Algebra & Geometry Seminar


Davide Frapporti (Politecnico di Milano)

Date19/03/2024           Time: 14:15                     Room: 713

Title: On mixed surfaces: construction and examples

Abstract: A product-quotient surface is a surface arising as (the minimal resolution of the singularities of) a product of two curves modulo the action of a finite group.

In the talk I will review some general facts about product-quotient surfaces and then consider the special case of mixed surfaces and report on a  work in progress with M. Alessandro and C. Gleißner.

Past Talks

Michele Bolognesi (Université de Montpellier)

Date: 13/09/2023            Time: 14:00                     Room: 704

Title: Finite dimensionality of motives of cubic fourfolds

Abstract: A cubic fourfold is a smooth hypersurface of degree 3 in P5. The rationality of such objects is a daunting open problem, that has been approached with several different techniques. In the first part of this talk I will introduce the birational geometry of C.F. and discuss some general classical stuff concerning their moduli space and its intersection theory. In the second part of the talk I will introduce Chow motives, and in particular define the transcendental motive of a C.F., highlighting the relation with the same object for K3 surfaces. Finally I will apply some intersection theory of the moduli space of cubic fourfolds to display new classes of cubic fourfolds with finite dimensional transcendental motive, and to show that these classes of cubics are quite ubiquitous in the moduli space.

Dumitru Stamate (University of Bucharest)

Date:  19/09/2023           Time: 14:00                     Room: 704

Title: Canonical trace ideals and Gorenstein related properties

Abstract: In this talk we present two classes of rings which emerged in the recent years from attempts to single out meaningful classes of rings sitting between the Cohen-Macaulay and the Gorenstein ones. Both have in common the trace of the canonical module ω of the Cohen-Macaulay local ring R. Loosely speaking, for the nearly Gorenstein property the canonical trace ideal contains the maximal ideal of R, whereas for the far flung Gorenstein rings, tr(ω) is in some sense, as small as possible. This reports on joint works with J. Herzog and T. Hibi, and with J. Herzog and S. Kumashiro respectively.

Andrea Petracci (Università di Bologna)

Date:  3/10/2023           Time: 14:00                     Room: 713

Title: Deformations of toric affine varieties and applications to K-moduli of Fano varieties

Abstract: Algebraic geometers are obsessed with moduli spaces, i.e. with algebraic varieties which parametrises certain algebro-geometric objects of a fixed type. The most interesting examples of moduli spaces are those which parametrises smooth (or mildly singular) varieties of fixed dimension, fixed volume and fixed sign (i.e. positive or negative) of the canonical bundle. Fano varieties are the varieties with negative canonical bundle, i.e. the varieties with positive curvature.
In this talk I will explain how toric geometry (which is a subject that relates combinatorics of polytopes to certain algebraic varieties) can be used to exhibit singular points on the moduli spaces of Fano varieties. This talk is partially based on joint work with Anne-Sophie Kaloghiros.

Volkmar Welker (Philipps-Universität Marburg)

Date:  10/10/2023           Time: 14:00                     Room: 713

Title: Homological and combinatorial versions of the Garland method

Abstract: In this talk we explain the classical Garland method for proving vanishing of the k-th cohomology group of a simplicial complex with coefficients in a field of char 0. The method is based on estimating the spectral gaps of the graph Laplacians of links of (k-1)-faces. We then show that the method is a special case of a results on pairs of three term complexes and explain how to approach the vanishing of cohomology of arbitrary three term complexes (over fields of char 0) with the generalized Garland method. This is joint work with Eric Babson

Alessio Corti (Imperial College London)

Date:  17/10/2023           Time: 14:00                     Room: 713

Title: Deformations and smoothings of toric Fano 3-folds

Abstract: I will speak of joint work with Paul Hacking and Andrea Petracci on a criterion for smoothing a Gorenstein toric Fano 3-fold

Rodrigo Iglesias González (Universidad de La Rioja)

Date24/10/2023           Time: 14:00                     Room: 713

Title: The Rees Algebra of tri-generated monomial ideals

Abstract: In this talk I will show an algorithm for finding the minimal generating set of the Rees Ideal of any monomial ideal generated by 3 minimal generators in two variables. This type of ideal generalizes the well known class of ideals associated to a parametrization of a monomial plane curve. Furthermore, I will give a description of the whole minimal free resolution of the Rees ideal via using Gröbner basis and involutive-like basis. Finally, we will see how the minimal free resolution can be read off directly from a simplicial graph. 

