A Cayley-Bacharach theorem and plane configurations

TL;DR

This paper explores how the Cayley-Bacharach condition imposes linear constraints on finite point sets in projective space, leading to geometric restrictions and applications to rational maps of complete intersections.

Contribution

It establishes bounds on point configurations satisfying Cayley-Bacharach, linking them to unions of low-dimensional linear spaces and analyzing rational maps for specific complete intersections.

Findings

Points satisfying Cayley-Bacharach lie on unions of low-dimensional linear spaces.

Bounds on the number of such points are derived.

Descriptions of fibers of rational maps for certain complete intersections.

Abstract

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.