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Grzegorz Kapustka

Publications and source records attributed to Grzegorz Kapustka.

At least 19 recordsLinked to original sources
arXiv2026

Generalized Nikulin surfaces and irreducible symplectic fourfolds

A Nikulin surface is the minimal resolution of the quotient of a $K3$ surface $S$ by a symplectic involution $ι_S$. Equivalently, it is the $2$-dimensional component of the fixed locus of the involution induced by $ι_S$ on the Hilbert scheme $S^{[2]}$. We study $K3$ surfaces $F$ that are the $2$-dimensional component of the fixed locus of a symplectic involution $ι$ on hyper-Kähler manifolds $X$ of $K3^{[2]}$-type; we call them generalized Nikulin surfaces. We show that a projective $K3$ surface is a generalized Nikulin surface if and only if its Néron-Severi lattice contains primitively the lattice $E_7(-2)$. Moreover, we show that the transcendental lattices $T_F$ and $T_{\widetilde{X/ ι}}$, where $\widetilde{X/ ι}$ is the terminalization of the quotient $X/ι$, are Hodge isometric. Finally, we describe projective models of generalized Nikulin surfaces of small degrees.

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arXiv2026

A symplectic fourfold

We present a method to construct irreducible symplectic varieties by studying terminalisations of quotient of hyper-Kähler manifolds by non-natural group actions. In particular, we construct irreducible symplectic varieties of dimension $4$ with $b_2 = 4$ and non-quotient singularities: this provides explicit examples of ISVs for which a global Torelli theorem is not known to hold.

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arXiv2025

Constructions of derived equivalent hyper-Kähler fourfolds

We study twisted derived equivalences of hyper-Kähler fourfolds. We describe when two hyper-Kähler fourfolds of $K3^{[2]}$-type of Picard rank $1$ with isomorphic transcendental lattices are derived equivalent. Then we present new constructions of pairs of twisted derived equivalent hyper-Kähler manifolds of Picard rank $\geq 2$.

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arXiv2024

Variety of apolar schemes to powers of quadrics

We study the variety $\rm{VAPS}_G(q_2^3, 10)$ (resp. $\rm{VAPS}_G(q_3^2, 10)$), a Grassmannian compactification of the variety of finite schemes of length $10$ apolar to $q_{2}^3$ (resp. $q_3^2$), where $\{q_{n} = 0\}\subset \mathbb P^{n} $ is a smooth quadric hypersurface. In particular, we show that $\rm{VAPS_G}(q_2^3, 10)$ is the tangent developable of a rational normal curve, while $\rm{VAPS}_G(q_3^2, 10)$ is reducible with three singular $5$-dimensional components.

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arXiv2024

Even nodal surfaces of K3 type

We study Fano fourfolds of K3 type with a conic bundle structure. We construct direct geometrical links between these fourfolds and hyperKähler varieties. As a result we describe families of nodal surfaces that can be seen as generalisations of Kummer quartic surfaces. Each of these families actually arises through two families of Fano fourfolds, whose conic bundle structures are related by hyperbolic reduction.

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arXiv2024

EPW sextics vs EPW cubes

We study a correspondence between double EPW cubes and double EPW sextics, two families of polarized hyper-Kähler manifolds related to Gushel--Mukai fourfolds. We infer relations between these families in terms of Hodge structures and moduli spaces of elliptic curves. As an application, we prove that a very general double EPW cube is the moduli space of stable objects with respect to a suitable stability condition on the Kuznetsov component of its corresponding Gushel--Mukai fourfolds; this answers a problem posed by Perry, Pertusi and Zhao.

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arXiv2022

Contractions of hyper-Kähler fourfolds and the Brauer group

We study the geometry of exceptional loci of birational contractions of hyper-Kähler fourfolds that are of K3$^{[2]}$-type. These loci are conic bundles over K3 surfaces and we determine their classes in the Brauer group. For this we use the results on twisted sheaves on K3 surfaces, on contractions and on the corresponding Heegner divisors. For a general K3 surface of fixed degree there are three (T-equivalence) classes of order two Brauer group elements. The elements in exactly two of these classes are represented by conic bundles on such fourfolds. We also discuss various examples of such conic bundles.

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arXiv2022

Symmetric locally free resolutions and rationality problems

We show that the birationality class of a quadric surface bundle over $\mathbb{P}^2$ is determined by its associated cokernel sheaves. As an application, we discuss stable-rationality of very general quadric bundles over $\mathbb{P}^2$ with discriminant curves of fixed degree. In particular, we construct explicit models of these bundles for some discriminant data. Among others, we obtain various birational models of a nodal Gushel-Mukai fourfold, as well as of a cubic fourfold containing a plane. Finally, we prove stable irrationality of several types of quadric surface bundles.

