In the words of the authors, “the Cayley-Bacharach theorem, in its classical form, may be seen as a result about the number of independent conditions imposed on polynomials of given degree by certain sets of points in the plane.” For example, if \(X\) is the complete intersection of two cubic curves in the projective plane, then any cubic curve passing through eight points of \(X\) also passes through the ninth point; in other words, the ninth point does not impose an extra condition on cubics. This theorem, and related theorems, have a long history in algebraic geometry.
The first part of this paper is purely expository and traces the evolution of this theorem from Pappus’ theorem to modern day, and it relates the geometric ideas behind it to algebraic notions including the Gorenstein property and duality theory. In particular, the authors introduce some of the basic developments of modern commutative algebra.
In the second part of the paper the authors propose a possible next step in Cayley-Bacharach theory, in the form of a conjecture. This is stated first for quadrics and then given in general form. Roughly, the conjecture concerns subschemes of a zero-dimensional complete intersection of hypersurfaces of given degrees \(d_1 ,\dots,d_n\) in \({\mathbb{P}}^n\), and the minimum possible degree of such a subscheme which does not impose independent conditions on certain hypersurfaces.