AGATES Workshop
Deformation Theory
December 5th9th, 2022
IMPAN, Warsaw, Poland
The workshop will discuss the recent advancements in deformation theory and its applications to tensors, in particular Hilbert schemes of points: components and singularities, border apolarity, multigraded Hilbert schemes, Hilbert schemes of higher dimensional subschemes in projective space, subloci inside the Hilbert schemes.
Venue
The workshop takes place at Institute of Mathematics of Polish Academy of Sciences (IMPAN). The address is Śniadeckich 8, 00656 Warsaw, Poland. The talks are in room 312, the coffee breaks in 409. In the evenings, you are in general free to use the room 409 and the coffee machine outside it.
Schedule Show/hide abstracts
Monday
 9:00  10:00 Joseph Landsberg From computer science to deformation theory
Lower complexity bounds in theoretical computer science are notoriously difficult to come by. For example, Chap. 14 of the famous text by Arora and Barak is titled "Circuit lower bounds, complexity theory's Waterloo". The best lower bounds so far have been obtained using methods from classical algebraic geometry, but these results have been limited. In recent years it was realized that the source of these limitations may be traced to the reducibility of the Hilbert scheme of points. Even more recently, it was realized that deformation theory provides a path to overcome this barrier. I will discuss recent work with J. Jelisiejew and A. Pal implementing representation theory in one special case to overcome the barriers. The case is that of tensors of minimal border rank, i.e., secant varieties of triple Segre products.
 10:00  10:45 coffee
 10:45  11:45 Balázs Szendrői Hilbert schemes of points of singular surfaces
I will explain what is known, and what is not known, about Hilbert schemes of points of singular surfaces, reporting in particular on joint work with Gyenge, Nemethi, Craw and Gammelgaard concerning rational double points.
 13:45  14:45 Ritvik Ramkumar An invitation to the fiberfull scheme
I will introduce the fiberfull scheme which can be seen as the parameter space that generalizes the Hilbert and Quot schemes by controlling the entire cohomological data. In particular, given a sequence of functions (h_0,...,h_n) the fiberfull scheme parameterizes subschemes X of P^n satisfying dim H^i(X,O_X(v)) = h_i(v). I will briefly sketch its construction, describe some of its local properties and discuss some applications. This is joint work with Yairon CidRuiz.
 14:55  16:00 Open problem session
 16:00  16:30 coffee
 16:30  17:30 Open problem session c.d., slides for Matteo Varbaro's problem
Tuesday
 9:00  10:00 Fabio Tanturri On the unirationality of moduli spaces of curves
Even though moduli spaces of curves of genus g are a central topic in modern algebraic geometry, their geometry and in particular their Kodaira dimension is not fully understood. They are known to be unirational for low genera and of general type for higher ones, but a few cases in between remain mysterious. The situation is to some extent similar when one considers curves equipped with additional structures, such as Hurwitz spaces or moduli spaces of curves with marked points.
A possible approach to find new unirationality results for these spaces is to provide an explicit construction of a family of projective models of curves in a suitable Hilbert scheme dominating the relevant moduli space. I will report on some advances in this direction, based on works in collaboration with F.O. Schreyer and H. Keneshlou.  10:00  10:45 coffee
 10:45  11:45 Michał Szachniewicz Nonreducedness of the Hilbert schemes of few points
We use generalised BiałynickiBirula decomposition, apolarity and obstruction theories to prove nonreducedness of the Hilbert scheme of 13 points on A^6 . Our argument doesn’t involve computer calculations and gives an example of a fractallike structure on this Hilbert scheme.
 13:45  14:15 Enrico Schlesinger Maximum genus, multiple lines and generic forms
I will report on the connection between the maximum genus problem for locally CohenMacaulay space curves, the construction of multiple lines with good cohomology, and generic weighted homogeneous forms.
 14:15  14:45 Alex Constantinescu Extensions of free pairs of semigroups
For a pair of semigroups, being free is equivalent to the flatness of the corresponding algebras. Motivated by the deformation theory of the associated toric singularity, we prove that the category of free extensions always contains an initial object, which we describe explicitly. When the semigroups arise from a rational polyhedron, the description is related to decompositions into Minkowski summands.
