Seminario de Geometría Algebraica

2017-07-19
14:00-15:20hrs.

Marcello Bernardara. Universidad de Toulose
Derived categories, cycles, and rationality I
Sala 1
Abstract:
In the last decades, semiorthogonal decompositions of derived categories of coherent sheaves have been considered as a possible tool to attack birationality questions. Since the seminal work of Bondal and Orlov, the work of many authors, Kuznetsov above all of them, has made clear what we should expect and which are the main technical problems. In particular, one would expect that the derived category of a rational variety must be decomposed in "codimension 2" subcategories.

The aim of these lectures is to set definitions and property that could give a sense to the above considerations, and to show that this conjectural obstruction is stronger than the known classical ones in cases such as surfaces or Fano threefolds, and closely related for some particular fourfold (cubic, Gushel'-Mukai). A particular attention will be put on constructions related to cycles and motives, such as intermediate Jacobians and decomposition of the diagonal.
https://raizdepie.wixsite.com/algebraic-geometry

2017-07-19
15:40-17:00hrs.

Paolo Stellari. Universidad de Milan
Cubic threefolds and fourfolds: geometry and homological algebra I
Sala 1
Abstract:
We revisit some classical results concerning the geometry of smooth cubic hypersurfaces of dimension 3 and 4 by means of modern techniques involving derived categories and Bridgeland stability conditions. The examples we want to examine are: the Fano variety of lines, twisted cubics curves on cubic fourfolds and the Torelli theorem for cubic threefolds and fourfolds.g derived categories and Bridgeland stability conditions. The examples we want toWe revisit some classical results concerning the geometry of smooth cubic hypersurfaces of dimension 3 and 4 by means of modern techniques involving derived categories and Bridgeland stability conditions. The examples we want to examine are: the Fano variety of lines, twisted cubics curves on cubic fourfolds and the Torelli theorem for cubic threefolds and fourfolds. are: the Fano variety of lines, twisted cubics curves on cubic fourfolds and the Torelli theorem for cubic threefolds and fourfolds.
https://raizdepie.wixsite.com/algebraic-geometry

2017-07-14
14:00-16:00hrs.

Sukhendu Mehrotra. PUC
Introducción a las categorías derivadas en la geometría algebraica I-II
Sala 2
Abstract:
Estas son charlas preparatorias para los mini-cursos de la próxima semana.

2017-07-04
16:00hrs.

Héctor Pastén. Harvard University
Avances recientes en la conjetura abc
Sala 2, Facultad de Matemáticas
Abstract:
Luego de un breve recuento de los métodos y resultados existentes en relación a la conjetura abc, voy a dar un esbozo una técnica nueva basada en curvas de Shimura y voy a explicar el tipo de resultados incondicionales que permite obtener.

2017-06-16
14:30hrs.

Sukhendu Mehrotra. PUC Chile
Exceptional Bundles From Surface Degenerations II
sala 2

2017-06-09
14:30hrs.

Sukhendu Mehrotra. PUC Chile
Exceptional Bundles From Surface Degenerations I
sala 2

2017-06-02
14:30hrs.

José Yáñez. PUC Chile
Puntos de Acumulación de K^2 en Superficies Estables
sala 2, Facultad de Matemáticas PUC

2017-05-26
14:30hrs.

Sergio Troncoso. PUC Chile
Theory of Peeling
sala 2

2017-05-19
14:30-16:00hrs.

Sönke Rollenske. U Marburg
Geometry of Stable Surfaces III
Sala 2, Facultad de Matemática

2017-05-12
16:00 - 17:00hrs.

Sönke Rollenske. U Marburg
Geometry of Stable Surfaces II
Sala 2, Facultad de Matemáticas

2017-05-12
14:30-15:45hrs.

