25/04, IMPA. Sala: Auditório 3
10:3011:30 
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. subsemigroups 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.
 11:3014:00  Lunch break
 14:0015:00 
Anthony VárillyAlvarado (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 "wellbehaved" varieties, like rational surfaces, and arithmetically "wild" varieties, like surfaces of general type.
 15:0015:30  Café

