  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.
