
Seminari de Geometria Algebraica 2017/2018 


Conferenciant

Títol 
Data i hora 
Eduard CasasAlvero
Universitat de Barcelona

On the analytic classification of irreducible plane curve
singularities
I will present new results regarding which Puiseux
coefficients the analytic type of a complex irreducible plane curve
singularity depends on.

Divendres 29 de setembre, 15h, Aula T2, FMIUB

Ignasi Mundet
Universitat de Barcelona

Symplectic finite group actions on \(\Large{S^2\times T^2}\)
Let \( X=S^2\times T^2\).
For any symplectic form \(\omega\) of \(X\) we denote by \(Symp(X,\omega)\) the group of symplectomorphisms of \( (X,\omega)\).
In this talk we will explain different results on the finite subgroups of \( Symp(X,\omega)\).
The main results are:
 for any \(\omega\) there exist infinitely many isomorphism classes of finite subgroups of
\(Diff(X)\) which are not represented by any finite subgroup of \(Symp(X,\omega)\);
 for any \(\omega\) there exists another symplectic form \(\omega'\)
and a finite subgroup of \(Symp(X,\omega')\) which is not isomorphic to any finite subgroup of \(Symp(X,\omega)\).
We will sketch the proofs, which use the theory of pseudoholomorphic curves.
We will make an effort to make the talk understandable without previous knowledge of symplectic geometry.

Divendres 6 d'octubre, 15h, Aula T2, FMIUB

Gianfranco Casnati
Politecnico di Torino

Ulrich bundles on some classes of surfaces in projective spaces
An Ulrich bundle on a variety in the projective space is a vector bundle whose associated module of sections has a linear resolution over the projective space.
Ulrich bundles have many interesting properties and their existence on a fixed variety has several geometric consequences for it.
Ulrich bundles on curves can be easily described.
This is no longer true for Ulrich bundles on a surface, though many results are known.
In the talk we focus our attention on this latter case proving the existence of Ulrich bundles on some classes of surfaces, giving some results on the size of the families of Ulrich bundles on them and sometimes dealing with their stability properties.

Divendres 27 d'octubre, 15h, Aula T2, FMIUB

Fatmanur Yıldırım
INRIA SophiaAntipolis & UB

Finite fibers of multigraded rational maps on \(\Large{\mathbb{P}^3}\)
I will present a new method to study the fibers of a rational multigraded map \(\Psi\) from \(\mathbb{P}^2\times\mathbb{P}^1\) (or \(\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\)) to \(\mathbb{P}^3\), which is a joint work with Nicolás Botbol, Laurent Busé and Marc Chardin.
My motivation is to compute the distance from a point \(p\in\mathbb{R}^3\) to an algebraic rational surface \(\mathcal{S}\in\mathbb{R}^3\).
Firstly, from a parametrization of \(\mathcal{S}\), I will construct a homogeneous parametrization \(\Psi\) for the normal lines to \(\mathcal{S}\), where \(\Psi\) is a multigraded rational map from \(\mathbb{P}^2\times\mathbb{P}^1\) (or \(\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\)) to \(\mathbb{P}^3\).
Then, I will describe the fibers over a point \(p\in\mathbb{P}^3\).
After that, I will state a matrix \(\mathcal{M}(\Psi)_{\nu}\) of a certain multidegree \(\nu\) obtained by the syzygies of ideal generated by the coordinates of \(\Psi\).

Divendres 3 de novembre, 15h, Aula T2, FMIUB

Meritxell Saez Cornellana
University of Copenhagen

Positive solutions to linear systems
slides
Usually in applications, where variables represent measurable quantities, only nonnegative solutions are meaningful.
Hence, criteria to decide the positivitity of the solutions to a system of equations are desired.
I will present some of the known results in this direction and a new criteria for linear systems based on a multidigraph associated with the equations.
The main motivation for this work has been on the application to biochemical reaction networks that I will briefly present.

Divendres 10 de novembre, 15h, Aula T2, FMIUB

Jose Ignacio Burgos Gil
ICMAT (CSIC)

Where do little elliptic curves go?
Let C be a curve over \(\mathbb{Q}\) provided with an integral model, an ample line bundle on the model and a semipositive metric.
To these data we can associate the height of the curve and the height of every algebraic point of the curve.
The essential minimum of the curve is the minimal accumulation point of the height of the algebraic points.
The essential minimum is a mysterious and elusive invariant.
A result of Zhang shows that the essential minimum has a lower bound in terms of the height of the curve, and an example of Zagier shows that there can be several isolated values of the height below the essential minimum.
When C is the modular curve, the line bundle agrees with the bundle of modular forms and the metric is the WeilPetersson metric, then the height of an algebraic point agrees with the stable Faltings height of the corresponding elliptic curve.
In this talk we will discuss methods of proving lower and upper bounds for the essential minimum and apply them to the modular curve, giving a partial description of the spectrum of the stable Faltings height of elliptic curves.
This is joint with with Ricardo Menares and Juan RiveraLetelier.

