Algebra & Geometry Seminar
Algebra & Geometry Seminar
Date: 06/05/2025 Time: 13:45 Room: 714
Title: Clifford's inequality for nodal curves
Abstract: Part of the classical Clifford theorem is an upper bound on the rank of linear series in the special range. It is easily seen that any similar bound purely in terms of the degree must fail for reducible curves. However, one can extend, quite surprisingly, the same bound as for smooth curves to nodal curves, if one restricts only to certain so-called multidegrees. In the talk I will give an overview of this circle of ideas and of what is known regarding possible upper bounds in the special range.
Date: 06/05/2025 Time: 14:50 Room: 714
Title: Brill-Noether loci of pencils with prescribed ramification
Abstract: The geometry of curves carrying pencils with prescribed ramification is regulated by the so called adjusted Brill-Noether number. In this talk I will discuss the problem of existence and dimension of Brill-Noether varities in this context and compare it to the classical one without imposed ramification. The new results are based on joint work with Andreas Leopold Knutsen.
Date: 13/05/2025 Time: TBA Room: 714
Title: The geometry of linear codes and some recent applications
Abstract: It is well-known that a nondegenerate linear code of length n and dimension k can be associated with a set of n points (with multiplicities) in a projective space of dimension k−1. Some coding-theoretical properties can be interpreted geometrically. This perspective connects MDS codes to problems involving arcs in projective spaces (the famous MDS conjecture was initially formulated as a problem in projective geometry by Segre), intersecting codes to non-2-cohyperplanar sets, Schur squares with evaluation of quadrics, and so on. In this talk, we will illustrate some recent results obtained by using this geometrical approach for Hamming-metric codes and outline how this can be generalized to the rank metric.
Date: 13/05/2025 Time: TBA Room: TBA
Title: Relations among P-twists
Abstract: On a K3 surface, rational curves and line bundles give rise to interesting autoequivalences of its derived category, so-called spherical twists. It was shown by P. Seidel and R. Thomas, that these spherical twists are mirror-dual to Dehn twists in symplectic geometry.
Moreover, they showed that given a chain of rational curves, the associated spherical twists satisfy braid relations. Generalising from K3 surfaces to hyperkähler varieties, D. Huybrechts and R. Thomas showed that the corresponding generalisation of the spherical twists are P^n-twists.
In this talk, I will introduce these autoequivalences, and speak about the possible relations among them. This talk is about joint work with Andreas Krug.
Date: 20/05/2025 Time: TBA Room: TBA
Title: Iarrobino scheme and Iarrobino symmetric decompositions
Abstract: Iarrobino's decomposition of the Hilbert function of an Artinian Gorenstein algebra is a very classical and useful invariant. In the first part of the talk I will review it. In the second part, I will present a moduli space such that k^*-limits on this space correspond to taking the Iarrobino decomposition.
Date: 29/05/2025 Time: TBA Room: TBA
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Date: 03/06/2025 Time: TBA Room: TBA
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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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Date: 07/02/2025 Time: 15:00 Room: 713
Title: Secant varietes of toric varieties and multigraded Hilbert scheme
Abstract: For a smooth toric projective variety $X \subset \mathbb{P}^N$ (in a torus equivariant embedding) and its Cox ring $S$, consider the secant varieties of $X$. The $r$-th secant variety is the closure of the union of all $\mathbb{P}^{r-1}$'s spanned by points of $X$. We present an algebraic criterion for a point to be in the $r$-th secant variety in terms of an action of $S$ called "apolarity action". The criterion heavily involves the geometry of multigraded Hilbert scheme of $S$, and provides a nice and uniform description of all points of the secant variety. Moreover, it is consistent with the action of the automorphism group $Aut(X)$ on the secant variety and on the multigraded Hilbert scheme. It provides a new technique for determining if a given point is in some specific secant variety, which is especially efficient if the point is invariant under a relatively large subgroup of $Aut(X)$. As an application, we are able to determine the minimal $r$ such that a prescribed point is in the $r$-th secant variety in a bunch of situations of interest to complexity theory.
The talk is based on a joint work with Weronika Buczyńska (Duke Mathematical Journal, 2021).
