Searcharxiv⌕ Search

arXiv subjects

Joachim Jelisiejew

Publications and source records attributed to Joachim Jelisiejew.

At least 37 records · Page 2Linked to original sources
arXiv2021

The Hilbert scheme of infinite affine space and algebraic K-theory

We study the Hilbert scheme $\mathrm{Hilb}_d(\mathbb{A}^\infty)$ from an $\mathbb{A}^1$-homotopical viewpoint and obtain applications to algebraic K-theory. We show that the Hilbert scheme $\mathrm{Hilb}_d(\mathbb{A}^\infty)$ is $\mathbb{A}^1$-equivalent to the Grassmannian of $(d-1)$-planes in $\mathbb{A}^\infty$. We then describe the $\mathbb{A}^1$-homotopy type of $\mathrm{Hilb}_d(\mathbb{A}^n)$ in a range, for $n$ large compared to $d$. For example, we compute the integral cohomology of $\mathrm{Hilb}_d(\mathbb{A}^n)(\mathbb{C})$ in a range. We also deduce that the forgetful map $\mathrm{FFlat}\to\mathrm{Vect}$ from the moduli stack of finite locally free schemes to that of finite locally free sheaves is an $\mathbb{A}^1$-equivalence after group completion. This implies that the moduli stack $\mathrm{FFlat}$, viewed as a presheaf with framed transfers, is a model for the effective motivic spectrum $\mathrm{kgl}$ representing algebraic K-theory. Combining our techniques with the recent work of Bachmann, we obtain Hilbert scheme models for the $\mathrm{kgl}$-homology of smooth proper schemes over a perfect field.

math.AG↗
arXiv2019

Pathologies on the Hilbert scheme of points

We prove that the Hilbert scheme of points on a higher dimensional affine space is non-reduced and has components lying entirely in characteristic p for all primes p. In fact, we show that Vakil's Murphy's Law holds up to retraction for this scheme. Our main tool is a generalized version of the Bialynicki-Birula decomposition.

math.AG↗
arXiv2019

Elementary components of Hilbert schemes

We generalize the Bialynicki-Birula decomposition to singular schemes and apply it to the Hilbert scheme of points on an affine space. We find an infinite family of small, elementary and generically smooth components of the Hilbert scheme of points of the affine four-space. Our method gives easily verifiable sufficient conditions for proving that a point of the Hilbert scheme is smooth and lies on an elementary component. We also present a necessary condition for smoothability of a finite subscheme given by a homogeneous ideal.

math.AG↗
arXiv2018

Białynicki-Birula decomposition for reductive groups

We generalize the Białynicki-Birula decomposition from actions of $G_m$ on smooth varieties to actions of linearly reductive group ${\bf G}$ on finite type schemes and algebraic spaces. We also provide a relative version and briefly discuss the case of algebraic stacks. We define the Białynicki-Birula decomposition functorially: for a fixed ${\bf G}$-scheme $X$ and a monoid $\overline{\bf G}$ which partially compactifies ${\bf G}$, the BB decomposition parameterizes ${\bf G}$-schemes over $X$ for which the ${\bf G}$-action extends to the $\overline{\bf G}$-action. The freedom of choice of $\overline{\bf G}$ makes the theory richer than the $G_m$-case.

math.AG↗
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 $\mathbb{P}^6$.

math.AG↗
arXiv2018

VSPs of cubic fourfolds and the Gorenstein locus of the Hilbert scheme of 14 points on A^6

This paper is concerned with the geometry of the Gorenstein locus of the Hilbert scheme of $14$ points on $\mathbb{A}^6$. This scheme has two components: the smoothable one and an exceptional one. We prove that the latter is smooth and identify the intersection of components as a vector bundle over the Iliev-Ranestad divisor in the space of cubic fourfolds. The ninth secant variety of the triple Veronese reembedding lies inside the Iliev-Ranestad divisor. We point out that this secant variety is set-theoretically a codimension two complete intersection and discuss the degrees of its equations.

