A Cayley–Bacharach theorem and plane configurations

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.

Matroids satisfying the matroidal Cayley--Bacharach property and ranks of covering flats

“…A finite subset Γ ⊂ P n satisfies the Cayley-Bacharach condition of degree r if a homogeneous polynomial of degree r vanishing on all but one point of Γ vanishes on all of Γ. In recent work, Levinson-Ullery [9] show that a finite subset Γ ⊂ P n satisfying the Cayley-Bacharach condition of degree r is covered by low-dimensional linear subspaces if |Γ| is not very large compared to r (Theorem 1.3 on p. 2 of [9]). The result was motivated by constructions relating to degrees of irrationality of smooth complete intersections.…”

“…If K X is sufficiently positive, then the fibers also lie in special positions. For example, a result of Bastianelli-Cortini-De Poi (Theorem 1.1 on p. 2 of [9]) states that a finite subset Γ ⊂ P n satisfying the degree r Cayley-Bacharach property of degree r such that |Γ| ≤ 2r + 1 lies on a line. The results of Levinson-Ullery (Theorem 1.3 on p. 2 of [9]) are analogues which show that Γ still lies on a union of low-dimensional linear subspaces when we impose a weaker linear upper bound in r on the size of Γ.…”

“…Since many of the arguments used in the result of Levinson-Ullery (Theorem 1.3 on p. 2 of [9]) are combinatorial in a way that can sometimes be rewritten in terms of matroid theory (p. 14 of [9]), the authors define a matroid-theoretic analogue of the Cayley-Bacharach property. Definition 1.1.…”

“…Definition 1.1. (p. 14 of [9]) A matroid M with underlying set E satisfies the matroidal Cayley-Bacharach property of degree a (denoted MC B (a)) if, whenever a union of a flats contains all but one point of E , the union contains the last point. In other words:…”

“…Question 1.2. (Levinson-Ullery, Question 7.6 on p. 14 of [9]) Let M be a matroid of rank r with underlying set E of size n. Suppose that M satisfies MC B (a). Must M (meaning the underlying set E ) be covered by a union of proper flats of specified ranks?…”