25/04, IMPA. Sala: Auditório 3
Ethan Cotterill (UFF),
Dimension counts for singular rational curves |
Resumo. Rational curves are essential tools for classifying algebraic varieties. Establishing dimension bounds for families of embedded rational curves that admit singularities of a particular type arises arises naturally as part of this classification. Singularities, in turn, are classified by their value semigroups. Unibranch singularities are associated to numerical semigroups, i.e. sub-semigroups of the natural numbers. These fit naturally into a tree, and each is associated with a particular weight, from which a bound on the dimension of the corresponding stratum in the Grassmannian may be derived. Understanding how weights grow as a function of (arithmetic) genus g, i.e. within the tree, is thus fundamental. We establish that for genus g \leq 8, the dimension of unibranch singularities is as one would naively expect, but that expectations fail as soon as g=9. Multibranch singularities are far more complicated; in this case, we give a general classification strategy and again show, using
semigroups, that dimension grows as expected relative to g when g \leq 5. This is joint work with Lia Fusaro Abrantes and Renato Vidal Martins.
|11:30-14:00|| Lunch break
Anthony Várilly-Alvarado (Rice), Note a mudança do horario!
Explicit Arithmetic on Algebraic Surfaces |
Resumo. The geometric complexity of a variety is a good proxy for its arithmetic complexity. Using the classification of algebraic surfaces as a guide for geometric complexity, we will discuss explicit techniques for computing cohomological obstructions to the existence and distribution of rational points on algebraic surfaces, with a view toward identifying a boundary between arithmetically "well-behaved" varieties, like rational surfaces, and arithmetically "wild" varieties, like surfaces of general type.