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Seminari de Geometria Algebraica 2021/2022 |
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Conferenciant
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Títol |
Data i hora |
Luis Narváez
Universidad de Sevilla
Contact:
josep.alvarez at upc.es
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Differentially admissible algebras on a field of positive characteristic
In the case of a base field of characteristic zero,
Núñez-Betancourt (2013) introduced a new class of algebras on such a field, which later became called
"differentially admissible algebras", which generalized a class of algebras that Mebkhout and the author had studied
in 1991 as a general framework where there is a good theory of Bernstein-Sato polynomials and holonomous D-modules.
In this session, we will propose a notion of "differentially admissible algebras" on a positive characteristic field
and show some of the properties they share.
Unfortunately, and as is well known, the theory of D-modules on a field of positive characteristic is more complicated
and "uncertain" than on characteristic zero, although recently some generalizations of the "classical theory" have been
introduced (holonomy, Bernstein-Sato polynomials, etc.) that are still being explored and that could provide interesting advances.
In principle, differentially admissible algebras on a field of positive characteristic would add to this area to be explored.
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Divendres 1 d'octubre, 15h00, Aula T2, FMI-UB.
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Roberto Gualdi
Universität Regensburg
Contact:
sombra at ub.edu
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On arithmetic divisors in Arakelov geometry
Arakelov geometry offers an enjoyable framework to develop an arithmetic
counterpart of the usual intersection theory. For varieties defined over
the ring of integers of a number field, and inspired by the geometric case, one
can define a suitable notion of arithmetic Chow groups and of an arithmetic
intersection product.
In a joint work with Paolo Dolce (Università degli Studi di Udine), we
prove an arithmetic analogue of the classical Shioda-Tate formula, relating
the dimension of the first Arakelov-Chow vector space of an arithmetic
variety to some of its geometric invariants. In doing so, we also characterize
numerically trivial arithmetic divisors, confirming part of a conjecture by
Gillet and Soulé.
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Divendres 8 d'octubre, 15h00, Aula T2, FMI-UB.
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Rick Miranda
Colorado State University, EEUU
Contact:
jroe at mat.uab.cat
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\((-1)\)-curves in \(\mathbb P^r\)
We'll review some ways to generalize \((-1)\)-curves to higher dimensions, and also review the Coxeter Group theory that applies to standard Cremona transformations based at points, and how the Weyl group is represented in the Chow ring.
The goal is to better understand criteria for when a general \((-1)\)-curve is a Cremona image of a line, and to apply these ideas to the Mori Dream Space cases of \(\mathbb P^r\) blown up at points.
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Divendres 22 d'octubre, 15h00, Aula T2, FMI-UB.
Sessió doble
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Ciro Ciliberto
Università di Roma Tor Vergata, Itàlia
Contact:
jroe at mat.uab.cat
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Extensions of canonical curves and double covers
A variety of dimension \(n\) is said to be extendable \(r\) times if it is the space section of a variety of dimension
\(n+r\) which is not a cone.
I will recall some general facts about extendability, with special regard for extensions of canonical curves to \(K3\) surfaces and Fano 3-folds.
Then I will focus on double covers and on their extendability properties.
In particular I will consider \(K3\) surfaces of genus 2, that are double covers of the plane branched over a general sextic.
A first result is that the general curve in the linear system pull back of plane curves of degree \(k\geq 7\) lies on a unique \(K3\) surface,
so it is only once extendable.
A second result is that, by contrast, if \(k\leq 6\) the general such curve is extendable to a higher dimensional variety.
In fact in the cases \(k=4,5,6\), this gives the existence of singular index \(k\) Fano varieties of dimensions 8, 5, 3, genera 17, 26, 37, and indices 6, 3, 1 respectively. For $k = 6$ one recovers the Fano
variety \(\mathbb P(3, 1, 1, 1)\), one of two Fano threefolds with canonical Gorenstein singularities with the maximal genus 37, found by Prokhorov.
A further result is that this latter variety is no further extendable.
For \(k=4\) and \(5\) these Fano varieties have been identified by Totaro.
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Divendres 22 d'octubre, 16h00, Aula T2, FMI-UB.
Sessió doble
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Organitzada per la RGAS
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Jornada de Jóvenes Doctores en Geometría Algebraica
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4-5 novembre, FMI-UB.
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Pablo González Mazón
INRIA, Nice, França
Contact:
cdandrea at ub.edu
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Trilinear birational maps
A trilinear rational map \(\phi: (\mathbb{P}_\mathbb{C}^1)^3 \dashrightarrow \mathbb{P}_\mathbb{C}^3\) is a rational map whose entries are trilinear polynomials.
If it admits an inverse rational map \(\phi^{-1}: \mathbb{P}_\mathbb{C}^3 \dashrightarrow (\mathbb{P}_\mathbb{C}^1)^3\), we say that \(\phi\) is a trilinear birational map. Similarly, a bilinear birational map is a birational map \(\psi: (\mathbb{P}_\mathbb{C}^1)^2 \dashrightarrow \mathbb{P}_\mathbb{C}^2\) with bilinear entries.