Sönke Rollenske. U Marburg
Geometry of Stable Surfaces I
Sala 2, Facultad de Matemáticas
Abstract:
Stable surfaces are the two-dimensional analogue of stable curves: they are the singular surfaces that are parametrised by a natural compactification of the Giesecker moduli space of surfaces of general type. In the lectures I will illustrate some basic techniques needed to deal with such surfaces. We will see that any closer look at examples quickly takes us to classical questions in algebraic geometry.

2017-05-05
14:30hrs.

Sergio Troncoso. PUC Chile
Descripción del Log Mmp
Sala 2, Facultad de Matemáticas PUC
http://www.mat.uc.cl/~urzua/

2017-04-21
14:30hrs.

Sergio Troncoso. PUC
Ejecutando Mmp Explícitamente y Su Resultado Final
Sala 2 (Facultad de Matemáticas PUC)

2017-04-07
14:30hrs.

José Ignacio Yáñez. PUC
Pares Log Canonical y Teorema del Cono
Sala 2 PUC

2017-03-31
14:30hrs.

José Ignacio Yáñez. PUC
Introducción al Minimal Model Program
Sala 2
Abstract:
Este semestre estudiaremos el Minimal Model Program (o teoría de Mori) para superficies algebraicas con borde. El objetivo principal es entender la demostración del teorema de boundedness dada por Alexeev a comienzos de los 90s. Este teorema implica que la compactificación del espacio de moduli de superficies algebraicas de tipo general, dada por Kollár y Shepherd-Barron y generalización de la compactificación de Deligne-Mumford para curvas, define una variedad proyectiva. En particular, las singularidades de las correspondientes superficies están acotadas a través de los números de Chern (de hecho  K^2 basta), formando una lista finita. ¿Cuál es esa lista para K^2 dado?

La idea es desarrollar todos los prerequicitos para poder entender los detalles de la demostración.

En esta charla se definirán superficies con bordes, las singularidades involucradas junto a sus discrepancias, el teorema del cono, y todo aspecto básico relacionado con MMP.

2017-03-24
14:30hrs.

Fabien Trihan. U Sophia, Japón
Abelian varieties over function fields and related conjecture
sala 2
Abstract:

2017-03-10
15:00hrs.

Dulip Piyaratne. Kavli Ipmu, University of Tokyo
Stability conditions on derived categories of varieties
Sala 2
Abstract:
The aim of this talk is to discuss Bridgeland stability conditions on smooth projective varieties. The notion of stability appears in many guises and it is fundamental to geometric invariant theory. There is a systematic way of studying stability conditions due to Bridgeland and his approach is essentially an abstraction of the usual slope stability for sheaves. This categorical stability notion was introduced in order to understand the work of Douglas on Pi-stability in superconformal field theories. However, construction of Bridgeland stability conditions on higher dimensional varieties is a challenging problem, and from string-theoretic point of view, stability conditions on smooth projective threefolds are the most interesting ones. In this talk, I will recall some important notions associated to derived categories of varieties and stability conditions, with special emphasis on curves and surfaces.

2017-03-10
16:00hrs.

Dulip Piyaratne. Kavli Ipmu, University of Tokyo
Stability conditions on derived categories of varieties II
Sala 2
Abstract:
In this talk I will discuss stability conditions on projective threefolds. A conjectural construction for any 3-fold was introduced by Bayer, Macri and Toda, and the problem is reduced to proving so-called Bogomolov-Gieseker type inequality holds for certain stable objects in the derived category. It has been shown to hold for some 3-folds including Fano 3-folds of Picard rank one. However, Schmidt and Martinez gave some counter-examples for Fano 3-fold of higher Picard rank. In this talk, I will explain how to modify the original conjectural inequality in order to get a family of Bridgeland stability conditions, and why it holds for general Fano 3-folds.

2016-12-01
16:15-17:15hrs.

Sukhendu Mehrotra. PUC Chile
The Franchetta Conjecture (Cont.)
Fac. de Matematicas, Universidad de Chile

2016-11-24
16:15-17:15hrs.

Sukhendu Mehrotra. Puc, Chile
The Franchetta Conjecture
Universidad de Chile
Abstract: