How to use intersection theory to count geometric objects

par Tangi Pasquer (Sorbonne Université, IMJ-PRG)

mercredi 17 déc. 2025, 14:00 → 15:15 Europe/Paris

Salle Pierre Grisvard (IHP - Bâtiment Borel)

For a given space X, understanding how to put other spaces Y into X is generally a natural and interesting problem. In the framework of algebraic geometry, there is usually a finite dimensional family of embeddings of an algebraic variety Y into some given variety X. It may then be reasonable to try and understand the structure of this family. One can for instance fix the degree of the map, or ask for the image to intersect some subvariety of X in a prescribed way. I will explain how to use the tools of intersection theory to get some information about the structure of these families, in the case X is a surface and Y is a curve, with fixed degree and some point incidence conditions. One example will be lines on a smooth cubic surface, and the other genus g, degree d curves on a surface, all over an algebraically closed field.