arXiv subjects

Michal Kapustka

Publications and source records attributed to Michal Kapustka.

At least 19 recordsLinked to original sources

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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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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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

Mirror symmetry for Pfaffian Calabi-Yau 3-folds via conifold transitions

In this note we construct conifold transitions between several Calabi-Yau threefolds given by Pfaffians in weighted projective spaces and Calabi-Yau threefolds appearing as complete intersections in toric varieties. We use the obtained results to predict mirrors following ideas of \cite{BCKS, Batsmalltoricdegen}. In particular we consider the family of Calabi--Yau threefolds of degree 25 in $\mathbb{P}^9$ obtained as a transverse intersection of two Grassmannians in their Plucker embeddings.

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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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arXiv2015

Calabi--Yau threefolds in $\mathbb{P}^6$

We study the geometry of $3$-codimensional smooth subvarieties of the complex projective space. In particular, we classify all quasi-Buchsbaum Calabi--Yau threefolds in projective $6$-space. Moreover, we prove that this classification includes all Calabi--Yau threefolds contained in a possibly singular 5-dimensional quadric as well as all Calabi--Yau threefolds of degree at most $14$ in $\mathbb{P}^6$.

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arXiv2015

Bilinkage in codimension $3$ and canonical surfaces of degree $18$ in $\mathbb{P}^5$

We study the behavior of the bilinkage process in codimension $3$. In particular, we construct a smooth canonically embedded and linearly normal surface of general type of degree $18$ in $\mathbb{P}^5$, this is probably the highest degree such surface may have. Next, we apply our construction to find a geometric description of Tonoli Calabi--Yau threefolds in $\mathbb{P}^6$.

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arXiv2015

Remarks on Mukai threefolds admitting $C^{*}$ action

We investigate geometric invariants of the one parameter family of Mukai threefolds that admit $\mathbb C^{*}$ action. In particular we find the invariant divisors in the anticanonical system, and thus establish a bound on the log canonical thresholds. Furthermore we find an explicit description of such threefolds in terms of the quartic associated to the variety-of-sum-of-powers construction. This yields that any such threefold admits an additional symmetry which anticommutes with the $\mathbb C^{*}$ action, a fact that was previously observed near the Mukai-Umemura threefold by Rollin, Simanca and Tipler. As a consequence the Kähler-Einstein manifolds in the class form an open subset in the standard topology.

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arXiv2015

Tonoli's Calabi--Yau threefolds revisited

We find a simple geometric construction of Tonoli's examples of Calabi--Yau threefolds of degree 17 in complex $\mathbb{P}^6$. We prove that the rank of the Picard group of elements of one of these families is at least $2$.

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arXiv2013

Projections of Mukai Varieties

This note is an answer to a problem proposed by Ranestad and Iliev. We prove that the projection of general nodal linear sections of suitable dimension of the Mukai varieties $M_g$ are linear sections of $M_{g-1}$.

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arXiv2011

Some degenerations of $G_2$ and Calabi-Yau varieties

We introduce a variety $\hat{G}_2$ parameterizing isotropic five-spaces of a general degenerate four-form in a seven dimensional vector space. It is in a natural way a degeneration of the variety $G_2$, the adjoint variety of the simple Lie group $\mathbb{G}_2$. It occurs that it is also the image of $\mathbb{P}^5$ by a system of quadrics containing a twisted cubic. Degenerations of this twisted cubic to three lines give rise to degenerations of $G_2$ which are toric Gorenstein Fano fivefolds. We use these two degenerations to construct geometric transitions between Calabi--Yau threefolds. We prove moreover that every polarized K3 surface of Picard number 2, genus 10, and admitting a $g^1_5$ appears as linear sections of the variety $\hat{G}_2$.

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arXiv2011

Geometric transitions between Calabi-Yau threefolds related to Kustin-Miller unprojections

We study Kustin-Miller unprojections between Calabi-Yau threefolds or more precisely the geometric transitions they induce. We use them to connect many families of Calabi-Yau threefolds with Picard number one to the web of Calabi Yau complete intersections. This enables us to find explicit description of a few known families of Calabi-Yau threefolds in terms of equations. Moreover we find two new examples of Calabi-Yau threefolds with Picard group of rank one, described by Pfaffian equations in weighted projective spaces.

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arXiv2010

Vector bundles on Fano varieties of genus ten

In this note we describe a unique linear embedding of a prime Fano 4-fold F of genus 10 into the Grassmannian G(3,6). We use this to construct some moduli spaces of bundles on linear sections of F. In particular the moduli space of bundles with Mukai vector (3,L,3) on a generic polarized K3 surface (S,L) of genus 10 is constructed as a double cover of the projective plane branched over a smooth sextic.

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arXiv2010

A primer on Seshadri constants

Seshadri constants express the so called local positivity of a line bundle on a projective variety. They were introduced by Demailly. The original idea of using them towards a proof of the Fujita conjecture failed but they quickly became a subject of intensive study quite in their own right. Lazarsfeld's book "Positivity in Algebraic Geometry" contains a whole chapter devoted to local positivity and serves as a very enjoyable introduction to Seshadri constants. Since this book has appeared, the subject witnessed quite a bit of development. It is the aim of these notes to give an account of recent progress as well as to discuss many open questions and provide some examples.

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arXiv2009

A cascade of determinantal Calabi--Yau threefolds

We study Kustin--Miller unprojections of Calabi--Yau threefolds. As an application we work out the geometric properties of Calabi--Yau threefolds defined as linear sections of determinantal varieties. We compute their Hodge numbers and describe the morphisms corresponding to the faces of their Kähler--Mori cone.

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arXiv2008

Fiber products of elliptic surfaces with section and associated Kummer fibrations

We investigate Calabi--Yau three folds which are small resolutions of fiber products of elliptic surfaces with section admitting reduced fibers. We start by the classification of all fibers that can appear on such varieties. Then, we find formulas to compute the Hodge numbers of obtained three folds in terms of the types of singular fibers of the elliptic surfaces. Next we study Kummer fibrations associated to these fiber products.

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arXiv2008

Correspondences between modular Calabi--Yau fiber products

We describe two ways to construct finite rational morphisms between fiber products of rational elliptic surfaces with section and some Calabi--Yau manifolds. We use them to construct correspondences between such fiber products that admit at most five singular fibers and rigid Calabi--Yau threefolds.

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