The relation type of point configurations in the projective plane

Abstract

We study the {\it relation type} of ideals of finite reduced sets of points in the projective plane; for a given ideal, this is the maximal $T$-degree of a minimal generator of the defining ideal of the Rees algebra. Our main focus is on point configurations whose defining ideals are not necessarily linearly presented, with an emphasis on almost collinear configurations. We prove that $\rt(X)\in\{1,3\}$ whenever $X\subseteq \PP^2_k$ is a finite set of at most ten points; and we characterize the configurations of relation type $3$ in this range. We then show that a configuration of eleven points in generic position has relation type $5$, thereby yielding the first occurrence of relation type larger than $3$. Finally, we exhibit a configuration of $17$ points with relation type $4$ and we formulate some questions regarding the spectrum of admissible relation types of point configurations.