Divendres 17 de novembre, 15h, Aula T2, FMIUB

Ana Belén de Felipe
Universitat de Barcelona

Topology of spaces of valuations and geometry of singularities
Given an algebraic variety X defined over a field k, the space of all valuations of the field of rational
functions of X extending the trivial valuation on k is a projective limit of algebraic varieties. This space
had an important role in the program of Zariski for the proof of the existence of resolution of singularities.
In this talk we will consider the subspace RZ(X,x) consisting of those valuations which are centered in a
given closed point x of X and we will focus on the topology of this space. In particular we will concentrate
on the relation between its homeomorphism type and the local geometry of X at x. We will characterize
this homeomorphism type for regular points and normal surface singularities. This will be done by studying
the relation between RZ(X,x) and the normalized nonArchimedean link of x in X coming from the point of
view of Berkovich geometry.

Divendres 24 de novembre, 15h, Aula T2, FMIUB

Julio José Moyano Fernández
Universitat Jaume I de Castelló

Some instances of generating series in algebraic geometry
The use of zeta functions has a long tradition in algebraic geometry and related fields, coming back to the classical works of Riemann, Dedekind or Hilbert. They offer an elegant way to encode many invariants of the typical objects under consideration. In this talk we will focus on certain series defined in local contexts, namely the socalled Poincaré series attached to curve singularities. We will describe the relation of those Poincaré series with some zeta functions coming from a rather numbertheoretical setting, and shed some light on the role that they play in global contexts. The talk will contain some pieces of collaborations with F. Delgado, A. Melle and W. Zúñiga.

Divendres 1 de desembre, 15h, Aula T2, FMIUB

Matthias Nickel
Goethe Universität Frankfurt am Main

NewtonOkounkov bodies of exceptional curve valuations
NewtonOkounkov bodies are convex bodies associated to line bundles on projective varieties that capture positivity properties of the line bundle in question. Let p be a closed point in \({\mathbb{P}}_{\mathbb{C}}^2\) and consider a surface obtained by a sequence of finitely many blowups of points where we start with p and always blow up a point in the exceptional divisor created last.
Our result is that the NewtonOkounkov body of the pullback of \(\mathcal{O} _{\mathbb{P}^2}(1)\) with respect to the flag given by the last exceptional divisor and a point on it is either a triangle or a quadrilateral. Furthermore a characterization of both cases can be given using the dual graph and the supraminimal curve.
This is joint work with Carlos Galindo, Francisco Monserrat and Julio José MoyanoFernández.

Divendres 15 de desembre, 15h, Aula de l'IMUB, FMIUB

Simone Marchesi
IMECC  UNICAMP, Brasil 
Line arrangements and vector bundles
One of the most famous and interesting conjectures regarding line arrangements (we will restrict to the projective plane case) is the so called Terao's conjecture, which basically states that the freeness of an arrangement depends on its combinatorics.
If the conjecture is not true, than the arrangement would be associated to a vector bundle whose jumping locus is related to a 0dimensional scheme in the projective plane.
In this talk we will focus on the case when such scheme is a point, characterizing the associated vector bundles and relating, through examples, this jumping point to the line arrangement.

Divendres 12 de gener, 15h, Aula T2, FMIUB

David Marín
Universitat Autònoma de Barcelona 
Foliations and webs with continuous symmetries on complex projective surfaces
We will describe the structure of foliations and webs on complex projective surfaces which are invariant by a germ of birational flow.
We will discuss in detail the case of the projective plane, characterizing planar projective webs with many infinitesimal symmetries.
This is a joint work with Marcel Nicolau.

Divendres 19 de gener, 15h, Aula T2, FMIUB

Antonio Macchia
Università degli Studi di Bari 
Slack ideals of Polytopes
We introduce a canonical realization space for a polytope that arises as the positive part of a real variety.
The variety is determined by the socalled slack ideal of the polytope, that encodes its combinatorics.
The slack ideal offers new ways to categorize polytopes in terms of complexity.
Our constructions provide a uniform computational framework for several classical questions about polytopes such as rational realizability, projectively uniqueness, nonprescribability of faces, and realizability of combinatorial polytopes.
The simplest slack ideals are toric.
We identify the toric ideals that arise from projectively unique polytopes: all dpolytopes with d+2 facets or vertices have such slack ideals but there are more interesting examples in this set.
We illuminate the relationship between projective uniqueness and toric slack ideals using new and classical examples.
(Joint work with João Gouveia, Rekha Thomas and Amy Wiebe)

Divendres 26 de gener, 15h, Aula T2, FMIUB


Workshop on Complex Algebraic Geometry  Pirola 60th

5  9 de febrer 2018, Aula B5, UB


