We use geometrically defined divisors on moduli spaces of pointed curves to compute the graded Betti numbers of general sets of points on any nonhyperelliptic canonically embedded curve. This gives a positive answer to the Minimal Resolution Conjecture in the case of canonical curves. But we prove that the conjecture fails on curves of large degree. These results are related to the existence of theta divisors associated to certain stable vector bundles. Nous utilisons des diviseurs définis pour des conditions géométriques sur des espaces de modules de courbes stables à points marqués pour calculer les nombres de Betti des ensembles généraux de points sur une courbe non hyperelliptique arbitraire, canoniquement plongée. Cela donne une réponse affirmative à la conjecture de résolution minimale dans le cas des courbes canoniques. Par ailleurs, nous prouvons que la conjecture est fausse pour les courbes de grand degré. Ces résultats sont liés à l'existence des diviseurs thêta associés à certains fibrés vectoriels stables.

We apply the theory of $M$ -regularity developed by the authors [ Regularity on abelian varieties , I, J. Amer. Math. Soc. 16 (2003), 285–302] to the study of linear series given by multiples of ample line bundles on abelian varieties. We define an invariant of a line bundle, called $M$ -regularity index, which governs the higher order properties and (partly conjecturally) the defining equations of such embeddings. We prove a general result on the behavior of the defining equations and higher syzygies in embeddings given by multiples of ample bundles whose base locus has no fixed components, extending a conjecture of Lazarsfeld [proved in Syzygies of abelian varieties , J. Amer. Math. Soc. 13 (2000), 651–664]. This approach also unifies essentially all the previously known results in this area, and is based on Fourier-Mukai techniques rather than representations of theta groups.

We introduce the notion of Mukai regularity ( $M$ -regularity) for coherent sheaves on abelian varieties. The definition is based on the Fourier-Mukai transform, and in a special case depending on the choice of a polarization it parallels and strengthens the usual Castelnuovo-Mumford regularity. Mukai regularity has a large number of applications, ranging from basic properties of linear series on abelian varieties and defining equations for their subvarieties, to higher dimensional type statements and to a study of special classes of vector bundles. Some of these applications are explained here, while others are the subject of upcoming sequels.

We compute the class of the compactification of the divisor of curves sitting on a $K3$ surface and show that it violates the Harris-Morrison Slope Conjecture. We carry this out using the fact that this divisor has four distinct incarnations as a geometric subvariety of the moduli space of curves. We also give a counterexample to a hypothesis raised by Harris and Morrison that the Brill-Noether divisors are essentially the only effective divisors on the moduli space of curves having minimal slope $6+12/(g+1)$ .

We study the Hodge filtration on the local cohomology sheaves of a smooth complex algebraic variety along a closed subschemeZin terms of log resolutions and derive applications regarding the local cohomological dimension, the Du Bois complex, local vanishing and reflexive differentials associated toZ.