Star configuration points and generic plane curves

Let ${\ell }_1,\dots ,{\ell }_l$ 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 $\{{\ell }_i\cap {\ell }_j|1\le i>j\le l\}\subseteq {\mathbb{P}}^2$ . 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.