# Encuentro de la red de Geometría Algebraica y Singularidades 2018

• #### Fechas:

Del 29/01/18 al 02/02/18

• #### Lugar:

Plaza de las Ciencias 3, Madrid, España (mapa)

Web del evento

## Descripción ↑ subir

Parte del encuentro puede verse en el canal de Youtube de la Facultad de Ciencias Matemáticas de la UCM

El encuentro consistirá en dos partes.

La primera parte estará dirigida a los jóvenes de la red. Las mañanas de lunes a jueves se dedicarán a un curso de Miles Reid (ver los detalles en el programa), y las tardes del lunes al miércoles se reservarán para conferencias de participantes jóvenes. Cada nodo propondrá un joven participante. Los jóvenes que no pertenezcan a ningún nodo podrán solicitar presentar una comunicación  escribiendo a los organizadores-

La segunda parte(jueves por la tarde y viernes por la mañana) se dedicará a conferencias de participantes senior, proponiendo cada nodo un participante.

AL final de la sesión del miércoles tendrá lugar una reunión de la red.

Se ruega registrarse en esta página.

Tenemos algunos fondos que podrían usarse para ayudar con los gastos  de aquellos participantes que necesiten financiación. Las solicitudes de comunicaciones o financiación deberán enviarse antes del 10 de enero.

Comité local: Enrique Arrondo (UCM) y Alejandro Melle (UCM)

Comité CIentífico: Enrique Arrondo (UCM), Alejandro Melle (UCM) y Luis Narváez (U. Sevilla).

Para cualquier información, escribir a arrondo@mat.ucm.es

## Programa ↑ subir

El programa en pdf puede descargarse aquí.

Lunes 29:

11:00-12:00 y 13:00-14:00 Curso de Miles Reid

16:00-17:00: Manuel González Villa (CIMAT,Guanajuato): Zeta Functions in Singularity Theory

18:00-18:45: Ángel González Prieto (Universidad Complutense de Madrid): Topological Quantum Field Theory for Hodge structures on character varieties
18:45-19:30: Juan Serrano de Rodrigo (Universidad de Zaragoza): A functorial representation of the Magnus representation to the category of 3-dimensional cobordisms.

Martes 30:

9:00-10:00 y 11:00-12:00 Curso de Miles Reid

15:30-16:30: Xavier Gómez Mont (CIMAT, Guanajuato): Algebra, Topology and Differential Structure in the
Brieskorn Lattice of an Isolated Hypersurface Singularity
16:30-17:15: Ángel Luis Muñoz Castañeda (Universidad de León): On the Compactification of the moduli space of principal G-bundles on nodal curves

17:30-18:15: Carlos Jesús Moreno Ávila (Universidad Jaume I): The cone of curves of a surface obtained by blowing-up a Hirzebruch surface
18:15-19:00: Guillem Blanco Fernández (Universidad Politécnica de Cataluña): Monomial generators of complete planar ideals

Miércoles 31:

9:00-10:00 y 11:00-12:00 Curso de Miles Reid

15:30-16:30: María de la Paz Tirado Hernández (Universidad de Sevilla): Integrability and Hasse-Schmidt derivations.

16:30-17:15: Martí Salat Moltò (Universidad de Barcelona): On the classification of monomial Togliatti systems.

18:00: Reunión de la red (en la sala 237)

Jueves 1:

9:00-10:00 y 11:00-12:00 Curso de Miles Reid

15:00-15:45: Luis Giraldo Suárez (Universidad Complutense de Madrid): On  vector fields with simply connected trajectories in the complex plane.
15:45-16:30: Jesús Soto Prieto (Universidad de Sevilla): (A little) Beyond the characteristic polytope.
16:30-17:15: Maria Alberich Carramiñana (Universidad Politécnica de Cataluña): Action of quadratic plane Cremona maps on planar polynomial differential systems and algebraic limit cycles

17:45-18:30: Enrique Artal Bartolo (Universidad de Zaragoza): Integrales impropias, polinomio de Bernstein-Sato y conjetura de Yano
18:30-19:15: Laura Costa (Universidad de Barcelona): Ulrich bundles on ruled surfaces
19:15-20:00: Raúl Oset Sinha (U. Valencia): The extra-nice dimensions

