Federico Mangano: Del Pezzo Surfaces And Points In The Plane

Abstract

This thesis will focus on the study of the relationship that exists between Del Pezzo surfaces, a kind of surface defined as "a smooth birationally trivial surface V on which the anticanonical sheaf is ample" and counting points in the projective plane P2(k) built over a finite field k.

The result allowing us to to link the two concepts says that a Del Pezzo surface of degree d bigger that 1 is isomorphic to the blowup up of 9-d points in P2(k).

Once we have settled the relationship between the two concepts the results will revolve around counting n-tuples of points in P2(k) both from a theoretical and computational point of view.

In particular the case of 8-tuples, corresponding to Del Pezzo surfaces of degree 1, will be the one around which most of the work will revolve, culminating with the statement of a degree 8 monic polynomial expressing the number of 8-tuples of points in general position as a function of the dimension of the base field.

The work concludes by considering further instances of counting points in a projective plane, in particular in the case of points on which the Frobenius morphism acts in a specific way.

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