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 $k\ge 1$ and $\Gamma =\{P_1,\dots ,P_d\}\subset {\mathbb{P}}^n$ is a set of distinct points satisfying the Cayley-Bacharach condition with respect to $|{\mathcal{O}}_{{\mathbb{P}}^n}(k)|$, with $d\le h(k-h+3)-1$ and $3\le h\le 5$, then $\Gamma$ 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 ${\mathbb{P}}^3$. Moreover, we discuss correspondences with null trace on smooth hypersurfaces of ${\mathbb{P}}^n$ and on codimension $2$ complete intersections.

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