Star configuration points and generic plane curves

Abstract:

Let $\ell _1,\ldots ,\ell _l$ be $l$ lines in $\mathbb {P}^2$ such that no three lines meet in a point. Let $\mathbb {X}(l)$ be the set of points $\{\ell _i \cap \ell _j ~|~ 1 \leq i < j \leq l\} \subseteq \mathbb {P}^2$. We call $\mathbb {X}(l)$ a star configuration. We describe all pairs $(d,l)$ such that the generic degree $d$ curve in $\mathbb {P}^2$ contains an $\mathbb {X}(l)$. Our proof strategy uses both a theoretical and an explicit algorithmic approach. We also describe how one may extend our algorithmic approach to similar problems.