Geometry of Points Satisfying Cayley-Bacharach Conditions and Applications

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 $k\geq 1$ 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 $d\leq h(k-h+3)-1$ and $3\leq h\leq 5$, 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.