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Jarosław Buczyński

Publications and source records attributed to Jarosław Buczyński.

At least 19 recordsLinked to original sources
arXiv2026

Cactus barriers

Determinantal methods for bounding the rank and border rank of tensors or polynomials are subject to a major barrier. For instance, it is known that using determinantal methods one cannot prove a lower bound for the border rank of a 3-way tensor of size m in each direction that exceeds 6m-4. We explain the precise geometric reason for this number (and analogous bounds in more general tensor spaces) using cactus varieties and, more generally, scheme theoretic methods in algebraic geometry.

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arXiv2026

Apolarity for border cactus decompositions

The border apolarity technique was introduced in our earlier work for secant varieties over complex numbers. We extend the theory to cactus varieties of toric varieties over any algebraically closed field. A border cactus decomposition is a mulithomogeneous ideal in the Cox ring (also called the total coordinate ring) of the toric variety that witnesses that a given point is in a specific cactus variety. The definition of such witness uses apolarity and we describe the set of ideals that are credible witnesses for this purpose in terms of a correspondence between the usual Hilbert scheme (parametrising all closed subschemes of the toric variety) and the multigraded Hilbert scheme (parametrising all multihomogeneous ideals in the Cox ring). We also take this opportunity to extend the border apolarity to linear subspaces (in non-border setting, this is equivalent to simultaneous decompositions).

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arXiv2025

Grassmann cactus variety and socle dimension

Grassmann cactus variety is a common generalisation of Grassmann secant variety and cactus variety. In their definitions one considers the vector spaces of fixed dimension that are contained in the linear span of some finite schemes. We prove that to characterise Grassmann cactus varieties it is enough to use finite schemes that locally have low socle dimension. This motivates the study of parameter spaces of such schemes and simplifies calculations of examples of Grassmann cactus varieties.

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arXiv2024

Cactus varieties of sufficiently ample embeddings of projective schemes have determinantal equations

For a fixed projective scheme X, a property P of line bundles is satisfied by sufficiently ample line bundles if there exists a line bundle L_0 on X such that P(L) holds for any L with (L - L_0) ample. As an example, sufficiently ample line bundles are very ample, moreover, for a normal variety X, the embedding corresponding to sufficiently ample line bundle is projectively normal. The grandfather of such properties and a basic ingredient used to study this concept is Fujita vanishing theorem, which is a strengthening of Serre vanishing to sufficiently ample line bundles. The r-th cactus variety of X is an analogue of secant variety and it is defined using linear spans of finite schemes of degree r. In this article we show that cactus varieties of sufficiently ample embeddings of X are set-theoretically defined by minors of matrices with linear entries. The topic is closely related to conjectures of Eisenbud-Koh-Stillman, which was proved by Ginensky in the case X a smooth curve. On the other hand Sidman-Smith proved that the ideal of sufficiently ample embedding of any projective scheme X is generated by 2 x 2 minors of a matrix with linear entries.

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arXiv2024

Cactus scheme, catalecticant minors, and scheme theoretic equations

The $r$-th cactus variety of a subvariety $X$ in a projective space generalizes the $r$-th secant variety of $X$ and it is defined using linear spans of finite subschemes of $X$ of degree $r$. One of its original purposes was to study the vanishing sets of catalecticant minors. In this article, we equip the cactus variety with a scheme structure, via ``relative linear spans'' of families of finite schemes over a potentially non-reduced base. In this way, we are able to study the vanishing scheme of the catalecticant minors. For a sufficiently high degree Veronese variety, we show that $r$-th cactus scheme and the zero scheme of appropriate catalecticant minors agree on a dense open subset which is the complement of the $(r-1)$-th cactus variety (or scheme). This article is the first part of a series. In the follow-up, as an application, we can describe the singular locus of (in particular) secant varieties to high degree Veronese varieties in terms of singularities of the Hilbert scheme. We will also generalize the result to high degree Veronese reembeddings of other varieties and schemes.