Alessio D'Alì (Politecnico di Milano)

Date:  31/10/2023           Time: 14:00                     Room: 713

Title: On a generalization of symmetric edge polytopes to regular matroids

Abstract: Symmetric edge polytopes are a class of reflexive lattice polytopes depending on the combinatorial data of a graph. Such objects arise in many different contexts, including finite metric space theory, physics and optimal transport, and have been studied extensively in the last few years.
The aim of this talk is to show that symmetric edge polytopes are special instances of a more general construction that associates a reflexive lattice polytope with every regular matroid. A matroid is called regular if it can be represented over every field; by work of Tutte, a matroid is regular if and only if it can be represented by a totally unimodular matrix, i.e. a matrix whose square submatrices of any size all have determinant equal to -1, 0 or 1. The class of regular matroids is a natural candidate for our extension as it contains the class of graphic matroids, is closed under matroid duality and enjoys many other good structural properties.
This is joint work with Martina Juhnke-Kubitzke and Melissa Koch.

Lorenzo Venturello (Università di Pisa)

Date7/11/2023           Time: 14:00                     Room: 713

Title: Two graph polytopes

Abstract: There are several meaningful ways to associate to a graph an integral polytope.  We focus on two constructions, namely symmetric edge polytopes and cosmological polytopes. Both objects have been introduced and studied in the context of physics, and the former plays a key role also in the study of finite metric spaces and optimal transport. The general goal is to understand how interesting geometric invariants of these polytopes (their Ehrhart theory, volume, triangulations) are related to the combinatorics of the underlying graph. In the first part I will present a conjecture due to Ohsugi and Tsuchiya on a numerical invariant called the h*-vector of symmetric edge polytopes, and present partial results obtained in a joint work with Alessio D’Alì, Martina Juhnke-Kubitzke and Daniel Köhne. In the second part I will focus on cosmological polytopes. In a joint work with Martina Juhnke-Kubitzke and Liam Solus we show that cosmological polytopes have regular unimodular triangulations. In particular, their volume can be computed combinatorially.

Barbara Betti (Max Planck Institute for Mathematics in the Sciences - Leipzig)

Date:  14/11/2023           Time: 14:00                     Room: 713

Title: F-threshold of determinantal rings

Abstract: The F-threshold of a positive characteristic ring is an important invariant that can describe the singularities of the ring. However, its computation is a challenging problem. In this talk we will present a new upper bound for the F-threshold of determinantal rings generated by maximal minors. The method used is purely combinatoric and it is independent of the characteristic of the ring. We will restrict to the 3xn and 4xn case and prove that the F-threshold coincides with the negative a-invariant. We conjecture such equality holds for maximal minors of all matrices.

Saverio Secci (Università di Milano)

Date21/11/2023           Time: 14:00                     Room: 713

Title: Fano fourfolds with large anticanonical base locus

Abstract: The anticanonical system is probably one of the most natural objects associated with a Fano manifold. In this talk I will present some examples of Fano manifolds with nonempty anticanonical base locus and discuss a new result on the anticanonical system of Fano fourfolds: if the base locus is a normal irreducible surface, then all of its members are singular. Joint work with Andreas Höring.

