Counting the number of trigonal curves of genus five over finite fields

Abstract

The trigonal curves form a closed subscheme of M5, the moduli space of smooth curves of genus five. The cohomological data of these spaces can be found by counting their number of points over finite fields. The trigonal curves of genus five can be represented by projective plane quintics that have one singularity that is an ordinary node or an ordinary cusp. We use a partial sieve method for plane curves to count the number of trigonal curves over any finite field. The result agrees with the findings of a computer program we have written that counts the number of trigonal curves over the finite fields of two and three elements. We also use the partial sieve method to count the number of smooth plane quintics. This result agrees with a previous result by Gorinov where he computed the cohomology of nonsingular plane quintics using a different method.