Cayley-Bacharach theorems with excess vanishing

Griffiths and Harris showed in 1978 that if E is a rank n vector bundle on a smooth projective variety of dimension n, and if s is a section of E vanishing simply on a finite set Z, then any section of (K_X + det E) vanishing at all but one of the points of Z must also vanish on the remaining one. This generalizes the classical theorem of Cayley-Bacharach, which appears when E is a direct sum of line bundles on projective space. In a recent paper, Mu-Lin Li proposed an extension allowing for the possibility that the zero-locus of s has positive dimensional components, but his result requires a splitting hypothesis that in practice is rarely satisfied. We show that multiplier ideals lead to a quite clean statement in the case of excess vanishing. Along the way, we give some analogues, that seem to be new, of Cayley-Bacharach for finite determinantal loci.