Viernes 2:

9:00-9:45: Carlos Galindo Pastor (Universidad Jaume I): Minimal valuations and Newton-Okounkov bodies of exceptional curve valuations.
9:45-10:30: Antonio Campillo  (Universidad de Valladolid): Poincaré series associated to non geometric filtrations
10:30-11:15: Miles Reid (Universidad de Warwick): Horikawa quintics

11:45-12:30: Enrique González Jiménez (Universidad Autónoma de Madrid): Growth of torsion groups of elliptic curve upon base change
12:30-13:15: Carlos Tejero Prieto (Universidad de Salamanca): Derived equivalences of abelian varieties and symplectic isomorphisms

## Ponentes ↑ subir

EL fichero pdf con los abstracts se puede descargar aquí.

Curso las mañanas de lunes 29 a jueves 1:

Miles Reid (Universidad de Warwick): On graded ring constructions of curves,surfaces and 3-folds.

 Abstract: There are some elementary parts to this, esp. how to use Riemann-Roch to calculate Hilbert series, and how to predict constructions. The version of RR for orbifolds is handled by Ice-Cream functions, which give beautiful and simple calculations that have many applictions. Constructing the varieties is also easy if you stick to codimension 1 or 2, but in codimension three you have to use the Buchsbaum-Eisenbud structure theorem. In codimension >= 4 this also breaks down, and one can only hope for partial results. The method of unprojection works in some cases, including the Tom and Jerry cases that construct many Fano 3-folds and can be applied with some modifications to canonical surfaces. The elementary introduction is contained in preprint form in my notes http://homepages.warwick.ac.uk/staff/Miles.Reid/surf/ + Graded rings. Most of the rest of the material is documented in one form or another on my website, for example, http://homepages.warwick.ac.uk/staff/Miles.Reid/codim4/ + Fun

Encuentro junior las tardes de lunes 29 a miércoles 31:

Conferenciantes senior:

-Xavier Gómez Mont (CIMAT, Guanajuato): Algebra, Topology and Differential Structure in the
Brieskorn Lattice of an Isolated Hypersurface Singularity

-Manuel González Villa (CIMAT, Guanajuato): Zeta Functions in Singularity Theory

Abstract: We will discuss some roles and relations of zeta functions of singularities of complex hypersurfaces and present some recent results involving motivic zeta functions and some generalizations.

Conferenciantes junior:

Abstract: Differential methods have been extensively used in the study of singularities of algebraic varieties. While many of these methods are well suited to study the singularities of an embedded variety, say $X \subset V$, defined over a perfect field $k$, they tend to fail when $k$ is non-perfect.
Consider a regular affine scheme of characteristic $p > 0$, say $V = \Spec(A)$. As we will see in this talk, some differential methods have a natural extension to the case in which the ring $A$ has a $p$-basis over $A^p$. However, in general, a regular ring $A$ does not admit a $p$-basis over $A^p$. In the talk we will prove a theorem of existence of $p$-basis for regular algebras of finite type over a non-necessarily perfect field $k$. This result will enable us to extend some of the differential methods to the case of varieties defined over a non-perfect field. In particular, given a regular affine variety $V = \Spec(A)$ defined over a non-perfect field $k$, and an ideal $I \subset A$, we will give a description of the set of points of $V$ where $I$ has order at least $n$ using differential operators on $A$.

-Guillem Blanco Fernández (Universidad Politécnica de Cataluña): Monomial generators of complete planar ideals

Abstract: In this talk we will show a geometric algorithm that computes a set of generators for any complete ideal in a smooth complex surface. These generators admit a presentation as monomials in a set of maximal contact elements associated to the minimal log-resolution of the ideal. As a result, we provide a geometric method to compute the integral closure of a planar ideal. We will see how this method applies to the study of families of complete planar ideals.
This is a joint work with Maria Alberich-Carramiñana and Josep Álvarez Montaner.

