A Cayley--Bacharach theorem and plane configurations

In this paper, we examine linear conditions on finite sets of points in projective space implied by the Cayley–Bacharach condition. In particular, by bounding the number of points satisfying the Cayley–Bacharach condition, we force them to lie on unions of low-dimensional linear spaces. These results are motivated by investigations into degrees of irrationality of complete intersections, which are controlled by minimum-degree rational maps to projective space. As an application of our main theorem, we describe the fibers of such maps for certain complete intersections of codimension two.