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arXiv2021

Duality and integrability on contact Fano manifolds

We address the problem of classification of contact Fano manifolds. It is conjectured that every such manifold is necessarily homogeneous. We prove that the Killing form, the Lie algebra grading and parts of the Lie bracket can be read from geometry of an arbitrary contact manifold. Minimal rational curves on contact manifolds (or contact lines) and their chains are the essential ingredients for our constructions. Along the way we collect several facts about bundles of projective lines admitting a contractible section, which might be of independent interest.

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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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arXiv2020

Apolarity, border rank and multigraded Hilbert scheme

We introduce an elementary method to study the border rank of polynomials and tensors, analogous to the apolarity lemma. This can be used to describe the border rank of all cases uniformly, including those very special ones that resisted a systematic approach. We also define a border rank version of the variety of sums of powers and analyse its usefulness in studying tensors and polynomials with large symmetries. In particular, it can be applied to provide lower bounds for the border rank of some very interesting tensors, such as the matrix multiplication tensor. We work in a general setting, where the base variety is not necessarily a Segre or Veronese variety, but an arbitrary smooth toric projective variety. A critical ingredient of our work is an irreducible component of a multigraded Hilbert scheme related to the toric variety in question.

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arXiv2020

Algebraic torus actions on contact manifolds

We prove the LeBrun-Salamon Conjecture in low dimensions. More precisely, we show that a contact Fano manifold X of dimension 2n+1 that has reductive automorphism group of rank at least n-2 is necessarily homogeneous. This implies that any positive quaternion-Kahler manifold of real dimension at most 16 is necessarily a symmetric space, one of the Wolf spaces. A similar result about contact Fano manifolds of dimension at most 9 with reductive automorphism group also holds. The main difficulty in approaching the conjecture is how to recognize a homogeneous space in an abstract variety. We contribute to such problem in general, by studying the action of algebraic torus on varieties and exploiting Bialynicki-Birula decomposition and equivariant Riemann-Roch theorems. From the point of view of T-varieties (that is, varieties with a torus action), our result is about high complexity T-manifolds. The complexity here is at most 1/2(dim X+5) with dim X arbitrarily high, but we require this special (contact) structure of X. Previous methods for studying T-varieties in general usually only apply for complexity at most 2 or 3.

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arXiv2020

Singular Curves of Low Degree and Multifiltrations from Osculating Spaces

In order to study projections of smooth curves, we introduce multifiltrations obtained by combining flags of osculating spaces. We classify all configurations of singularities occurring for a projection of a smooth curve embedded by a complete linear system away from a projective linear space of dimension at most two. In particular, we determine all configurations of singularities of non-degenerate degree d rational curves in $\mathbb{P}^n$ when $d - n \leq 3$ and $d < 2n$. Along the way, we describe the Schubert cycles giving rise to these projections. We also reprove a special case of the Castelnuovo bound using these multifiltrations: under the assumption $d < 2n$, the arithmetic genus of any nondegenerate degree $d$ curve in $\mathbb{P}^n$ is at most $d - n$.

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arXiv2019

On Strassen's rank additivity for small three-way tensors

We address the problem of the additivity of the tensor rank. That is for two independent tensors we study if the rank of their direct sum is equal to the sum of their individual ranks. A positive answer to this problem was previously known as Strassen's conjecture until recent counterexamples were proposed by Shitov. The latter are not very explicit, and they are only known to exist asymptotically for very large tensor spaces. In this article we prove that for some small three-way tensors the additivity holds. For instance, if the rank of one of the tensors is at most 6, then the additivity holds. Or, if one of the tensors lives in $C^k \otimes C^3 \otimes C^3$ for any $k$, then the additivity also holds. More generally, if one of the tensors is concise and its rank is at most 2 more than the dimension of one of the linear spaces, then additivity holds. In addition we also treat some cases of the additivity of border rank of such tensors. In particular, we show that the additivity of the border rank holds if the direct sum tensor is contained in $C^4 \otimes C^4 \otimes C^4$. Some of our results are valid over an arbitrary base field.