Davide Bricalli (Università di Pavia)

Date28/11/2023           Time: 14:00                     Room: 713

Title: On the Hessian of cubic hypersurfaces

Abstract: The Hessian loci associated with projective hypersurfaces have been studied by many authors in the last decades. We mainly focus on the case of smooth cubic hypersurfaces. In the middle of the 20th century, B. Segre analyzed the Hessian of a cubic surface in P^3, while in 1996 Adler published a very deep description of the Hessian associated with a general smooth cubic threefold in P^4. In this talk, after generalizing some results and constructions to every dimension, we focus on the case of smooth cubic fourfolds, describing, in the general case, the singular locus of the Hessian, for which a natural unramified double cover comes into the picture. By exploiting an algebraic description coming from the apolar ring associated with the cubic hypersurface, we also discuss reducedness and reducibility of these Hessian loci. This is based on a work (also in progress) in collaboration with F.F.Favale and G.P.Pirola. 

Irene Spelta (Centre de Recerca Matemàtica)

Date5/12/2023           Time: 14:00                     Room: 713

Title: Dihedral coverings and Prym maps

Abstract: In this talk we consider étale cyclic covers \tilde C-> C of odd degree d=2k+1 of genus two curves. It is well-known that the lift of the hyperelliptic involution, combined with the deck transformations of the étale cover, generates a dihedral group D_d of automorphisms of \tilde C. In this way, several Jacobians and Pryms arise in the picture and the geometry turns out to be very interesting. We will look at the Prym map P_d that sends  \tilde C-> C to its associated Prym variety P(\tilde C, C). We will show that P_d is generically injective for d\geq 11 and k prime. Further literature will also be discussed for lower values of d. This is joint work with J.C. Naranjo and A. Ortega.

Cristina Bertone (Università di Torino)

Date:  12/12/2023           Time: 14:00                     Room: 713

Title: Marked bases for some quotient rings and applications

Abstract: Marked bases in a polynomial ring R are a powerful tool for the investigation of a Hilbert scheme on P^n. In a joint work with F. Cioffi, M. Orth, W.M. Seiler, we generalize marked bases to some quotient rings. We can do this in two different ways. The first one works for quotient rings R/I, where I is a homogeneous ideal, and exploits the good behaviour of quasi-stable ideals with respect to saturations. The second one is especially conceived for quotient rings R/I with I a quasi-stable ideal and adapts the algorithms for marked bases over a polynomial ring to this setting.
We apply the second strategy in two different directions: we give an open cover of a Hilbert scheme on Proj(R/I), when I is quasi-stable and Cohen Macaulay, generalizing the open cover for a Hilbert scheme on P^n; secondly, we explicitly investigate the smoothness of the Lex point of some Hilbert schemes on Proj(R/I), with R/I a Macaulay-Lex ring with I quasi-stable.

Elena Caviglia (University of Leicester)

Date20/12/2023           Time: 14:00                     Room: 713

Title: Generalizing principal bundles and quotient stacks

Abstract: Principal bundles over topological spaces are an important and useful notion in geometry. We will capture this notion in a categorical way and produce a new concept of principal bundle that makes sense in any nice category equipped with a Grothendieck topology. The topological group involved in the standard notion of principal bundle is generalized with a group object in the category. And locally trivial morphisms are generalized considering pullbacks along the morphisms of a covering family for the Grothendieck topology.
We will then see how to use generalized principal bundles to construct generalized quotient prestacks. When the Grothendieck topology is subcanonical and the category is nice enough, generalized quotient prestacks are stacks.
We will then move to dimension 2, introducing a notion of principal 2-bundle that makes sense in any nice 2-category equipped with a Grothendieck bitopology. In this context the group object involved in the definition is generalized with an internal 2-group and pullbacks are replaced with comma objects.
Finally, we will use principal 2-bundles to construct a generalization of quotient prestacks one dimension higher. When the the Grothendieck bitopology is subcanonical, these objects are instances of an original notion of 2-stack.

