 |
|
Seminari de Geometria Algebraica 2022/2023 |
|
| |
|
Conferenciant
|
Títol |
Data i hora |
Kiran Kedlaya
University of California San Diego
Contact:
francesc.fite at gmail.com
|
The relative class number one problem for function fields
Building on my lecture from ANTS-XV, we classify extensions of function fields (of curves over finite fields) with relative class number 1. Many of the ingredients come from the study of the maximum number of points on a curve over a finite field, such as the function field analogue of Weil's explicit formulas (a/k/a the "linear programming method"). Additional tools include the classification of abelian varieties of order 1 and the geometry of moduli spaces of curves of genus up to 7.
|
Divendres 9 de setembre, 15h10, Aula T1, FMI-UB.
|
Alexandre Turull
University of Florida
Contact:ignasi.mundet at ub.edu
|
Representations of finite groups over small fields
Let \(G\) be a finite group and let \(F/K\) be a field extension. A representation of \(G\) over
\(F\) is a group homomorphism from \(G\) to \(GL_n(F)\), the general linear group
of all automorphisms of a vector space of dimension \(n\) over \(F\).
What representations of \(G\) over \(F\) arise from representations of \(G\) over \(K\)
by simply composing with a natural homomorphism \(GL_n(K) \to GL_n(F)\)?
We will discuss how to answer such questions in many cases.
Along the way we will discuss \(p\)-adic numbers, Brauer groups,
Brauer characters, and how to calculate certain invariants associated with
irreducible representations of finite groups.
These invariants may be different for Galois conjugate representations, but
are nevertheless uniquely defined in the same sense that Brauer characters
are uniquely defined.
|
Divendres 30 de setembre, 15h10, Aula B1, FMI-UB.
|
Barbara Fantechi
SISSA
Contact:marchesi at ub.edu
|
MINI-CURS: Abelian covers and their moduli spaces
One of the oldest methods to construct new smooth projective
varieties from those we already know is to take cyclic covers branched
over a smooth divisor. We will introduce this "bottom up" approach and
apply to study moduli problems.
In the first lecture, we will review this construction, and outline
how to generalize it to abelian covers (i.e., finite Galois covers
with abelian Galois group), following Pardini's work. In particular,
we will show how geometric information about the cover can be
described in terms of the base of the cover and the "branching data"
(line bundles and divisors).
In the second lecture we will introduce some key notions in
infinitesimal deformation theory, and then apply them to study
infinitesimal deformations of abelian covers, following a joint work
with Pardini.
In the third lecture we will briefly recall what a moduli space for
varieties with ample canonical divisor is, and apply what we learned
to construct examples of such moduli spaces. In particular we will
discuss Vakil's lovely "Murphy's Law in Algebraic Geometry" paper, as
well as work in progress with Pardini on moduli of surfaces of general
type fibered in hyperelliptic curves.
|
Dimarts 11, 18 i 25 d'octubre, 15h00, Aula IMUB, IMUB-UB.
|
Irene Spelta
Universitat de Barcelona
Contact:irene.spelta01 at universitadipavia.it
|
Prym maps and generic Torelli theorems: the case of plane quintics
The talk deals with Prym varieties and Prym maps. Prym varieties are polarized abelian varieties associated with finite morphisms between smooth curves. Prym maps are accordingly defined as maps from the moduli space of coverings to the moduli spaces of polarized abelian varieties.
Once recalled the classical generic Torelli theorem for the Prym map of étale double coverings, we will move to the more recent results on the ramified Prym map \(P_{g,r}\) associated with ramified double coverings. For most of the values of \((g,r)\) a generic Torelli theorem holds and, furthermore, a global Torelli theorem holds when \(r\) is greater (or equal to) 6. At the same time, it is known that \(P_{g,2}\) and \(P_{g,4}\) have positive dimensional fibres when restricted to the locus of coverings of hyperelliptic curves. But this is not a characterization: the study of the differential \(dP_{g,r}\) shows that there are also other configurations to be considered. We will focus on the case of degree 2 coverings of plane quintics ramified in 2 points. We will show that the restriction of \(P_{g,r}\) here is generically injective. This is joint work with J.C. Naranjo.
|
Divendres 21 d'octubre, 15h10, Aula B1, FMI-UB.