Bilinear birational maps can be efficiently used in Computer Aided Design, and are already understood well.
More explicitly, the set of bilinear birational maps is an irreducible locally closed hypersurface in \(\mathbb{P}^{11}\).
Moreover, the base scheme of \(\psi\) is always a closed point, and the minimal bi-graded free resolution of its base ideal is Hilbert-Burch, with two linear syzygies.
In our work, we answer the analogous questions about the algebraic structure, geometry of the base scheme, and resolution of the base ideal, for the 3-dimensional counterpart.
Namely, we give the set (of classes) of \(\text{Bir}_{(1,1,1)}\) of trilinear birational maps the structure of an algebraic set, and describe its irreducible components.
Secondly, we provide the complete list of possible base schemes for \(\phi\), up to isomorphism between these.
Finally, we classify all the possible tri-graded minimal free resolutions of the base ideal of \(\phi\).
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Divendres 12 de novembre, 15h00, Aula T2, FMI-UB.
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Francesc Fité
UB
Contact:
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On a local-global principle for quadratic twists of abelian varieties
Let \(A\) and \(A'\) be abelian varieties defined over a number field \(k\).
In the talk I will consider the following question: Is it true that \(A\) and \(A'\) are quadratic twists of one another if and only if
they are quadratic twists modulo \(p\) for almost every prime \(p\) of \(k\)?
Serre and Ramakrishnan have given a positive answer in the case of elliptic curves and a result of Rajan implies
the validity of the principle when \(A\) and \(A'\) have trivial geometric endomorphism ring. Without restrictions on the endomorphism ring,
I will show that the answer is affirmative up to dimension \(3\), but that it becomes negative in dimension \(4\).
The proof builds on Rajan's result and uses a Tate module tensor decomposition of an abelian variety geometrically isotypic
(the latter obtained in collaboration with Xavier Guitart).
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Divendres 17 de desembre, 15h00, Aula T2, FMI-UB.
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Pietro Pirola
Università di Pavia, Itàlia
Contact:
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A new proof of a theorem of Gordan and Noether
In a fundamental paper of 1876 Gordan and Noether fixed Hesse’s claim by showing that a complex projective hypersurface V(F) of the projective space of dimension n<4 is a cone if and only if the determinant of the Hessian of F is zero. One of the applications of this theorem is a Lefschetz-type result for Standard Gorenstein Artinian Algebras (SAGA) of codimension less than 5. Here we reverse the logical line of this implication. We first give a direct geometric and elementary proof the Lefschetz property, then we deduce Gordan Noether theorem using Macaulay’s theory. We also give some other application of our method. This work is a collaboration with Davide Bricalli and Filippo Favale.
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Divendres 18 de febrer, 15h10, Aula T2, FMI-UB.
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Paola Fredriani
Università di Pavia, Itàlia
Contact:
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Second fundamental form and Gaussian maps.
We will first overview some results obtained in collaboration with Colombo, Ghigi and Pirola on the second fundamental form of the Torelli map and its relation with the second Gaussian map. Then we will discuss some work in progress on higher Gaussian maps.
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Divendres 25 de febrer, 15h10, Aula T2, FMI-UB.
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Ignasi Mundet
UB
Contact:
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Actions of finite abelian groups vs actions of tori
If a space \(X\) supports and effective action of the \(n\)-torus \(T=(S^1)^n\), then
it also supports effective actions of all finite subgroups of \(T\). Is the
converse true? If \(X\) is a compact Kaehler manifold and all actions under
consideration are holomorphic, then the answer is YES. But if \(X\) is a
topological manifold and we consider continuous actions, then the answer
is most likely NO. In fact, there exists compact topological manifolds supporting
effective actions of \(Z/d\) for all odd integers d but supporting no effective
action of the circle. Somewhat in contrast to this, we will discuss the
following result. Suppose that \(X\) is a closed connected topological \(n\)-dimensional
manifold and that there exists a map of nonzero degree from \(X\) to the \(n\)-torus.
Suppose also that the fundamental group of \(X\) is virtually solvable. If \(X\) supports
effective actions of \((Z/r)^n\) for arbitrary large values of \(r\), then \(X\) is homeomorphic
to the \(n\)-torus. Hence \(X\) certainly supports an effective action of the n-torus.
The proof uses the topological rigidity of tori and some commutative
algebra.
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Divendres 4 de març, 15h10, Aula T2, FMI-UB.
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Lissa Nicklasson
Università di Genova, Itàlia
Contact:
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Toric ideals of polymatroids and White's conjecture
It was conjectured by White in 1980 that the toric ring associated to a matroid is defined by symmetric exchange relations. These are degree two relations reflecting the combinatorial structure of a matroid. This conjecture was extended to discrete polymatroids by Herzog and Hibi.
Several special cases of the conjecture has been proved, but it remains open in full generality. In this talk I will give an introduction to discrete polymatroids, White's conjecture, and related problems.
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Divendres 18 de març, 15h10, Aula T2, FMI-UB.