Martin Kreuzer (University of Passau)

Date9/01/2024           Time: 14:00                     Room: 713

Title: Points and Codes

Abstract: Evaluation codes are obtained from finite sets of points in affine or projective spaces. The parameters of these codes are related to the Hilbert function and the uniformity of the corresponding sets of points. Many classical results about finite sets of points in affine or projective space have been rediscovered in recent years in the context of Coding Theory. The purpose of this talk is to first recall and explain the classical theory of finite sets of points, as developed starting from the 1980s and 1990s, and then connect it to recent trends in Coding Theory. A particular emphasis will be placed on providing a translation table between modern notions in Coding Theory and the corresponding classical notions in Algebraic Geometry. We also supply extensive references, so that researchers in Coding Theory have an easier access to this part of Algebraic Geometry.

Luca Schaffler (Università degli Studi Roma Tre)

Date30/01/2024           Time: 14:00                     Room: 713

Title: The non-degeneracy invariant of Enriques surfaces: a computational approach

Abstract:  For an Enriques surface S, the non-degeneracy invariant nd(S) retains information about the elliptic fibrations of S and its projective realizations. While this invariant is well understood for general Enriques surfaces, computing it becomes challenging when specializing our Enriques surface. In this talk, we introduce a combinatorial version of the non-degeneracy invariant that depends on S along with a configuration of smooth rational curves, and gives a lower bound for nd(S). We also provide a SageMath code that computes this combinatorial invariant and we apply it in several examples where nd(S) was previously unknown. In particular, we study the families of Enriques surfaces introduced by Brandhorst and Shimada. The results presented are joint works and ongoing projects with Riccardo Moschetti and Franco Rota.

Alessio Moscariello (Università degli Studi di Catania)

Date20/02/2024           Time: 14:15                     Room: 715

Title: Open problems on relations of numerical semigroups

Abstract: Numerical semigroups are cofinite additive submonoid of the set of natural numbers. Despite their elementary nature, numerical semigroups appear in various areas of mathematics, including combinatorics, number theory, and commutative algebra. In this talk we present some open problems about minimal presentations of numerical semigroups (that is, minimal collections of relations among the generators of a numerical semigroup) and the Betti numbers of their semigroup rings.
This talk is based on a joint work with Alessio Sammartano.

Simone Murro (Università di Genova)

Date:  27/02/2024           Time: 14:15                     Room: 715

Title: Noncommutative Gelfand duality: a pathway to noncommutative spacetimes

Abstract: The duality between algebraic structures and geometric spaces is of paramount importance in mathematics and physics, because provides a dictionary to describe manifolds and variaties in a purely algebraic fashion. In his seminal paper, Gelfand showed that a topological space can be functorially reconstructed from its Banach algebra of continuous functions. Conversely, the Gelfand spectrum of the algebra of continuous functions is homeomorphic to the underlying topological space.

Motivated by a quantum theory of gravitation, it would be desirable to look for a noncommutive analogue of the Gelfand duality. In this talk, we restrict ourselves to the category of rings. Using state-of-the-art techniques from derived algebraic geometry, we show that category of rings is anti-equivalent to a subcategory of homotopically ringed spaces, reminiscent of Grothendieck’s work on commutative rings.

Laura Pertusi (Università di Milano)

Date05/03/2024           Time: 14:15                     Room: 713

Title: Non-commutative abelian surfaces and generalized Kummer varieties

Abstract: Examples of non-commutative K3 surfaces arise from semiorthogonal decompositions of the bounded derived category of certain Fano varieties. The most interesting cases are those of cubic fourfolds and Gushel-Mukai varieties of even dimension. Using the deep theory of families of stability conditions, locally complete families of hyperkahler manifolds deformation equivalent to Hilbert schemes of points on a K3 surface have been constructed from moduli spaces of stable objects in these non-commutative K3 surfaces. On the other hand, an explicit description of a locally complete family of hyperkahler manifolds deformation equivalent to a generalized Kummer variety is not yet available.

In this talk we will construct families of non-commutative abelian surfaces as equivariant categories of the derived category of K3 surfaces which specialize to Kummer K3 surfaces. Then we will explain how to induce stability conditions on them and produce examples of locally complete families of hyperkahler manifolds of generalized Kummer deformation type. This is a joint work in progress with Arend Bayer, Alex Perry and Xiaolei Zhao.

Previous years