|
Martí Salat Moltó
Universitat de Barcelona
Contact:marti.salat at ub.edu
|
Equivariant sheaves and vector bundles on toric varieties
In this talk, we consider the theory of equivariant sheaves on a toric variety described via (multi)filtrations of a vector space, as introduced by Klyachko in the 90s. Starting with torsion-free equivariant sheaves, we will apply this theory to give a description of monomial ideals, generalizing the classical staircase diagrams and suited to non-standard gradings of a polynomial ring. Afterwards, we will focus on equivariant reflexive sheaves and vector bundles. We introduce a family of lattice polytopes encoding their global sections. In the case of Picard rank 2 smooth projective toric varieties, this description allows us to compute explicitly the Hilbert polynomial of an equivariant reflexive sheaf.
|
Divendres 28 d'octubre, 15h10, Aula B1, FMI-UB.
|
Luis Núñez-Betancourt
CIMAT-CRM
Contact:josep.alvarez at upc.es
|
Nash blowup in prime characteristic
The Nash blowup is a natural modification of algebraic varieties that replace singular points by limits of certain vector spaces associated to the variety at non-singular points. For several decades it has been studied whether it is possible to resolve singularities of algebraic varieties by iterating Nash blowups. This problem has mostly been treated in characteristic zero due to an example given by Nobile. In this talk, we will discuss a new approach in prime characteristic using differential operators, and an application to resolution of singularities of toric varieties.
|
Divendres 4 de novembre, 15h10, Aula B1, FMI-UB.
|
Raheleh Jafari
Kharazmi University
Contact:szarzuela at ub.edu
|
On the Gorenstein locus of simplicial affine semigroup rings
The Gorenstein locus of simplicial affine semigroup rings is studied by an analysis of Cohen-Macaulay type of homogeneous localizations at monomial prime ideals. In particular, we discuss a geometrical characterization of the semigroup rings that are Gorenstein on the punctured spectrum.
|
Divendres 18 de novembre, 15h10, Aula B1, FMI-UB.
|
Lothar Göettsche
ICTP - International Centre for Theoretical Physics
Contact:marchesi at ub.edu
|
(Refined) Verlinde and Segre formulas for Hilbert schemes of points
This is joint work with Anton Mellit.
Segre and Verlinde numbers of Hilbert schemes of points have been studied for a long time.
The Segre numbers are evaluations of top Chern and Segre classes of so-called tautological bundles on Hilbert schemes of points.
The Verlinde numbers are the holomorphic Euler characteristics of line bundles on these Hilbert schemes.
We give the generating functions for the Segre and Verlinde numbers of Hilbert schemes of points.
The formula is proven for surfaces with \(K_S^2=0\), and conjectured in general. Without restriction on \(K_S^2\) we prove the conjectured
Verlinde-Segre correspondence relating Segre and Verlinde numbers of Hilbert schemes. Finally we find a generating function for finer invariants,
which specialize to both the Segre and Verlinde numbers, giving some kind of explanation of the Verlinde-Segre correspondence.
|
Divendres 25 de novembre, 15h10, Aula B1, FMI-UB.
|
Giuseppe Ancona
Université de Strasbourg
Contact:ffite at ub.edu
|
Quadratic forms arising from geometry
The cup product on topological manifolds or the intersection product on algebraic varieties induce quadratic forms which turn out to be a fine invariant of these geometric objects. We will discuss some old theorems on the signature of these quadratic forms and some applications both of geometric and arithmetic origins. Finally we will study an old conjecture of Grothendieck about those signatures and explain some new evidences.
|
Dimarts 29 de novembre, 17h00, Aula B7, FMI-UB.
|
Daniel Macias Castillo
UAM-ICMAT
Contact:francesc at mat.uab.edu
|
Towards a refined class number formula for Drinfeld modules
This is joint work with María Inés de Frutos Fernández and Daniel Martínez
Marqués.
In 2012, Taelman proved an analogue of the Analytic Class Number Formula,
for the Goss L-functions that are associated to Drinfeld modules. He also explicitly
stated that "it should be possible to formulate and prove an equivariant version" of
this formula.
We report on work in progress motivated by Taelman's statement. We formulate a precise equivariant, or `refined', generalisation of Taelman's formula, and we
provide very strong evidence to support it.
In the general case, we reduce the validity of our refined class number formula to
a purely algebraic problem in the K-theory of positive characteristic group rings.
As a concrete consequence of our general approach, we also prove (unconditionally) a natural non-abelian generalisation of a Theorem of Anglès and Taelman,
regarding the Galois structure of Taelman's class group.
|
Divendres 2 de desembre, 15h10, Aula B1, FMI-UB.
|
Barbara Fantechi
SISSA
Contact:marchesi at ub.edu
|
Smoothability for some Gorenstein non normal Godeaux surfaces
This talk reports on two papers with M. Franciosi and R. Pardini.