-Ángel González Prieto (Universidad Complutense de Madrid): Topological Quantum Field Theory for Hodge structures on character varieties

Abstract: Topological Quantum Field Theory is a powerful categorical tool that provides deep insight into the computation of topological invariants. In this talk, we will construct a lax monoidal TQFT that computes the mixed Hodge structure on the cohomology of representation varieties i.e.\ varieties of representations $\rho: \pi_1(M) \to G$, where $G$ is a complex algebraic group and $M$ is a compact manifold. For that, we will use a general quantization procedure by means of Saito's mixed Hodge modules.
Moreover, we will show how this TQFT gives a method of computation of $E$-polynomials of character varieties that improves the stratification technique developed by Logares, Mu\~noz and Newstead. Furthermore, it offers a new framework in which mirror symmetry conjectures for $E$-polynomials can be addressed.
Joint work with M. Logares and V. Muñoz.

-Carlos Jesús Moreno Ávila (Universidad Jaume I): The cone of curves of a surface obtained by blowing-up a Hirzebruch surface

Abstract: A complex rational surface $X$ can be obtained by blowing-up the projective plane, $\mathbb{P}^2_\mathbb{C}$, or a Hirzebruch surface, $\mathbb{F}_\delta$, where $\delta =0$ or $\delta\geq 2.$ There is no characterization of finite generation of the cone of curves of rational surfaces. In this talk, we will consider complex rational surfaces $X$ defined by plane valuations and will give valuative conditions which will be equivalent to the finite polyhedrality of the cone of curves $NE(X)$ of those surfaces.
This work is part of my PhD thesis that is being supervised by C. Galindo and F. Monserrat.

-Ángel Luis Muñoz Castañeda (Universidad de León): On the Compactification of the moduli space of principal G-bundles on nodal curves

-Martí Salat Moltò (Universidad de Barcelona):

Abstract: The study of Lefschetz properties has established multiple connections among different areas in mathematics such as commutative algebra, combinatorics or algebraic geometry. In particular it was recently discovered by Ottaviani, Mezzetti and Miró-Roig a link between ideals failing the weak Lefschetz property and varieties satisfying a Laplace Equation. These ideals are called Togliatti systems. There have been many efforts in classifying these ideals and partial results have been achieved in the monomial case. Here, we adress the problem on classifying monomial Togliatti systems focusing on the number of generators and give some results.

-Juan Serrano de Rodrigo (Universidad de Zaragoza): A functorial representation of the Magnus representation to the category of 3-dimensional cobordisms.

Abstract: Let $R$ be an integral domain and $G$ be a subgroup of its group of units. We consider the category $\mathsf{Cob}_G$ of $3$-dimensional cobordisms between oriented surfaces with connected boundary, equipped with a representation of their fundamental group  in $G$. Under some mild conditions on $R$, we construct a  monoidal functor from  $\mathsf{Cob}_G$ to the category $\mathsf{pLagr}_R$ consisting of pointed Lagrangian relations'' between skew-Hermitian $R$-modules. We call it the Magnus functor'' since it contains the Magnus representation of mapping class groups as a special case. Our construction is inspired from the work of Cimasoni and Turaev on the extension of the Burau representation of braid groups to the category of tangles. The study and computation of the Magnus functor is carried out using classical techniques of low-dimensional topology.

-María de la Paz Tirado Hernández (Universidad de Sevilla): Integrability and Hasse-Schmidt derivations.

Abstract: Given a k-algebra A, decide whether a k-derivation of A is integrable or not (in the sense of Hasse-Schmidt) is a difficult question. In this talk, we will defined the Hasse-Schmidt derivations and we will give some results to solve, in some cases, this problem

Encuentro senior jueves 1 por la tarde y viernes 2 por la mañana:

-Maria Alberich Carramiñana (Universidad Politécnica de Cataluña): Action of quadratic plane Cremona maps on planar polynomial differential systems and algebraic limit cycles

Abstract: The Cremona group of birational transformations of the complex projective plane acts on the space of planar polynomial differential systems. This action is not compatible with the degree of the differential system. When the degree of a differential system is invariant under the action of a plane Cremona map $\Phi$, we say that the differential system is numerically invariant by $\Phi$. We will characterize numerically invariant planar polynomial differential systems by quadratic plane Cremona map. As a consequence, we will provide a new family of quadratic systems having an algebraic limit cycle of degree 5.

