Cayley-Bacharach theorems with excess vanishing

Lawrence Ein; Robert Lazarsfeld

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arXiv · 1909.08493

Cayley-Bacharach theorems with excess vanishing

Lawrence Ein·Robert Lazarsfeld

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Abstract

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 simplified and somewhat strengthened accounts of results of Tan-Viehweg and Sun.

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Lawrence Ein, Robert Lazarsfeld. 2019-09-18. Cayley-Bacharach theorems with excess vanishing. https://arxiv.org/abs/1909.08493

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