We take several families of non normal Gorenstein surfaces and show that each is contained in a unique component of the KSBA moduli of surfaces of general type, which is generically smooth of the expected dimension.
|
Divendres 16 de desembre, 15h10, Aula B1, FMI-UB.
|
Eriola Hoxhaj
Johannes Kepler Universitat Linz
Contact:cdandrea at ub.edu
|
Reconstructing Surfaces from their Silhouettes
This talk will be a summary of my work research and other related
papers on the subject. The main tools that relate all of these are
ramification curves (contour), branching curve (silhouette),
singularities of the branching curve, the conductor ideals of all
singularities and in the end the construction of the surface.
|
Divendres 20 de gener, 15h10, Aula T2, FMI-UB.
|
Robert Auffarth
Universidad de Chile
Contact:marti.lahoz at ub.edu
|
Smooth quotients of complex tori by finite groups
A famous open problem asked by Ekedahl and Serre is if there exist infinite genera for which there exist completely decomposable Jacobians; that is, Jacobians that are isogenous to a product of elliptic curves. It was proved quite recently that if an abelian variety has a finite group action that fixes the origin and its quotient is isomorphic to projective space, then the abelian variety must be isogenous to the self product of an elliptic curve. This then leads one to ask if there exist Jacobians of any genera with a finite group action whose quotient is projective space. In this talk, we will give a disappointing answer to an intriguing problem that nevertheless opens the door to a beautiful and deep topic: smooth quotients of complex tori.
This is joint work with Giancarlo Lucchini Arteche.
|
Divendres 20 de gener, 16h20, Aula T2, FMI-UB.
|
Carles Checa
National Kapodistrian University of Athens
Contact:marchesi at ub.edu
|
Toric Sylvester forms and applications in elimination theory
We investigate the structure of the saturation of ideals generated by systems of sparse homogeneous polynomials over a toric variety with respect to the irrelevant ideal of its Cox ring. As our main results, we establish a duality property and make it explicit by introducing toric Sylvester forms, under a certain positivity assumption on the variety. In particular, we prove that toric Sylvester forms yield bases of some graded components of \(I^{sat}/I\), where \(I\) denotes an ideal generated by n+1 generic forms, n is the dimension of the variety and \(I^{\sat}\) is the saturation with respect to the irrelevant ideal. We illustrate the relevance of toric Sylvester forms by providing three consequences regarding elimination matrices, sparse resultants and toric residues. This is joint work with Laurent Buse.
|
Divendres 3 de febrer, 15h10, Aula T2, FMI-UB.
|
Vincenzo Antonelli
Politecnico di Torino
Contact:marchesi at ub.edu
|
Vector bundles and representations of hypersurfaces
Let \(X\) be an integral hypersurface in the projective space. The description of hypersurfaces as zero loci of suitable square matrices (possibly with some further properties) is a very classical topic in algebraic geometry. In this talk we show how the existence of particular classes of vector bundles supported on \(X\) yields the description of its defining equation as the degeneracy locus of a map between bundles. Then we show that every smooth surface in \(\mathbb{P}^3\) is the pfaffian of a map between Steiner bundles and finally we discuss some examples in higher dimensions. This is a joint work with Gianfranco Casnati.
|
Divendres 17 de febrer, 15h10, Aula T2, FMI-UB.
|
Souvik Goswami
Universitat de Barcelona
Contact:sombra at ub.edu
|
Height Pairing on Bloch's higher Chow groups
Heights are ubiquitous in algebraic geometry and number theory. Roughly
speaking, height of a point on an algebraic variety in projective space mea-
sures the arithmetic complexity of the point. One such theory of heights
is given by Néron, and is called the Néron-Tate height. For more general
subvarieties in complimentary codimensions, Beilinson developed the theory
of height pairing. The height pairing is connected to a series of celebrated
conjectures, and should have a number of interesting properties. In this
seminar we review this height pairing and introduce a new pairing on higher
Chow groups, which is akin to the one given by Beilinson.
|
Divendres 3 de març, 15h10, Aula T2, FMI-UB.
|
Alicia Dickenstein
Universidad de Buenos Aires
Contact:cdandrea at ub.edu
|
Iterated and mixed discriminants
Classical work by Salmon and Bromwich classified singular intersections of two quadric surfaces. The basic idea of these results was already pursued by Cayley in connection with tangent intersections of conics in the plane and used by Schäfli for the study of hyperdeterminants. More recently, the problem has been revisited with similar tools in the context of geometric modeling and a generalization to the case of two higher dimensional quadric hypersurfaces was given by Ottaviani. In joint work with Sandra di Rocco and Ralph Morrison, we propose and study a generalization of this question for systems of Laurent polynomials with support on a fixed point configuration.
