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 9d points in P2(k). Once we have settled the relationship between the two concepts the results will revolve around counting ntuples of points in P2(k) both from a theoretical and computational point of view. In particular the case of 8tuples, 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.
Del Pezzo surfaces and points in the plane
Mangano, Federico
2021/2022
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 9d points in P2(k). Once we have settled the relationship between the two concepts the results will revolve around counting ntuples of points in P2(k) both from a theoretical and computational point of view. In particular the case of 8tuples, 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.File  Dimensione  Formato  

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https://hdl.handle.net/20.500.12608/21220