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arXiv2021

Projective models of Nikulin orbifolds

We study projective fourfolds of $K3^{[2]}$-type with a symplectic involution and the deformations of their quotients, called orbifolds of Nikulin types; they are IHS orbifolds. We compute the Riemann--Roch formula for Weil divisors on such orbifolds and describe the first complete family of orbifolds of Nikulin type with a polarization of degree $2$ as double covers of special complete intersections $(3,4)$ in $\mathbb{P}^6$.

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arXiv2020

Special lines on contact manifolds

In a series of two articles Kebekus studied deformation theory of minimal rational curves on contact Fano manifolds. Such curves are called contact lines. Kebekus proved that a contact line through a general point is necessarily smooth and has a fixed standard splitting type of the restricted tangent bundle. In this paper we study singular contact lines and those with special splitting type. We provide restrictions on the families of such lines, and on contact Fano manifolds which have reducible varieties of minimal rational tangents. We also show that the results about singular lines naturally generalise to complex contact manifolds, which are not necessarily Fano, for instance, quasi-projective contact manifolds or compact contact manifolds of Fujiki class C. In particular, in many cases the dimension of a family of singular lines is at most 2 less than the dimension of the contact manifold.

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arXiv2019

On Morin configurations of higher length

This paper studies finite Morin configurations $F$ of planes in $\mathbb P^5$ having higher length. The uniqueness of the configuration of maximal cardinality $20$ is proven. This is related to the stable canonical genus $6$ curve $C_{\ell}$ union of the $10$ lines of a smooth quintic Del Pezzo surface $Y$ in $\mathbb P^5$ and to the Petersen graph. Families of length $\geq 16$, previously unknown, are constructed by smoothing partially $C_{\ell}$. A more general irreducible family of special configurations of length $\geq 11$, we name as Morin-Del Pezzo configurations, is considered and studied. This depends on $9$ moduli and is defined via the family of nodal and rational canonical curves of $Y$. The special relations between Morin-Del Pezzo configurations and the geometry of special threefolds, like the Igusa quartic or its dual Segre primal, are focused.

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arXiv2018

Smoothable zero dimensional schemes and special projections of algebraic varieties

We study the degrees of generators of the ideal of a projected Veronese variety $v_2(\mathbb{P}^3)\subset \mathbb{P}^9$ to $\mathbb{P}^6$ depending on the center of projection. This is related to the geometry of zero dimensional schemes of length $8$ in $\mathbb{A}^4$, Cremona transforms of $\mathbb{P}^6$, and the geometry of Tonoli Calabi-Yau threefolds of degree $17$ in $\mathbb{P}^6$.

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arXiv2018

Equivalence of K3 surfaces from Verra threefolds

We study (2,2) divisors in $P^2 \times P^2$ giving rise to pairs of non-isomorphic, derived equivalent and L-equivalent K3 surfaces of degree 2. In particular, we confirm the existence of such fourfolds as predicted by Kuznetsov and Shinder in \cite{KS}.

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arXiv2017

Verra fourfolds, twisted sheaves and the last involution

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 $U(2)\oplus D_4(-1)$ or $U(2)\oplus E_8(-2)$. This complements the results obtained in [Mongardi and Wandel 2015], [Bossiere et al 2016], and the results from [arXiv:1603.00403] about the geometry of IHS fourfolds constructed using the Hilbert scheme of $(1,1)$ conics on Verra fourfolds. As a byproduct we find that IHS fourfolds of $K3^{[2]}$-type with Picard lattice $U(2)\oplus E_8(-2)$ naturally contain non-nodal Enriques surfaces.

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arXiv2017

Hyperkähler fourfolds and Kummer surfaces

We show that a Hilbert scheme of conics on a Fano fourfold double cover of $\mathbb{P}^2\times\mathbb{P}^2$ ramified along a divisor of bidegree $(2,2)$ admits a $\mathbb{P}^1$-fibration with base being a hyper-Kähler fourfold. We investigate the geometry of such fourfolds relating them with degenerated EPW cubes, with elements in the Brauer groups of $K3$ surfaces of degree $2$, and with Verra threefolds studied in [Ver04]. These hyper-Kähler fourfolds admit natural involutions and complete the classification of geometric realizations of anti-symplectic involutions on hyper-Kähler $4$-folds of type $K3^{[2]}$. As a consequence we present also three constructions of quartic Kummer surfaces in $\mathbb{P}^3$: as Lagrangian and symmetric degeneracy loci and as the base of a fibration of conics in certain threefold quadric bundles over $\mathbb{P}^1$.

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arXiv2016

Arithmetically Gorenstein Calabi-Yau threefolds in $\mathbb{P}^7$

We present a list of arithmetically Gorenstein Calabi-Yau threefolds in $\mathbb{P}^7$ and give evidence that this is a complete list. In particular we construct three new families of arithmetically Gorenstein Calabi-Yau threefolds in $\mathbb{P}^7$ for which no mirror construction is known.

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arXiv2016

EPW Cubes

We construct a new 20-dimensional family of projective 6-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 3 subvarieties of the Grassmanian G(3,6). These codimension 3 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.

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