 14:55  16:00 Working groups
 16:00  16:30 coffee
 16:30  17:30 Working groups (informal)
 18:00  ... conference dinner (same building)
Wednesday
 9:00  10:00 Matteo Varbaro Gröbner deformations
We will discuss various aspects of Gröbner degenerations, from classical facts to the most recent developments.
 10:00  10:45 coffee
 10:45  11:45 Alessio Sammartano On nested Hilbert schemes of points in the plane
Nested Hilbert schemes of A^2, also known as flag Hilbert schemes, parametrize chains of finite subschemes of the affine plane. In this talk, I will give a brief overview of what is known and what is not known about the geometry of these spaces. Then, I will focus on the case of nested pairs, and show how to reduce the study of its singularities to certain affine schemes with explicit defining equations. In particular, using techniques from commutative algebra, I will show that some of these nested Hilbert schemes have rational singularities. This is a joint work with Ritvik Ramkumar.
 13:45  14:45 Jerzy Weyman Structure of finite free resolutions of length 3
I will give an overview of the theory of higher structure maps for finite free resolutions of length 3 which saw important advances in recent years. They lead to some interesting open questions, including some that are related to deformation theory.
 14:55  16:00 Computer algebra session The package and examples from Paolo Lella's presentation.
 16:00  16:30 coffee
 16:30  17:30 Computer algebra session / working groups
Thursday
 9:00  10:00 Klemen Šivic Quot schemes and varieties of commuting matrices
Let C_n(M_d) denote the affine variety of all ntuples of commuting d\times d matrices. The ADHM construction relates these varieties to Quot schemes, and in particular to Hilbert schemes. On the more applied side, varieties C_n(M_d) are directly connected to the question whether a tensor has minimal border rank. Although C_n(M_d) is usually reducible for n>2 and d>3, very few irreducible components are known. In the talk we classify irreducible components for small d and all n. Moreover, we show that C_n(M_d), viewed as a scheme defined by the quadratic commutativity relations, has generically nonreduced components whenever d\ge 8 and n\ge 4, while it is generically reduced for d\le 7. Our results give the corresponding results for Quot schemes of points. In particular, the Quot scheme parametrizing degree 8 quotients of a free module of rank 4 over polynomial ring in 4 variables has a generically nonreduced component. This is joint work with Joachim Jelisiejew (University of Warsaw).
 10:00  10:45 coffee
 10:45  11:45 Cristina Bertone Marked bases for some quotient rings and applications
Inspired by some recent papers [2,3], we generalize the notion of marked basis over a quasistable ideal in a polynomial ring R to some quotient rings, and apply this tool to study the Hilbert scheme parameterizing subschemes of the projective scheme defined by the quotient ring with a prescribed Hilbert polynomial.
More precisely, we can generalize the main properties of marked bases in R to marked bases on R/I under the hypothesis that I is quasistable. The computational tools we define allow us to prove the existence of an open cover made of marked schemes of a Hilbert scheme defined on Proj(R/I), for I a quasistable CohenMacaulay ideal.
Furthermore, if I is quasistable and R/I is a MacaulayLex ring, we investigate the smoothness of the Lex point of the Hilbert scheme of Proj(R/I), exhibiting both cases of smoothness and of singularity of the Lex point.
This talk is based on a joint work with F. Cioffi, M. Orth, W.M. Seiler [1].
[1] C. Bertone F. Cioffi, M. Orth, W.M. Seiler, "Hilbert schemes over quotient rings via relative marked bases", arXiv:2203.11770 [math.AC].
[2] G. Caviglia, A. Sammartano, "Syzygies in Hilbert schemes of complete intersections", arXiv:1903.08770 [math.AC].