math.AG↗
arXiv2017

Equations and tropicalization of Enriques surfaces

In this article we explicitly compute equations of an Enriques surface via the involution on a K3 surface. We also discuss its tropicalization and compute the tropical homology, thus recovering a special case of the result of \cite{IKMZ}, and establish a connection between the dimension of the tropical homology groups and the Hodge numbers of the corresponding algebraic Enriques surface.

math.AG↗
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.

math.AG↗
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.

math.DG↗
arXiv2016

Classifying local Artinian Gorenstein algebras

The classification of local Artinian Gorenstein algebras is equivalent to the study of orbits of a certain non-reductive group action on a polynomial ring. We give an explicit formula for the orbits and their tangent spaces. We apply our technique to analyse when an algebra is isomorphic to its associated graded algebra. We classify algebras with Hilbert function (1, 3, 3, 3, 1), obtaining finitely many isomorphism types, and those with Hilbert function (1, 2, 2, 2, 1, 1, 1). We consider fields of arbitrary, large enough, characteristic.

math.AC↗
arXiv2016

On polynomials with given Hilbert function and applications

Using Macaulay's correspondence we study the family of Artinian Gorenstein local algebras with fixed symmetric Hilbert function decomposition. As an application we give a new lower bound for cactus varieties of the third Veronese embedding. We discuss the case of cubic surfaces, where interesting phenomena occur.

math.AC↗
arXiv2015

Irreducibility of the Gorenstein loci of Hilbert schemes via ray families

We analyse the Gorenstein locus of the Hilbert scheme of $d$ points on $\mathbb{P}^n$ i.e. the open subscheme parameterising zero-dimensional Gorenstein subschemes of $\mathbb{P}^n$ of degree $d$. We give new sufficient criteria for smoothability and smoothness of points of the Gorenstein locus. In particular we prove that this locus is irreducible when $d\leq 13$ and find its components when $d = 14$. The proof is relatively self-contained and it does not rely on a computer algebra system. As a by--product, we give equations of the fourth secant variety to the $d$-th Veronese reembedding of $\mathbb{P}^n$ for $d\geq 4$.

math.AG↗
arXiv2014

Deformations of zero-dimensional schemes and applications

In this thesis we consider the geometry of the Hilbert scheme of points in P^n, concentrating on the locus of points corresponding to the Gorenstein subschemes of P^n. New results are given, most importantly we provide tools for constructing flat families and analysis of finite Gorenstein algebras and expose their efficiency by proving smoothability of certain families of algebras. Much of the existing theory and folklore is reviewed, providing a micro-encyclopaedic reference.

math.AC↗
arXiv2014

On the rationality of Poincaré series of Gorenstein algebras via Macaulay's correspondence

Let $A$ be a local Artinian Gorenstein ring with algebraically closed residue field $A/{\frak M}=k$ of characteristic 0, and let $P_A(z) := \sum_{p=0}^{\infty} ({\mathrm{ Tor}}_p^A(k,k))z^p $ be its Poincaré series. We prove that $P_A(z)$ is rational if either $\dim_k({{\frak M}^2/{\frak M}^3}) \leq 4 $ and $ \dim_k(A) \leq 16,$ or there exist $m\leq 4$ and $c$ such that the Hilbert function $H_A(n)$ of $A$ is equal to $ m$ for $n\in [2,c]$ and equal to 1 for $n > c$. The results are obtained thanks to a decomposition of the apolar ideal $\mathrm {Ann}(F)$ when $F=G+H$ and $G$ and $H$ belong to polynomial rings in different variables.

math.AC↗
arXiv2013

An upper bound for the Waring rank of a form

In this paper we introduce the open Waring rank of a form of degree d in n variables and prove the that this rank in bounded from above by \binom{n+d-2}{d-1} - \binom{n+d-6}{d-3} whenever n, d >= 3. This proves the same upper bound for the classical Waring rank of a form, improving the result of [BBS] and giving, as far as we know, the best upper bound known.

math.AC↗
arXiv2013

On commutativity of ideal extensions

In this paper we examine the commutativity of ideal extensions. We introduce methods of constructing such extensions, in particular we construct a noncommutative ring T which contains a central and idempotent ideal I such that T/I is a field. This answers a question from [2]. Moreover we classify fields of characteristic 0 which can be obtained as T/I for some T.

math.RA↗