In the non-defective case, the closure of the locus of coefficients giving a non-degenerate multiple root of the system is defined by a polynomial called the mixed discriminant. We define a related polynomial called the multivariate iterated discriminant. This iterated discriminant is easier to compute and we prove that it is always divisible by the mixed discriminant. We show that tangent intersections can be computed via iteration if and only if the singular locus of a corresponding dual variety has sufficiently high codimension. We also study when point configurations corresponding to Segre-Veronese varieties and to the lattice points of planar smooth polygons, have their iterated discriminant equal to their mixed discriminant.
|
Divendres 17 de març, 15h10, Aula T2, FMI-UB.
|
Sebastià Xambó-Descamps
Universitat Politècnica de Catalunya
Contact:josep.alvarez at upc.es
|
The discrete charm of algebraic geometry
June Huh, in his Fields Lecture (July 6, 2022), presented the results he and his coworkers had obtained in the realm of combinatorics by means of a discrete variant of what he calls the Kähler package. The essence of this package was outlined by Alexander Grothendieck in his "standard conjectures" in intersection theory, which were proposed as a way to prove Weil's conjectures concerning the zeta function (a discrete object!) of smooth projective algebraic varieties defined over a finite field. The goal of this talk is to present the Kähler package with emphasis in its roots in intersection theory, in some of its discrete instances that have yielded proofs of a host of combinatorial conjectures, and in its potential for approaching problems in other areas. Slides
|
Divendres 24 de març, 15h10, Aula T2, FMI-UB.
|
Sebastian Casalaina-Martin
University of Colorado
Contact:marti.lahoz at ub.edu
|
Moduli spaces of cubic hypersurfaces
In this talk I will give an overview of some recent work, joint with Samuel Grushevsky, Klaus Hulek, and Radu Laza, on the geometry and topology of compactifications of the moduli spaces of cubic threefolds and cubic surfaces. A focus of the talk will be on some results regarding non-isomorphic smooth compactifications of the moduli space of cubic surfaces, showing that two natural desingularizations of the moduli space have the same cohomology, and are both blow-ups of the moduli space at the same point, but are nevertheless, not isomorphic, and in fact, not even K-equivalent.
|
Divendres 14 d'abril, 15h10, Aula T2, FMI-UB.
|
Paolo Stellari
Università degli Studi di Milano
Contact:marti.lahoz at ub.edu
|
Stability conditions in the trivial canonical bundle case: Hilbert
schemes of points and very general abelian varieties
The construction of stability conditions on the bounded derived
category of coherent sheaves on smooth projective varieties is
notoriously a difficult problem, especially when the canonical bundle
is trivial. In this talk, I will review some results and techniques
related to the latter setting. I will specifically concentrate on the
case of Hilbert schemes of points on K3 surfaces and very general
abelian varieties of any dimension. The latter result follows from a
new and very effective technique involving deformations of bounded
t-structures. This is joint work in progress with C. Li, E. Macri' and
X. Zhao.
|
Divendres 21 d'abril, 15h10, Aula T2, FMI-UB.
|
Roser Homs
Centre de Recerca Matemàtica
Contact:josep.alvarez at upc.es
|
Third order moment varieties of linear non-Gaussian graphical models
We study non-Gaussian graphical models from a perspective of algebraic statistics. Our focus is on algebraic relations among second- and third-order moments in graphical models given by linear structural equations. We show that when the graph is a tree these relations form a toric ideal. Using combinatorial tools, such as the multi-trek rule introduced by Robeva and Seby, we provide a Markov basis of the vanishing ideal of models given by trees, generalizing results on Gaussian models obtained by Sullivant.
|
Divendres 28 d'abril, 15h10, Aula T2, FMI-UB.
|
Elisabetta Colombo
Università degli Studi di Milano
Contact:jcnaranjo at ub.edu
|
The second fundamental form on the moduli space of cubic threefolds in \(\mathcal{A}_5\)
We study the second fundamental form of the Siegel metric in \(\mathcal{A}_5\) restricted to the locus of intermediate Jacobians of cubic threefolds. We prove that the image of this second fundamental form, which is known to be non-trivial, is contained in the kernel of a suitable multiplication map. Some ingredients are: the conic bundle structure of cubic threefolds, Prym theory, Gaussian maps and Jacobian ideals. This is a joint work with P. Frediani, J.C. Naranjo and G.P. Pirola.
|
Divendres 5 de maig, 15h10, Aula T2, FMI-UB.
|
Paola Frediani
Università degli Studi di Pavia
Contact:jcnaranjo at ub.edu
|
A canonical Hodge theoretic projective structure on compact Riemann surfaces.
