Algebra & Geometry Seminar
Valentina Bais (SISSA)
Date: 10/12/2024 Time: 14:15 Room: 713
Title: Constructions or irreducible exotic 4-manifolds with free abelian fundamental group
Abstract: Smooth structures on 4-dimensional manifolds behave very differently with respect to any other dimension. Indeed, while any topological n-manifold for n<4 is smoothable in a unique way, there are many instances of topological 4-manifolds supporting infinitely many pairwise inequivalent smooth structures. Such smooth structures are called exotic. On the other hand, one can have at most finitely many smooth structures on a topological n-manifold for n>4. I will present a joint work with Rafael Torres and Daniele Zuddas on the construction of irreducible exotic 4-manifolds with free abelian fundamental group and small Euler characteristics, in which we determine the homeomorphism type via the study of the equivariant intersection form.
Previous Talks
Dumitru Stamate (University of Bucharest)
Date: 1/10/2024 Time: 14:15 Room: 713
Title: New asymptotic properties for shifted families of numerical semigroups
Abstract: Given the positive integers r_1<r_2<…<r_k, their associated shifted family consists of the affine semigroups M_n=<n, n+r_1, …, n+r_k> for all n>0. Algebraic properties of the semigroup rings K[M_n] are commonly explored in terms of the semigroups M_n. Answering a conjecture of Herzog and Srinivasan, Vu proved that the Betti numbers of K[M_n] become eventually periodic in n, for n large enough. This opened a path of finding periodic properties for large n. In this talk I will report on joint work in progress with M. Cimpoeas, and with F. Strazzanti, respectively. The former discusses Betti numbers for intersections of the defining ideals of semigroup rings in the same shifted family, while the latter is concerned with the behaviour of a more subtle invariant, the residue, and the nearly Gorenstein property.
Rosanna Laking (Università degli Studi di Verona)
Date: 9/10/2024 Time: 9:00 Room: 714
Title: Grothendieck hearts of t-structures
Abstract: In this talk we discuss the notion of a cosilting t-structure and show how they parametrise a large class of t-structures with Grothendieck hearts. We will present joint work with Lidia Angeleri Hügel and Francesco Sentieri in which we show that those cosilting t-structures that are “nearby” a heart that is equivalent to a category of modules over a finite-dimensional algebra are parametrised by closed sets of a topological space called the Ziegler spectrum. A well-known example of this phenomenon is given by the category of quasi-coherent sheaves over P^1 that sits “nearby” the category of modules over the Kronecker algebra.
Jorge Vitória (Università degli Studi di Padova)
Date: 9/10/2024 Time: 10:00 Room: 714
Title: Grothendieck categories with a commutative flavour
Abstract: The representation theory of a commutative noetherian ring is tightly controlled by its prime spectrum. This is evident not only when studying modules, but also when studying complexes of modules. In this talk, we discuss some Grothendieck abelian categories that are full subcategories (in fact, hearts of special t-structures) of the derived category of modules over commutative noetherian rings. In particular, we discuss the role played by the residue fields and their shifts in the structure of these categories. This talk is based in joint work with Sergio Pavon, and in ongoing joint work with Michal Hrbek and Sergio Pavon.
Sora Miyashita (Osaka University)
Date: 15/10/2024 Time: 14:15 Room: 713
Title: On nearly Gorenstein Veronese subalgebras
Abstract: The nearly Gorenstein property is known as a generalization of the Gorenstein property. Herzog, Hibi, and Stamate proved that every Veronese subalgebra of a standard graded "Gorenstein" ring is nearly Gorenstein. Additionally, they posed the question: Does this statement still hold if the assumption is weakened from "Gorenstein" to "nearly Gorenstein"? In this talk, I will introduce a generalized notion of nearly Gorenstein rings in the broader context of semi-standard graded rings and provide an answer to this question. Furthermore, I will present results from my joint work with Hall, Kölbl, and Matsushita, where we extend our findings on Veronese subrings of Ehrhart rings to a broader class of semi-standard graded domains, including Ehrhart rings.
Giancarlo Rinaldo (Università degli Studi di Messina)
Date: 22/10/2024 Time: 14:15 Room: 713
Title: Sequentially Cohen-Macaulay binomial edge ideals
Abstract: Binomial edge ideals have been introduced in 2011 by Herzog, Hibi and others and, independently, by Ohtani. They are associated to finite simple graphs, in fact they arise from the 2-minors of a 2 × n matrix related to the edges of a graph with n vertices. The problem of finding a characterization of Cohen–Macaulay binomial edge ideals has been studied intensively by many authors. A nice and deeply studied generalization of the Cohen-Macaulay ring is the sequentially Cohen-Macaulay one defined by Stanley, SCM for short. In 2014, Schenzel and Zafar classified the complete bipartite graphs that are SCM. Recently, in [1], the authors classify the binomial edge ideals with quadratic Gröbner bases that are SCM. In this talk we show that cycles, wheels, and block graphs are SCM. Moreover, we provide a construction of new families of SCM graphs by cones (see [3]). The main ingredient to obtain the results in [1] and [3] is a characterization of SCM rings provided by Goodarzi [2].
Bibliography:
[1] V. Ene, G. Rinaldo, N. Terai, Sequentially Cohen-Macaulay binomial edge ideals of closed graphs. Res. Math. Sci. 9, 39 (2022)
[2] A. Goodarzi, Dimension filtration, sequential Cohen–Macaulayness and a new polynomial invariant of graded algebras. J. Algebra 456, 250–265 (2016)
[3] E. Lax, G. Rinaldo, F. Romeo, Sequentially Cohen-Macaulay binomial edge ideals, arXiv:2405.08671, 2024.
