LECTURE SERIES ANNOUNCEMENT - "Inclusion-exclusion in topology and (kind(s) of) arithmetic”
The first talk will cover all the prerequisites. The other talks should be accessible to anyone who knows cohomology and has heard of spectral sequences.
3:30 - 4:30 pm, Seminar Hall
Points and lines on cubic surfaces
Ronno Das
University of Chicago.
29-08-19

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Abstract

The Cayley-Salmon theorem states that every smooth cubic surface in CP^3 has exactly 27 lines. Their proof is that marking a line on each cubic surface produces a 27-sheeted cover of the 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 of incredibly singular `discriminant' hypersurfaces. I will explain how to compute the rational cohomology of these spaces. I'll also explain how these purely topological theorems have purely arithmetic consequences: the average smooth cubic surface over a finite field F_q contains 1 line and q^2 + q + 1 points.