Moduli spaces of algebraic curves with rational maps

Abstract

Let ℳg be the coarse moduli scheme of curves of genus g. For an algebraically closed field k define is a quasiprojective algebraic variety over k, its dimension being 3g – 3 for g ≥ 2, 1 for g = 1, and 0 for g = 0. It can be considered as the moduli variety for the classes of birationally equivalent curves of genus g over k. For 0 < g, g′ and n ≥ 1 let be the subset of those points of whose corresponding curves possess a rational map of degree n into a curve of genus g′ over k.