[3] A. Hashemi, M. Orth, W.M. Seiler, "Relative Gröbner and Involutive Bases For Ideals In Quotient Rings", Math. Comput. Sci. 15 (2021), no. 3, 453482.  13:45  14:15 Hanieh Keneshlou On the construction of regular maps to Grassmannians
A continuous map f : C^n → C^N is called kregular, if the image of any k distinct points in C^N are linearly independent. The study of existence of regular map was initiated by Borsuk 1957, and later attracted attention due to its connection with the existence of interpolation spaces in approximation theory, and certain inverse vector bundles in algebraic topology. In this talk, based on a joint work with Joachim Jelisiejew, we consider the general problem of the existence of regular maps to Grassmannians C^n → Gr(τ, C^nN). We will discuss the tools and methods of algebra and algebraic geometry to provide an upper bound on N, for which a regular map exists.
 14:15  14:45 Simone Marchesi Moduli spaces of logarithmic tangent sheaves
Given an effective reduced Cartier divisor D on a smooth projective variety X, it is possible to define its logarithmic tangent sheaf. In this talk, we will consider particular families for which the associated sheaf is stable. We will then study the moduli map from the family parametrizing the divisor to the component of the moduli space of stable sheaves that contains the logarithmic one. This is a joint work with Daniele Faenzi.
 14:55  16:00 Working groups
 16:00  16:30 coffee
 16:30  17:30 Working groups (informal)
Friday
 9:00  10:00 Tomasz Mańdziuk Limits of saturated ideals
We consider the multigraded Hilbert scheme parametrizing all ideals in the coordinate ring of product of projective spaces that define a zerodimensional subscheme and have a fixed Hilbert function. We address the problem of identifying those points (called saturable ideals) which are limits of saturated ideals. This problem is motivated by border apolarity. We present some necessary and one sufficient criterion for an ideal to be saturable.
The talk is based on results from my PhD thesis prepared under the supervision of Jarosław Buczyński and Joachim Jelisiejew and a later joint work with Joachim Jelisiejew on extending some of the results.  10:00  10:45 coffee
 10:45  11:45 Diane Maclagan The spine of the Tgraph of the Hilbert scheme
The torus T of projective space also acts on the Hilbert scheme of subschemes of projective space, and the Tgraph of the Hilbert scheme has vertices the fixed points of this action, and edges the closures of onedimensional orbits. In general this graph depends on the underlying field. I will discuss joint work with Rob Silversmith, in which we construct of a subgraph, which we call the spine, of the Tgraph of Hilb^N(A^2) that is independent of the choice of field. A key technique is an understanding of the tropical ideal, in the sense of tropical scheme theory, of the ideal of the universal family of an edge in the spine.
 13:45  14:45 Andrew Staal New Examples of Elementary Components of Hilbert Schemes of Points
I will present some recent progress in the study of Hilbert schemes Hilb^d(A^n) of d points in affine space. Specifically, I will give new examples of elementary components of Hilbert schemes of points. One infinite family of these answers a question posed by Iarrobino in the 80's: does there exist an irreducible component of the (local) punctual Hilbert scheme Hilb^d(O_{A^n,p}) of dimension less than (n1)(d1)? A different family of elementary components arises from the Galois closure operation introduced by BhargavaSatriano. In both situations, secondary families of elementary components also arise, providing further new examples of elementary components of Hilbert schemes of points.
 14:55  16:00 Working groups summary
 16:00  16:30 coffee
Organizers
 Joachim Jelisiejew (University of Warsaw, Poland)
 Paolo Lella (Politecnico di Milano, Italy)
Confirmed speakers so far
 Cristina Bertone
 Joseph Landsberg
 Diane Maclagan
 Tomasz Mańdziuk
 Ritvik Ramkumar
 Alessio Sammartano
 Klemen Šivic
 Andrew Staal
 Michał Szachniewicz
 Balázs Szendrői
 Fabio Tanturri
 Matteo Varbaro
 Jerzy Weyman
Institutional organizer:
 Banach Center (Warsaw)
 University of Warsaw
Registration
To register, please fill in the AGATES registration form, indicating that you plan to attend the workshop in the section "Interest in events".
Financial support and institutions involved
The workshop is supported by
 Mathematics Department of Politecnico di Milano and PRIN2020 project Squarefree Gröbner degenerations, special varieties and related topics

National Science Center of Poland.
The workshop is a part of a semester long program Algebraic Geometry with Applications to TEnsors and Secants (AGATES), which is supported by the Simons Foundation and by Institute of Mathematics of Polish Academy of Sciences.