Star configuration points and generic plane curves

Abstract

Let ℓ 1 , … , ℓ l \ell _1,\ldots ,\ell _l be l l lines in P 2 \mathbb {P}^2 such that no three lines meet in a point. Let X ( l ) \mathbb {X}(l) be the set of points { ℓ i ∩ ℓ j | 1 ≤ i > j ≤ l } ⊆ P 2 \{\ell _i \cap \ell _j ~|~ 1 \leq i > j \leq l\} \subseteq \mathbb {P}^2 . We call X ( l ) \mathbb {X}(l) a star configuration. We describe all pairs ( d , l ) (d,l) such that the generic degree d d curve in P 2 \mathbb {P}^2 contains an X ( l ) \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.