Date: 18/02/2025 Time: 14:15 Room: 714
Title: Elliptic fibrations on K3 surfaces with symplectic involutions
Abstract: In this talk, we will relate the geometry of two classes of projective surfaces: K3 surfaces and del Pezzo surfaces. This allows us to classify elliptic fibrations on such K3 surfaces using the theory of conic bundles on del Pezzo surfaces. This is joint work with Paola Comparin, Pedro Montero, and Yulieth Prieto-Montañez.
Date: 04/03/2025 Time: 15:00 Room: 714
Title: On the subadditivity of shifts
Abstract: In this talk we explain two constructions on the simplicial chain complex
of the order complex of a poset:
(i) the shuffle product (in case the poset is a lattice)
(ii) the synor complex, which is a subcomplex of the simplicial chain complex
chose chains have a well controlled structure. The synor complex is also shown
to give rise to minimal free resolutions of monomial ideals.
Then we show how (i) and (ii) can be used to answer a question
by Avramov, Conca and Iyengar
on the subadditivity of shifts in the minimal free resolution for the case
of monomial ideals.
Date: 18/03/2025 Time: 14:15 Room: 714
Title: Effective (inconceivably) big bounds for small subalgebras.
Abstract: This talk is based on a joint work with Yihui Liang and Cheng Meng. I will present explicit bounds for the projective dimension and the Castelnuovo-Mumford regularity of homogeneous ideals in the setting of Stillman's Conjecture and its solution by Ananyan and Hochster. As a corollary, in the spirit of the Eisenbud-Goto conjecture and its counterexamples, one derives explicit bounds for the Castelnuovo-Mumford regularity of homogeneous prime ideals solely in terms of their multiplicity.
Date: 01/04/2025 Time: 14:15 Room: 714
Title: Automorphisms on relative Prym varieties induced by K3 surfaces.
Abstract: The relative Prym construction is a way to produce examples of irreducible symplectic varieties, starting from a K3 surface S covering a del Pezzo (or Enriques) surface T. For some special choices of a divisor D on S, the moduli space M = Mv(S) with v = (0, D, 1 - g(D)) is singular, and it has a non-natural, non-symplectic regular involution that acts by taking the duals of stable sheaves. The composition of this involution with the one induced on M by the cover involution ι of S → T is symplectic, and the biggest component of its fixed locus, the relative Prym variety, can be an irreducible symplectic variety.
In this talk, I'm going to assume that a bigger group G acts on S, such that the action of G/ι is symplectic; I'll discuss the conditions under which this latter action extends to a symplectic action on the relative Prym variety, and give some examples.
This is a joint work in progress with Annalisa Grossi and Sasha Viktorova.
Date: 08/04/2025 Time: 14:15 Room: 714
Title: Equivariant spaces of matrices of constant rank
Abstract: A space of matrices of constant rank is a vector subspace V, say of dimension n+1, of the set of matrices of size axb over a field k, such that any nonzero element of V has fixed rank r. It is a classical problem to look for different ways to construct such spaces of matrices. In this talk I will report on my latest joint project (in progress) with D. Faenzi and D. Fratila, where we give a classification of all spaces of matrices of constant corank one associated to irreducible representation of a reductive group.
Date: 22/04/2025 Time: 14:15 Room: 714
Title: Nested Hilbert schemes on Hirzerbruch surfaces and quiver varieties
Abstract: For n greater or equal than 1 we show that the length 1 nested Hilbert scheme of the total space Xn of the line bundle OP1(-n), parameterizing pairs of nested 0-cycles in Xn, is a quiver variety associated with a suitable quiver with relations. This generalizes previous work about nested Hilbert schemes on C2 in one direction, and about the Hilbert schemes of points of Xn in another direction.
Date: 29/04/2025 Time: 14.15 Room: 714
Title: Differentials and differential operators
Abstract: In the first part, I will describe Kähler differentials and derivations and some of the history behind the classic Lipman-Zariski Conjecture, as well as a generalized question proposed by Graf. Together with Vassiliadou, we give a partial answer to Graf’s question for a certain class of varieties.
In the second portion, we turn our attention to higher order differential operators, finding explicit generators and free resolutions in some low orders for these for the hypersurfaces studied by Bernstein-Gel’fand-Gel’fand and Vigué. This is joint work with Diethorn, Jeffries, Packauskas, Pollitz, Rahmati, and Vassiliadou.