of a primitive K3 surface S of degree 2g - 2 = m2(2n - 2). Our example provides an alternative proof in the case m = 2, n = 3. Furthermore we identify the general fiber with the Hilbert scheme of twisted cubic curves in a Fano 3-fold of genus 9, and interpret the addition law on this variety.

scholarly journals Motivic decompositions for the Hilbert scheme of points of a K3 surface

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Andrei Neguţ ◽  
Georg Oberdieck ◽  
Qizheng Yin
Keyword(s):  
Lie Algebra ◽  
Hilbert Scheme ◽  
K3 Surface ◽  
Hilbert Scheme Of Points

Abstract We construct an explicit, multiplicative Chow–Künneth decomposition for the Hilbert scheme of points of a K3 surface. We further refine this decomposition with respect to the action of the Looijenga–Lunts–Verbitsky Lie algebra.

scholarly journals Obstructions to deforming curves on a 3-fold, III: Deformations of curves lying on a K3 surface

2017 ◽  
Vol 28 (13) ◽  
pp. 1750099 ◽  
Author(s):  
Hirokazu Nasu
Keyword(s):  
Elliptic Curves ◽  
Hilbert Scheme ◽  
Smooth Surface ◽  
Smooth Curve ◽  
K3 Surface ◽  
Sufficient Condition ◽  
Infinitesimal Deformation ◽  
First Order ◽  
Irreducible Components

We study the deformations of a smooth curve [Formula: see text] on a smooth projective [Formula: see text]-fold [Formula: see text], assuming the presence of a smooth surface [Formula: see text] satisfying [Formula: see text]. Generalizing a result of Mukai and Nasu, we give a new sufficient condition for a first order infinitesimal deformation of [Formula: see text] in [Formula: see text] to be primarily obstructed. In particular, when [Formula: see text] is Fano and [Formula: see text] is [Formula: see text], we give a sufficient condition for [Formula: see text] to be (un)obstructed in [Formula: see text], in terms of [Formula: see text]-curves and elliptic curves on [Formula: see text]. Applying this result, we prove that the Hilbert scheme [Formula: see text] of smooth connected curves on a smooth quartic [Formula: see text]-fold [Formula: see text] contains infinitely many generically non-reduced irreducible components, which are variations of Mumford’s example for [Formula: see text].

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.

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