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Martín Sombra
UB
Contact:
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The mean height of the solution set of a system of polynomial equations
Bernstein’s theorem allows to predict the number of solutions of a system of Laurent polynomial equations in terms of combinatorial invariants. When the coefficients of the system are algebraic numbers, one can ask about the height of these solutions. Based on an on-going project with Roberto Gualdi (Regensburg), I will explain how one can approach this question using tools from the Arakelov geometry of toric varieties.
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Divendres 1 d'abril, 15h10, Aula T2, FMI-UB.
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Joaquim Roé
UAB
Contact:
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Irrational nef rays at the boundary of the Mori cone for very
general blowups of the plane
We develop techniques to find irrational rays at the boundary
of the Mori cone for linear systems on a general blowup of the plane,
and give examples of such irrational rays. This is joint work with Ciro
Ciliberto and Rick Miranda.
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Divendres 22 d'abril, 15h10, Aula T2, FMI-UB.
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Valentina Beorchia
Università degli Studi di Trieste, Itàlia
Contact:
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Eigenvector configurations of tensors: equations and geometry
The notion of eigenvector for a tensor fits very well in the context of classical algebraic geometry. We shall present some results obtained in collaboration with F. Galuppi, L. Venturello and R. M. Miro' Roig regarding configurations of eigenvectors,
which study has been motivated by a question posed by Ch. Ranestad and B. Sturmfels (Question 16 of the website http://cubics.wikidot.com/question:all)
From the projective point of view, the sets of eigenvectors correspond to certain zero-dimensional standard determinantal schemes. It is possible to give a characterization of the determinantal generators of their ideal, and to study the projective map associated with the linear system of the generators. We shall see that such a map is a generically finite cover in the plane case, and, classically, it has been first studied by Geiser and Laguerre. Its study allows to characterize the geometric properties of planar eigenconfigurations. In the case of eigenconfigurations in the space, the eigenschemes turn out to be complete intersection schemes on suitable smooth surfaces. For n=3 also the converse of this result holds.
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Divendres 6 de maig, 15h10, Aula T2, FMI-UB.
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François Ballay
UB
Contact:
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Lower bounds for Seshadri constants and successive minima
Let \(L\) be an ample line bundle on a complex projective variety \(X\). The successive minima of \(L\) form a series of invariants introduced by Ambro and Ito in 2020, that measure the local positivity of \(L\) at a very general point \(x\). The smallest minimum coincides with the Seshadri constant of \(L\) at \(x\), while the largest one is the width of \(L\) at \(x\). In this talk, I will present some lower bounds for Seshadri constants in terms of gaps between successive minima.
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Divendres 3 de juny, 15h10, Aula T2, FMI-UB.
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Andreas Mihatsch
Universität Bonn
Contact:
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Linear Arithmetic Fundamental Lemmas
Arithmetic Fundamental Lemmas (AFLs) are certain identities in arithmetic geometry that relate intersection numbers on moduli spaces of \(p\)-divisible groups with derivatives of orbital integrals. They form the local building blocks of global conjectures that relate cycles on Shimura varieties and derivatives of L-functions. In this talk, I will motivate and explain some new AFL identities for \(GL_n\), which is joint work with Qirui Li.
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Divendres 17 de juny, 15h10, Aula B6, FMI-UB.
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Angela Ortega
Humboldt University
Contact:
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Klein coverings of genus 2 curves
We consider étale 4 : 1 coverings of smooth genus 2 curves with the
monodromy group the Klein group. Depending on the values of the Weil
pairing restricted to the group defining the covering, we distinguish the
isotropic and non-isotropic case. In this talk we will discuss the
correspondence between the non-isotropic Klein coverings and the
(1,4)-polarised abelian surface.
As a consequence of this, one can show the existence of exactly four
hyperelliptic curves in a general (1,4)-polarised abelian surface. We
will also give several characterisations of the Klein coverings
(isotropic and non-isotropic) leading to the result that the
corresponding Prym maps are generically injective in both cases.
This is a joint work with Pawel Borówka.
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Dimarts 28 de juny, 15h30, Aula T1, FMI-UB.
Sessió doble
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Sukmoon Huh
Sungkyunkwan University
Contact:
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Torelli problem on generalized logarithmic sheaves
The logarithmic sheaf is, roughly speaking, a sheaf of differential 1-forms with logarithmic poles along a divisor on a variety. Ever since the notion is introduced by P. Deligne to define a mixed Hodge structure on the complement of the divisor with normal crossings, there has been several generalizations/modifications on this notion, for example by K. Saito, I. Dolgachev and F. Catanese. In this talk, we suggest a definition of its generalized version associated to a pair of curve and points over smooth projective surfaces, where the points are chosen to be contained in the curve and we blow up the surface along the points. On the other hand, we may also consider the blow up of the surface along the points off the curve and consider the logarithmic sheaf associated to the strict transform of the curve. We report some answers to the Torelli question on these two different logarithmic sheaves. This is a joint work with S. Marchesi, J. Pons-Llopis and J. Valles.
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Dimarts 28 de juny, 17h00, Aula T1, FMI-UB.
Sessió doble
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