Georgia Institute of Technology College of Sciences

Given a finite set of points $\Gamma$ in $\mathbb{P}^n$, we say that $\Gamma$ satisfies the Cayley-Bacharach condition with respect to degree r polynomials, or is CB(r), if any degree r homogeneous polynomial F vanishing on all but one point of $\Gamma$ must vanish at the last point. In recent literature, the condition has played an important role in computing a birational invariant called the degree of irrationality of complex projective varieties. However, the condition itself has not been studied extensively, and surprisingly little is known about the geometric properties of CB(r) points. 

In this talk, I will discuss a new combinatorial approach to the study of the CB(r) condition, using matroid theory, and present some examples of how matroid theory can shed light on the underlying geometry of such sets.