Title: Del Pezzo Surfaces Over Finite Fields
Abstract:
In his book Cubic Forms, Manin proves that over any field k a degree d del
Pezzo surface V is rational when d>4 and unirational when d>1 if there exists
a point in V(k) which is not on any exceptional curve. Koll\'ar later proved
that cubic surfaces over finite fields are always unirational. This leads one
to wonder if degree two del Pezzo surfaces over finite fields are unirational.
One way of starting to answer this question is to determine how large a finite
field must be in order for Manin's unirational construction to always hold.
A del Pezzo surfaces over a finite field k is called "full" if all of the lines
on V(\bar{k}) are actually defined over the ground field k and all of the point
of V(k) are on these lines. Hirschfeld has classified full del Pezzo surfaces
up to degree 3. In this talk I will present work towards a classification of
full degree 2 del Pezzo surfaces. When the characteristic of k is not 2, these
are double covers of quartic plane curves Q whose 28 bitangents are defined
over k and the number of solutions to w^2=Q in P(1,1,1,2)(F_q) is q^2+8q+1.
This is joint work with Kris Reyes and Zach Scherr.