arXiv subjects

Joachim Jelisiejew

Publications and source records attributed to Joachim Jelisiejew.

At least 19 recordsLinked to original sources

arXiv2026

Unrestrictions and concise secant varieties

We introduce the concise secant varieties, which are, informally speaking, modular partial desingularisations of secant varieties to Segre embeddings. More precisely, they are projective and birational to the abstract secant varieties, yet each of their points corresponds to a concise tensor of appropriate border rank (that is, to a minimal border rank tensor). We discuss implications throughout the theory of tensors, including a characterisation of border rank $\leq r$ tensors as unrestrictions of minimal border rank $r$ tensors, a characterisation of tensors with cactus rank $\leq r$, a concise versions of border apolarity including the fixed point theorem, concise Varieties of Sums of Powers, connections to defectivity and identifiability in the Segre case, to the Salmon conjecture etc.

math.AG↗

arXiv2026

The Iarrobino scheme: a self-dual analogue of the Hilbert scheme of points

For a fixed quasi-projective scheme $X$ we introduce a self-dual analogue of ${\mathrm{Hilb}}_d(X)$ which we call the Iarrobino scheme of $X$. It is a fine moduli space of zero-dimensional subschemes of $X$ together with a self-dual filtration. It contains an open subset that parameterises oriented Gorenstein subschemes. We provide also self-dual analogues of the Quot scheme of points and of the stacks of coherent sheaves and finite algebras. We prove that the resulting analogues of Hilbert and Quot schemes are smooth for $X$ a smooth curve. The Iarrobino scheme is a natural ambient space for enumerative invariants of algebras as introduced by Michałek. When $X$ is an affine space, it also admits a commuting matrices description and related virtual structure.

math.AG↗

arXiv2026

Open problems in deformations of Artinian algebras, Hilbert schemes and around

We review the open problems in the theory of deformations of zero-dimensional objects, such as algebras, modules or tensors. We list both the well-known ones and some new ones that emerge from applications. In view of many advances in recent years, we can hope that all of them are in the range of current methods.

math.AG↗

arXiv2026

Hankel and Multiplication Tensor Completions for Cactus Rank

We show that the Hankel flat extension formulation of the cactus algorithm is equivalent to a completion problem for multiplication tensors of Artinian Gorenstein algebras. The unknown Hankel moments are canonically identified with the undetermined tensor coefficients, and under this identification the symbolic multiplication matrices and their commutation equations coincide. This shows that the usual degree extension formulation is a coordinate realization of a variable extension problem with marked generators. We further use Borel-fixed and squat staircases to reduce the family of candidate basis shapes in the resulting algorithm.

math.AC↗

arXiv2026

On a conjecture by Michael Wemyss regarding the calculation of GV invariants

Contraction algebras are noncommutative algebras introduced by Donovan and Wemyss to classify of 3-dimensional flops. Wemyss conjectures that contraction algebras can be deformed to a single semisimple algebra. This gives an intrinsic method of calculating Gopakumar-Vafa invariants of the flop. The main result is a proof of Wemyss' conjecture for types A and D. In the course of the proof, we recall and introduce new techniques for constructing flat deformations of associative algebras and compare various notions of deformations. We also put forward two conjectures which hint towards a deeper theory.

math.RA↗

arXiv2026

The Hilbert scheme of points on a threefold: broken Gorenstein structures and linkage

We investigate the Hilbert scheme of points on a smooth threefold. We introduce a notion of broken Gorenstein structure for finite schemes, and show that its existence guarantees smoothness on the Hilbert scheme. Moreover, we conjecture that it is exhaustive: every smooth point admits a broken Gorenstein structure. We give an explicit characterization of the smooth points on the Hilbert scheme of A^3 corresponding to monomial ideals. We investigate the nature of the singular points, and prove several conjectures by Hu. Along the way, we obtain a number of additional results, related to linkage classes, nested Hilbert schemes, and a bundle on the Hilbert scheme of a surface.

math.AG↗

arXiv2025

Symmetric powers: structure, smoothability, and applications

We investigate border ranks of twisted powers of polynomials and smoothability of symmetric powers of algebras. We prove that the latter are smoothable. For the former, we obtain upper bounds for the border rank in general and prove that they are optimal under mild conditions. We give applications to complexity theory. Many of the results rest on the notion of an encompassing polynomial, which we introduce.

math.AG↗

arXiv2025

Classification and degenerations of small minimal border rank tensors via modules

We give a self-contained classification of $1_*$-generic minimal border rank tensors in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m$ for $m \leq 5$. Together with previous results, this gives a classification of all minimal border rank tensors in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m$ for $m \leq 5$: there are $107$ isomorphism classes (only $37$ up to permuting factors). We fully describe possible degenerations among the tensors. We prove that there are no $1$-degenerate minimal border rank tensors in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m $ for $m \leq 4$.

math.AG↗

arXiv2025

Limits of saturated ideals

We investigate the question whether a given homogeneous ideal is a limit of saturated ones. We provide cohomological necessary criteria for this to hold and apply them to a range of examples. Our motivation comes from the theory of border apolarity and varieties of sums of powers, where the question above is tightly connected to proving new lower bounds for border ranks of tensors.

math.AC↗

arXiv2024

On construction of k-regular maps to Grassmannians via algebras of socle dimension two

