On the Cayley-Bacharach property and the construction of vector bundles

Abstract

We study the Cayley-Bacharach property on complex projective smooth varieties of dimension n 2 for zero dimensional subscheme defined by the zero set of the wedge of r − n + 1 global sections of a rank r n vector bundle, and give a construction of high rank reflexive sheaves and vector bundles from codimension 2 subschemes.

Cayley-Bacharach theorems with excess vanishing

“…, s e ∈ Γ X, E drop rank along Z, then any section of O X (K X + L) vanishing at all but one of the points of Z vanishes at the remaining one. (This is a special case of [11,Theorem 4.1]. It also can be deduced directly from the theorem of Griffiths-Harris.)…”

“…We conclude this section by sketching an application of Theorem 1.2 to statements, essentially due to Sun [11], of Cayley-Bacharach type for determinantal loci. The present approach is somewhat different than that of [11], which uses Eagon-Northcott complexes.…”

“…Concerning the organization of this note, we start in §1 with a review of the theorem of Griffiths-Harris, and the extension in the spirit of Tan and Viehweg. As an application of the latter, we give at the end of the section a somewhat simplified and strengthened account of some results of Sun [11] concerning finite determinantal loci: see Theorem 1.5. 2 In §2 we derive the results involving multiplier ideals by applying the classical theorems on a log resolution of the base ideal.…”

Virtual residue and a generalized Cayley-Bacharach theorem

“…Cayley-Bacharach Theorem said that if C ⊂ P 2 is any plane curve of degree d + e − 3 containing all but one point of Γ, then C contains all of Γ, see [3,Theorem CB4]. The extension of Cayley-Bacharach property on projective manifolds had been proved to be related to Fujita conjecture, and the construction of special bundles, see [6], [7], [8] and [9].…”