Geometry of Points Satisfying Cayley-Bacharach Conditions and Applications

Abstract

In this paper, we study the geometry of points in complex projective space that satisfy the Cayley-Bacharach condition with respect to the complete linear system of hypersurfaces of given degree. In particular, we improve a result by Lopez and Pirola and we show that, if k ≥ 1 and Γ = {P1, . . . , P d } ⊂ P n is a set of distinct points satisfying the Cayley-Bacharach condition with respect to |O P n (k)|, with d ≤ h(k − h + 3) − 1 and 3 ≤ h ≤ 5, then Γ lies on a curve of degree h − 1. Then we apply this result to the study of linear series on curves on smooth surfaces in P 3 . Moreover, we discuss correspondences with null trace on smooth hypersurfaces of P n and on codimension 2 complete intersections.