A continuous map $\mathbb{C}^n\to Gr(τ, N)$ is $k$-regular if the $τ$-dimensional subspaces corresponding to images of any $k$ distinct points span a $τk$-dimensional space. For $τ= 1$ this essentially recovers the classical notion of a $k$-regular map $\mathbb{C}^n\to \mathbb{C}^N$. We provide new examples of $k$-regular maps, both in the classical setting $τ= 1$ and for $τ\geq 2$, where these are the first examples known. Our methods come from algebraic geometry, following and generalizing Buczyński-Januszkiewicz-Jelisiejew-Michałek. The key and highly nontrivial part of the argument is proving that certain loci of the Hilbert scheme of points have expected dimension. As an important side result, we prove irreducibility of the punctual Hilbert scheme of $k$ points on a threefold, for $k\leq 11$.

math.AG↗

arXiv2024

Concise tensors of minimal border rank

We determine defining equations for the set of concise tensors of minimal border rank in $C^m\otimes C^m\otimes C^m$ when $m=5$ and the set of concise minimal border rank $1_*$-generic tensors when $m=5,6$. We solve this classical problem in algebraic complexity theory with the aid of two recent developments: the 111-equations defined by Buczyńska-Buczyński and results of Jelisiejew-Šivic on the variety of commuting matrices. We introduce a new algebraic invariant of a concise tensor, its 111-algebra, and exploit it to give a strengthening of Friedland's normal form for $1$-degenerate tensors satisfying Strassen's equations. We use the 111-algebra to characterize wild minimal border rank tensors and classify them in $C^5\otimes C^5\otimes C^5$.

math.AG↗

arXiv2024

Generically nonreduced components of Hilbert schemes on fourfolds

We exhibit generically nonreduced components of the Hilbert scheme of at least $21$ points on a smooth variety of dimension at least four. The result was announced in~[Jelisiejew__open_problems] and answers a question~[Problem~3.8, AIMPL]. The method is similar to the one of~[\S6, Jelisiejew-Šivic].

math.AG↗

arXiv2024

The variety of polar simplices II

We discuss the space $VPS(Q,H)$ of ideals with Hilbert function $H=(1,n,n, \ldots)$ that are apolar to a full rank quadric $Q$. We prove that its components of saturated ideals are closely related to the locus of Gorenstein algebras and to the Slip component in border apolarity. We also point out an important error in~[RS] and provide the necessary corrections.

math.AG↗

arXiv2023

On the Hilbert function of Artinian local complete intersections of codimension three

In singularity theory or algebraic geometry, it is natural to investigate possible Hilbert functions for special algebras $A$ such as local complete intersections or more generally Gorenstein algebras. The sequences that occur as {the} Hilbert functions of standard graded complete intersections are well understood classically thanks to Macaulay and Stanley. Very little is known in the local case except in codimension two. In this paper we characterise the Hilbert functions of quadratic Artinian complete intersections of codimension three. Interestingly we prove that a Hilbert function is admissible for such a Gorenstein ring if and only if is admissible for such a complete intersection. We provide an effective construction of a local complete intersection for a given Hilbert function. We prove that the symmetric decomposition of such a complete intersection ideal is determined by its Hilbert function.

math.AC↗

arXiv2022

Hermitian K-theory via oriented Gorenstein algebras

We show that the hermitian K-theory space of a commutative ring R can be identified, up to A^1-homotopy, with the group completion of the groupoid of oriented finite Gorenstein R-algebras, i.e., finite locally free R-algebras with trivialized dualizing sheaf. We deduce that hermitian K-theory is universal among generalized motivic cohomology theories with transfers along oriented finite Gorenstein morphisms. As an application, we obtain a Hilbert scheme model for hermitian K-theory as a motivic space. We also give an application to computational complexity: we prove that 1-generic minimal border rank tensors degenerate to the big Coppersmith-Winograd tensor.

math.AG↗

arXiv2022

Hilbert schemes of points and applications

The thesis concerns Hilbert schemes of points and apart from mathematical results, contains small open problems and history sections, see the introduction for more details. The thesis has not been edited since 2017, see first page for more information. It is shared "as is" (in the language of GPL) solely due/thanks to several people who decided to refer to it.

math.AG↗

arXiv2022

Bounds on complexity of matrix multiplication away from CW tensors

We present three families of minimal border rank tensors: they come from highest weight vectors, smoothable algebras, or monomial algebras. We analyse them using Strassen's laser method and obtain an upper bound $2.431$ on $ω$. We also explain how in certain monomial cases using the laser method directly is less profitable than first degenerating. Our results form possible paths in the search for valuable tensors for the laser method away from Coppersmith-Winograd tensors.

math.AG↗

arXiv2022

Components and singularities of Quot schemes and varieties of commuting matrices

We investigate the variety of commuting matrices. We classify its components for any number of matrices of size at most 7. We prove that starting from quadruples of size 8 matrices, this scheme has generically nonreduced components, while up to degree 7 it is generically reduced. Our approach is to recast the problem as deformations of modules and generalize an array of methods: apolarity, duality and Białynicki-Birula decompositions to this setup. We include a thorough review of our methods to make the paper self-contained and accessible to both algebraic and linear-algebraic communities. Our results give the corresponding statements for the Quot schemes of points, in particular we classify the components of $Quot_d(O_{\mathbb{A}^n}^{\oplus r})$ for $d\leq 7$ and all $r$, $n$.

math.AG↗