Emiliano Ambrosi (University of Strasbourg)
Date: 29/10/2024 Time: 14:15 Room: 713
Title: Reduction modulo p of the Noether problem
Abstract: Let k be an algebraically closed field of characteristic p≥0 and V a faithful k-rational representation of an l-group G. The Noether's problem asks whether V/G is (stably) birational to a point. If l is equal to p, then Kuniyoshi proved that this is true, while, if l is different from p, Saltman constructed l-groups for which V/G is not stably rational. Hence, the geometry of V/G depends heavily on the characteristic of the field. We show that for all the groups G constructed by Saltman, one cannot interpolate between the Noether problem in characteristic 0 and p. More precisely, we show that it does not exist a smooth proper R-scheme X---->Spec(Z_p) whose special fiber and generic fiber are both stably birational to V/G. The proof combines the integral p-adic Hodge theoretic results of Bhatt-Morrow-Scholze, with the study of the Cartier operator on differential forms in positive characteristic. This is a joint work with Domenico Valloni.
Federico Fallucca (Università degli Studi di Milano-Bicocca)
Date: 5/11/2024 Time: 14:15 Room: 715
Title: Classification of (k,p)-covers of the plane with geometric genus 3
Abstract: A (k,p)-cover of the plane is a compact complex smooth surface X with an automorphism subgroup isomorphic to Z_p^k, where p is a prime number, such that the quotient is P^2. There are several motivations for investigating these surfaces. For instance, (k,p)-covers of the plane are always regular, and those with geometric genus 3 are good candidates for surfaces with a high degree of the canonical map, of which there are few examples in the literature.
Furthermore, they are of interest as potential "Triple K3 Burgers", a class of surfaces due to Laterveer where Voisin's conjecture (on the pullback of 0-cycles on a surface) can be examined.
Additionally, bidouble and iterated bidouble covers of rational surfaces have been studied by Garbagnati and Penegini to produce examples for investigating the Infinitesimally Torelli Property, the Mumford-Tate conjecture, and the Tate conjecture.
In this seminar, I will briefly discuss the description of (k,p)-covers of P^2 in the language of abelian coverings by Pardini. I will then present the classification theorem of (k,2)-covers of P^2 with geometric genus 3 (a joint work with Pignatelli), and the more recent classification of (k,p)-covers of P^2 with p>2. Furthermore, I will show the degree of the canonical map for each case and discuss which ones are Triple K3 Burgers. Finally, time permitting, I will outline the approach used to achieve this classification.
Lisa Seccia (University of Neuchâtel)
Date: 12/11/2024 Time: 14:15 Room: 713
Title: F-purity of geometrically vertex decomposable ideals
Abstract: In this talk we explore F-purity of geometrically vertex decomposable ideals. After a brief overview of some basic concepts in F-singularity theory and geometric vertex decomposition, we present a way to iteratively construct Frobenius splittings for geometric vertex decomposable ideals. This technique will be illustrated with examples.
This is ongoing project with De Negri, Gorla, Klein, Mayada, Rajchgot, Shahada.
Michael Loenne (University of Bayreuth)
Date: 19/11/2024 Time: 14:15 Room: 713
Title: Discriminant knot groups of linear systems on surfaces
Abstract: Linear system of curves on surfaces with smooth generic members give naturally rise to a discriminant, the locus corresponding to singular curves, and the discriminant knot group which is the fundamental group of the discriminant complement.
The naturally associated family of curves gives rise to a monodromy map with values in the mapping class group, which by recent results is completely understood in important cases. However, the kernel might be quite large.
In the case of linear systems on Segre-Hirzebruch surfaces, I will give another monodromy map, which may capture the full discriminant knot group. I will provide supporting evidence and give a finite presentation of some discriminant knot groups.
Antonio Rapagnetta (Università di Roma Tor Vergata)
Date: 26/11/2024 Time: 14:15 Room: 713
Title: Singular moduli spaces of sheaves on K3 surfaces
Abstract: In this talk I will present some results, obtained in collaboration with A. Perego and C. Onorati, on the geometry of the moduli spaces of sheaves with non-primitive Mukai vector on K3 surfaces.
Matteo Ruggiero (Université Paris Cité)
Date: 26/11/2024 Time: 15:15 Room: 713
Title: On the Dynamical Manin-Mumford problem for plane polynomial endomorphisms
Abstract: The Dynamical Manin-Mumford problem is a dynamical question inspired by classical results from arithmetic geometry. Given an algebraic dynamical system (X,f), where X is a projective variety and f is a polarized endomorphism on X, we want to determine if a subvariety Y containing "unusually many" periodic points must be itself preperiodic.
In a work in collaboration with Romain Dujardin and Charles Favre, we prove this property to hold when f is a regular endomorphism of P^2 coming from a polynomial endomorphism of C^2 of degree d>=2, under the additional condition that the action of f at the line at infinity doesn't have periodic superattracting points.
The proof is an interesting blend of techniques coming from arithmetic geometry, holomorphic and non-archimedean dynamics.
Alessio Sammartano (Politecnico di Milano)
Date: 3/12/2024 Time: 14:15 Room: 713
Title: Irrational components of Hilbert schemes of points
Abstract: The Hilbert scheme of points in affine n-dimensional space parametrizes finite subschemes of a given length. It is smooth and irreducible if n is at most 2, singular and reducible if n is at least 3. Understanding its irreducible components, their singularities and birational geometry, has long been an inaccessible problem. In this talk, I will describe substantial progress on this problem achieved in recent years. In particular, I will focus on the problem of rationality of components. This is based on a joint work with Gavril Farkas and Rahul Pandharipande.