There are no projective surfaces in M_4
We answer the first non-classical case of a question of J. Harris from
the 1983 ICM: "what is the largest possible dimension of a
complete subvariety of M_g ?" Working over a base field with
characteristic 0 or greater than 3, we prove that there are no
projective surfaces in the moduli space of curves of genus 4; thus
proving that the largest possible dimension of a projective
subvariety in M_4 is 1.