Brill–Noether theory on a moving curve

Abstract

In this chapter we go back to the classical Brill–Noether theory of special divisors on curves, and we complete the work that was initiated in the first volume. We start by defining the relative versions of the Brill–Noether varieties we introduced in Chapter IV of Volume I. A fundamental tool in our study is the description of the tangent spaces to these varieties; we present these tangent space computations in Sections 5 and 6. After a digression on Looijenga’s proof of the vanishing of the tautological ring of M _g in degree greater than g−2, we then proceed to give Lazarsfeld’s proof of Petri’s conjecture, which is the basic result that was announced in Chapter V of Volume I. The second part of the chapter is devoted to the projective realizations of algebraic curves as ramified covers of ℙ¹ and as plane curves, and to a number of unirationality results for moduli spaces in low genus.