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Abstract

This thesis is a compilation of three papers. The Cayley–Salmon theorem implies the existence of a 27-sheeted covering space of the parameter space of smooth cubic surfaces, marking each of the 27 lines on each surface. In Chapter 2 we compute the rational cohomology of the total space of this cover, using the spectral sequence in the method of simplicial resolution developed by Vassiliev. The covering map is an isomorphism in cohomology (in fact of mixed Hodge structures) and the cohomology ring is isomorphic to that of PGL(4, C). We derive as a consequence that over the finite field F_q the average number of lines on a smooth cubic surface equals 1 (away from finitely many characteristics); this average is 1 + O(q^{-1/2}) by a standard application of the Weil conjectures. In Chapter 3 we compute the rational cohomology of the universal family of smooth cubic surfaces using the same method of simplicial resolution. Modulo embedding, the universal family has cohomology isomorphic to that of P^2. It again follows that over the finite field F_q, away from finitely many characteristics, the average number of points on a smooth cubic surface is q^2 + q + 1. In Chapter 4 we compute the distributions of various other markings on smooth cubic surfaces defined over the finite field F_q, for example the distribution of pairs of points, ‘tritangents’ or ‘double sixes’. We also compute the (rational) cohomology of some of the associated bundles and covers over complex numbers.

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