Star configuration points and generic plane curves

TL;DR

This paper characterizes the pairs (d, l) where a generic degree d plane curve contains all intersection points of l lines in P^2, assuming no three lines intersect at a single point.

Contribution

It provides a complete description of when generic degree d curves in P^2 contain the intersection points of l lines in general position.

Findings

Identifies all pairs (d, l) for which the intersection points are contained in a generic degree d curve.

Establishes conditions under which these points lie on such curves.

Advances understanding of configurations of points on algebraic curves.

Abstract

Consider l lines in P^2 such that no three lines meet in a point. Let X(l) denote all points of intersections of these l lines. We describe all pairs (d,l) such that generic degree d curve in P^2 contains a X(l).