-Enrique Artal Bartolo (Universidad de Zaragoza): Improper integrals, Bernstein-Sato polynomial and Yano's conjecture, (joint work with Pi. Cassou-Nogués, I. Luengo, A. Melle-Hernández)

Abstract: Bernstein-Sato polynomial of a singularity is an analytic, but not topological, invariant of the singulary and related with the monidromy and extremely dificult to compute. In 1982 and for irreducible  plane singularities Yano conjectured how was the Bernstein-Sato of a germ  which is generic equisingular.  In  1986 P. Cassou-Nogues proved the conjecture in the 1-Puisex case and recently the authors proved in case 2-Puiseux and monodromy having distint  eigenvalues using improper intregrals.
Such integrals allow to generalice Yano's conjecture fir reduced curves.

-Antonio Campillo  (Universidad de Valladolid): Poincaré series associated to non geometric filtrations

Abstract: En los últimos 20 años se han calculado sistemáticamente series de Poincaré en varias variables, siguiendo los trabajos iniciales conjuntos con F.Delgado y S.Gusein-Zade, obteniendo de ellas la suficiente información para determinar la parte de topología o de geometría de numerosos tipos de  variedades o de singularidades que sirven para clasificarlas. Los métodos de cálculo han sido posibles gracias a que las series de Poincaré utilizadas están asociadas a filtraciones por multi-índices de ideales de valoración, es decir por filtraciones 'geométricas'. Sin embargo se han encontrado otras filtraciones naturales, que no son geométricas, cuyas series de Poincaré juegan un papel similar en otros problemas de clasificación. Con énfasis en dichas filtraciones no geométricas, ofrecemos una panorámica general sobre las series de Poincaré.

-Laura Costa (Universidad de Barcelona): Ulrich bundles on ruled surfaces

Abstract: An Ulrich bundle on a smooth projective variety is a vector bundle that admits a completely linear resolution as a sheaf on the projective space. They appeared in commutative algebra, being associated to maximal Cohen Macaulay graded modules with maximal number of generators. Years ago, Eisenbud and Schreyer posted a question concerning  the existence of Ulrich bundles on any projective variety and they also asked for their minimal rank.
We are far from a general solution to this question and the existence is proved in many particular cases using the special geometry of the base variety. Particular  interest fall on the existence of  special Ulrich bundles on a variety X due to their connection with pfaffian
presentations of the Cayley-Chow form of X . During my talk I will explain some results concerning the existence of special rank two Ulrich bundles on ruled surfaces.

-Carlos Galindo Pastor (Universidad Jaume I): Minimal valuations and Newton-Okounkov bodies of exceptional curve valuations

Abstract: Bouckson, K\"{u}ronya,  Maclean and Szemberg introduced an analogue of the Seshadri constant $\hat{\mu}(\nu)$ for real valuations $\nu$ of the projective plane. It is always true that  $\hat{\mu}(\nu) \geq \sqrt{1/\mathrm{vol}(\nu)}$, where $\mathrm{vol}(\nu)$ means volume of the valuation. When the equality is satisfied, $\nu$ is called minimal. We state a Nagata conjecture involving minimal valuations which implies the original conjecture and it is implied by the Greuel-Lossen-Shustin conjecture. We also provide some evidence supporting the veracity of that conjecture.
We also give a complete description of the Newton-Okounkov bodies of flags $\{X \supset E_r \supset \{q\}\}$ attached to surfaces $X$ and exceptional divisors $E_r$ defined by divisorial valuations $\nu$ of the projective plane. We prove that the fact that $\nu$ is minimal can be characterized in terms of the vertices of the corresponding Newton-Okounkov body. We also show that these bodies are triangles or quadrilaterals and explain the reason for this difference.
Results in this talk have been obtained together with F. Monserrat, J. Moyano and M. Nickel.

