Ronno Das : Points and lines on cubic surfaces

Posted by Izzet Coskun , part of the Algebraic Geometry Seminar.

At

Feb. 6, 2019, 4 p.m.

In

427 SEO

Abstract

The Cayley-Salmon theorem states that every smooth cubic surface S in $\mathbb{C}\mathbb{P}^3$ has exactly 27 lines. Their proof is that marking a line on each cubic surface produces a 27-sheeted cover of the moduli space M of smooth cubic surfaces. Similarly, marking a point produces a 'universal family' of cubic surfaces over M. One difficulty in understanding these spaces is that they are complements in affine space of incredibly singular hypersurfaces. In this talk I will explain how to compute the rational cohomology of these spaces. I'll then explain how these purely topological theorems have (via the machinery of the Weil Conjectures) purely arithmetic consequences: the typical smooth cubic cubic surface over a finite field $F_q$ contains 1 line and $q^2 + q + 1$ points.