The Gieseker-Petri divisor in $${\mathcal{M}_g}$$ for g ≤ 13

Abstract

The Gieseker-Petri locus G P g is defined as the locus inside M g consisting of curves which violate the Gieseker-Petri Theorem. It is known that G P g has always some divisorial components and it has been conjectured that it is of pure codimension 1 inside M g . We prove that this holds true for genus up to 13.

Stability of rank-3 Lazarsfeld-Mukai bundles on K 3 surfaces

“…Thus, if h 0 (S, N ) 2, then the linear series N ⊗ O C contributes to the Clifford index of C ∈ |L| s . Furthermore, inequality (18), together with the fact that c 1 (N 2 ) 2 < 0 c 1 (N 1 ) 2 , implies that c 1 (N 2 ) · c 1 (N ) > c 1 (N 1 ) · c 1 (N ). We obtain c 1 (N 2 ) · c 1 (N ) > 1 2 c 1 (N ) · (c 1 (N 1 ) + c 1 (N 2 )), and we are in case (c) It remains to treat the case where both h 0 (S, N 2 ) < 2 and h 0 (S, N ) < 2.…”

“…Recall that GP 11 has pure codimension 1 in M 11 (cf. [18]) and decomposes in the following way: GP 11 = M 2 11,9 ∪ GP 2 11,10 ∪ 10 d=7 GP 1 11,d , where M 2 11,9 is a Brill-Noether divisor. Therefore, proving the transversality of N L 4 11,13 and GP 11 is equivalent to showing that in the above situation, if C ∈ |L| is general, then C has no g 2 9 and the varieties G 2 10 (C) and G 1 d (C) for 7 d 10 are smooth of the expected dimension.…”

A codimension 2 component of the Gieseker-Petri locus

“…We summarize some inclusions holding in any genus (cf. Section 2 in [LC1]): Proposition 2.1. One has that:…”

“…and M r+1 g,d+1 ⊂ GP r g,d (proceed as in the proof of [LC1,Lem. 2.5] and, in the case where the g r d−1 or the g r+1 d+1 on the curve C is not primitive, use a general point of C in order to construct a g r d ).…”

“…[Ca]), and the author herself in the range 9 ≤ g ≤ 13 (cf. [LC1]). However, in general not very much is known about the dimension of the loci GP r g,d and their reciprocal position.…”