-Luis Giraldo Suárez (Universidad Complutense de Madrid): On  vector fields with simply connected trajectories in the complex plane

Abstract

-Enrique González Jiménez (Universidad Autónoma de Madrid): Growth of torsion groups of elliptic curve upon base change

Abstract: One of the main goals in the theory of elliptic curves is to characterize the possible torsion structures over a given number field, or over all number fields of a given degree. One of the milestone in this subject was the characterization of the rational case given by Mazur in 1978. Later, the quadratic case was obtained by Kamienny, Kenku and Momose in 1992. For greater degree a complete answer for this problem is still open, although there have been some advances in the last years.
The purpose of this talk is to shed light on how the torsion group of an elliptic curve defined over the rationals grows upon base change.
This is an ongoing project partially joint with Á. Lozano-Robledo, F. Najman and J. M. Tornero.

-Raúl Oset Sinha (Universidad de Valencia): The extra-nice dimensions

Abstract: We define the extra-nice dimensions and prove that the subset of stable 1-parameter families in $C^{\infty}(N\times [0,1],P)$ is dense if and only if the pair of dimensions $(\dim N,\dim P)$ is in the extra-nice dimensions. This result is parallel to Mather's characterization of the nice dimensions as the pairs $(n,p)$ for which stable maps are dense. The extra-nice dimensions are characterized by the property that discriminants of stable germs in one dimension higher have $\mathscr A_e$-codimension 1 hyperplane sections. They are also related to the simplicity of $\mathscr A_e$-codimension 2 germs. We give a sufficient condition for any $\mathsc A_e$-codimension 2 germ to be simple and give an example of a corank 2 codimension 2 germ which is not simple. Then we establish the boundary of the extra-nice dimensions. Finally we answer a question posed by Wall about the codimension of non-simple maps.

-Miles Reid (Universidad de Warwick): Horikawa quintics

Abstract: A Horikawa quintic is a polarised n-fold (X, A) with A^n = 5, h^0(A) = n+2 and K_X = (3-n)A.  Assume that X is nonsingular, or has mild singularities (for example, at worst ordinary double locus in codimension >= 3).  One expects initially the linear system |A| to be free, defining an isomorphism with a quintic hypersurface X = X_5 in PP^{n+1} (Type I), but there is another possibility: |A| may have a single transverse base point P in X, and define a double cover X - -> Q to a quadric Q in PP^{n+1}.  It turns out that Q can only have rank 4 (Type II_a) or rank 3 (Type II_b).  The most interesting case is the deformation theory of X of Type II_b: this has small deformations to Type I and to Type II_a, and for n >= 3 these are topologically different (they have different Betti numbers). When n = 3 this provides an interesting case of "conifold transition".

-Jesús Soto Prieto (Universidad de Sevilla): (A little) Beyond the characteristic polytope.

Abstract: In this talk, I will present the concept of blurry Newton-Hironaka polytope and we will see what information we can extract from it.
We recently introduced the blurry polytope with the try and find relevant information for the combinatoric resolution of singularities.
This is joint work with H. Cobo and José M. Tornero, recently accepted in the Kyoto Journal of Mathematics.

-Carlos Tejero Prieto (Universidad de Salamanca): Derived equivalences of abelian varieties and symplectic isomorphisms

Abstract: We study derived equivalences of Abelian varieties in terms of their associated symplectic data. For simple Abelian varieties over an algebraically closed field of characteristic zero we prove that the natural correspondence introduced by Orlov, which maps equivalences to symplectic isomorphisms, is surjective.

## Vídeos ↑ subir

• 1_REID_On graded ring constructions of curves,surfaces and 3-folds
• 2_REID_On graded ring constructions of curves,surfaces and 3-folds
• 3_REID_On graded ring constructions of curves,surfaces and 3-folds
• 4_REID_On graded ring constructions of curves,surfaces and 3-folds
• 5_REID_On graded ring constructions of curves,surfaces and 3-folds
• 6_REID_On graded ring constructions of curves,surfaces and 3-folds
• 7_REID_On graded ring constructions of curves,surfaces and 3-folds
• 8_REID_On graded ring constructions of curves,surfaces and 3-folds

• ago

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### Matemàtiques per als graus de Biologia i Bioquímica - Bloc I

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### Curs en línia de preparació per al grau: Matemàtiques

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### Matemàtiques per als graus de Biologia i Bioquímica - Bloc II

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