Del Pezzo surfaces over finite fields

Degree of Unirationality for Del Pezzo Surfaces over Finite Fields

We address the question of the degree of unirational parameterizations of degree four and degree three del Pezzo surfaces. Specifically we show that degree four del Pezzo surfaces over finite fields admit degree two parameterizations and minimal cubic surfaces admit parameterizations of degree 6. It is an open question whether or not minimal cubic surfaces over finite fields can admit degree 3 ...

Quartic Del Pezzo Surfaces over Function Fields of Curves

The geometry of spaces of rational curves of low degree on Fano threefolds is a very active area of algebraic geometry. One of the main goals is to understand the Abel-Jacobi morphism from the base of a space of such curves to the intermediate Jacobian of the threefold. For instance, does a given space of curves dominate the intermediate Jacobian? If so, what are the fibers of this morphism? Do...

Rational Point Counts for del Pezzo Surfaces over Finite Fields and Coding Theory

The goal of this thesis is to apply an approach due to Elkies to study the distribution of rational point counts for certain families of curves and surfaces over finite fields. A vector space of polynomials over a fixed finite field Fq gives rise to a linear code, and the weight enumerator of this code gives information about point count distributions. The MacWilliams theorem gives a relation b...

Nonnormal Del Pezzo Surfaces

0.1 Throughout this paper, a del Pezzo surface is by definition a connected, 2-dimensional, projective k-scheme X,OX(1) that is Gorenstein and anticanonically polarised; in other words, X is Cohen–Macaulay, and the dualising sheaf is invertible and antiample: ωX ∼= OX(−1). For example, X = X3 ⊂ P 3 an arbitrary hypersurface of degree 3. Under extra conditions, del Pezzo surfaces are interesting...

Del Pezzo surfaces and minimal models for surfaces over nonclosed fields