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arXiv2019

Complex contact manifolds, varieties of minimal rational tangents, and exterior differential systems

Complex contact manifolds arise naturally in differential geometry, algebraic geometry and exterior differential systems. Their classification would answer an important question about holonomy groups. The geometry of such manifold $X$ is governed by the contact lines contained in $X$. These are related to the notion of a variety of minimal rational tangents. In this review we discuss the partial classification theorems of projective complex contact manifolds. Among such manifolds one finds contact Fano manifolds (which include adjoint varieties) and projectivised cotangent bundles. In the first case we also discuss a distinguished contact cone structure, arising as the variety of minimal rational tangents. We discuss the repercussion of the aforementioned classification theorems for the geometry of quaternion-Kähler manifolds with positive scalar curvature and for the geometry of second-order PDEs imposed on hypersurfaces.

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arXiv2018

Immaculate line bundles on toric varieties

We call a sheaf on an algebraic variety immaculate if it lacks any cohomology including the zero-th one, that is, if the derived version of the global section functor vanishes. Such sheaves are the basic tools when building exceptional sequences, investigating the diagonal property, or the toric Frobenius morphism. In the present paper we focus on line bundles on toric varieties. First, we present a possibility of understanding their cohomology in terms of their (generalized) momentum polytopes. Then we present a method to exhibit the entire locus of immaculate divisors within the class group. This will be applied to the cases of smooth toric varieties of Picard rank two and three and to those being given by splitting fans. The locus of immaculate line bundles contains several linear strata of varying dimensions. We introduce a notion of relative immaculacy with respect to certain contraction morphisms. This notion will be stronger than plain immaculacy and provides an explanation of some of these linear strata.

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arXiv2017

Maps of Mori Dream Spaces in Cox coordinates. Part I: existence of descriptions

Any rational map between affine spaces, projective spaces or toric varieties can be described in terms of their affine, homogeneous, or Cox coordinates. We show an analogous statement in the setting of Mori Dream Spaces. More precisely (in the case of regular maps) we show there exists a finite extension of the Cox ring of the source, such that the regular map lifts to a morphism from the Cox ring of the target to the finite extension. Moreover the extension only involves roots of homogeneous elements. Such a description of the map can be applied in practical computations.

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arXiv2017

Finite schemes and secant varieties over arbitrary characteristic

We present scheme theoretic methods that apply to the study of secant varieties. This mainly concerns finite schemes and their smoothability. The theory generalises to the base fields of any characteristic, and even to non-algebraically closed fields. In particular, the smoothability of finite schemes does not depend on the embedding into a smooth variety or on base field extensions. Independent of the base field, secant varieties to high degree Veronese reembeddings behave well with respect to the intersection and they are defined by minors of catalecticants whenever a suitable smoothability condition for Gorenstein subschemes holds. The content of the article is largely expository, although many results are presented in a stronger form than in the literature.

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arXiv2017

On the locus of points of high rank

Given a closed subvariety X in a projective space, the rank with respect to X of a point p in this projective space is the least integer r such that p lies in the linear span of some r points of X. Let W_k be the closure of the set of points of rank with respect to X equal to k. For small values of k such loci are called secant varieties. This article studies the loci W_k for values of k larger than the generic rank. We show they are nested, we bound their dimensions, and we estimate the maximal possible rank with respect to X in special cases, including when X is a homogeneous space or a curve. The theory is illustrated by numerous examples, including Veronese varieties, the Segre product of dimensions (1,3,3), and curves. An intermediate result provides a lower bound on the dimension of any GL_n orbit of a homogeneous form.

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arXiv2016

Constructions of k-regular maps using finite local schemes

A continuous map from R^m to R^N or from C^m to C^N is called k-regular if the images of any $k$ points are linearly independent. Given integers m and k a problem going back to Chebyshev and Borsuk is to determine the minimal value of N for which such maps exist. The methods of algebraic topology provide lower bounds for N, however there are very few results on the existence of such maps for particular values m and k. Using the methods of algebraic geometry we construct k-regular maps. We relate the upper bounds on N with the dimension of the locus of certain Gorenstein schemes in the punctual Hilbert scheme. The computations of the dimension of this family is explicit for k<10, and we provide explicit examples for k<6. We also provide upper bounds for arbitrary m and k.

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