After a brief introduction on projective structures, we will describe the construction of a canonical projective structure on every compact Riemann surface, coming from Hodge theory. We will show that this projective structure differs from the canonical projective structure produced by the uniformisation theorem. These are results obtained in collaboration with I. Biswas, E. Colombo and G.P. Pirola.
|
Divendres 12 de maig, 15h10, Aula T2, FMI-UB.
|
Angelica Torres
Centre de Recerca Matemàtica
Contact:josep.alvarez at upc.es
|
Multiview varieties: A bridge between Algebraic Geometry and Computer Vision
The 3D image reconstruction problem aims to build a 3D model of a scene depicted in multiple 2D images. The reconstruction process relies on the identification and triangulation of point and line features appearing in the images. In this talk we will focus on the triangulation of points and lines that satisfy certain incidence relations. We will start by defining the point and line multiview varieties, then we will mention some of their basic properties, and finally we will explore how they are used to triangulate incident points and lines and give a measure of the algebraic complexity of their use to fit real data.
|
Divendres 26 de maig, 15h10, Aula T2, FMI-UB.
|
Carlos D'Andrea
Universitat de Barcelona
Contact:
|
El converso de Euler-Jacobi en el toro
The classical Euler-Jacobi theorem states that if a system of n polynomials in n variables does not have zeroes at the infinity of projective space, the global residue of a "controlled" form vanishes. There is an analogue of this result for sparse polynomials and toric varieties. However, the converse of this statement (if the residues of all controlled forms vanish, does this mean that the system has no roots at infinity?) was only known in projective space. In this talk we will present the problem and show an approach to deal with the converse of this theorem in the toric setting. This is a joint work with Alicia Dickenstein from the University of Buenos Aires.
|
Dijous 29 de juny, 15h30, Aula T2, FMI-UB.
Sessió doble
|
Eduardo Sáenz de Cabezón
Universidad de La Rioja
Contact:
|
La combinatoria de las bases involutivas
Las bases involutivas tienen su origen en la teoría de Janet-Riquier de sistemas diferenciales. Son bases de Gröbner con ciertas propiedades combinatorias adicionales. En esta charla examinaremos algunas de esas propiedades adicionales y nos centraremos en el estudio de bases de Pommaret de ideales monomiales.
|
Dijous 29 de juny, 17h00, Aula T2, FMI-UB.
Sessió doble
|
Le Tuan Hoa
VIASM, Hanoi, Vietnam
Contact:
|
Maximal generating degrees of powers of an homogeneous ideal
This talk has two parts.
The first one is dealing with the degree excess function
\(\epsilon(I;n)\), which is the difference between the maximal
generating degree \(d(I^n)\) of a homogeneous ideal \(I\) of
a polynomial ring and \(p(I)n\), where \(p(I)\) is the leading
coefficient of the asymptotically linear function \(d(I^n)\).
It is shown that any non-increasing numerical function
can be realized as a degree excess function, and there is
a monomial ideal \(I\) whose \(\epsilon(I;n)\) has exactly
a given number of local maxima. In the case of monomial
ideals, an upper bound on \(\epsilon(I;n)\) is provided. As
an application it is shown that in the worst case, the
so-called stability index of the Castelnuovo-Mumford
regularity of a monomial ideal \(I\) must be at least an
exponential function of the number of variables.
The second part is dealing with the maximal generating
degree \(d(\overline{I^n})\) of the integral closure of a
power \(I^n\) of a monomial ideal \(I\) for all \(n\). We give
effective lower and upper bounds on \(d(\overline{I^n})\).
A number \(n_0\) is also given such that \(d(\overline{I^n})\)
becomes a linear function of \(n\) when \(n\geq n_0\).
|
Dijous 18 de juliol, 12h00, Aula T1, FMI-UB.
|
| |
| |
