Rational Maps between Moduli Spaces of Curves and Gieseker-petri Divisors

is injective. The theorem, conjectured by Petri and proved by Gieseker [G] (see [EH3] for a much simplified proof), lies at the cornerstone of the theory of algebraic curves. It implies that the variety Gd(C) = {(L, V ) : L ∈ Pic (C), V ∈ G(r + 1,H0(L))} of linear series of degree d and dimension r is smooth and of expected dimension ρ(g, r, d) := g− (r+1)(g−d+ r) and that the forgetful mapGd(C) → W r d (C) is a rational resolution of singularities (see [ACGH] for many other applications). It is an old open problem to describe the locus GPg ⊂ Mg consisting of curves [C] ∈ Mg such that there exists a line bundle L on C for which the Gieseker-Petri theorem fails. Obviously GPg breaks up into irreducible components depending on the numerical types of linear series. For fixed integers d, r ≥ 1 such that g − d + r ≥ 2, we define the locus GPg,d consisting of curves [C] ∈ Mg such that there exist a pair of linear series (L, V ) ∈ G r d(C) and (KC ⊗ L ∨,W ) ∈ G 2g−2−d (C) for which the multiplication map μ0(V,W ) : V ⊗W → H (C,KC )