On the unique unexpected quartic in $\mathbb{P}^2$

TL;DR

This paper investigates the phenomenon of unexpected quartic curves in the projective plane, classifies low degree cases, and proves the uniqueness of a special nine-point configuration that produces such an unexpected quartic.

Contribution

It provides a classification of low degree unexpected curves and establishes the uniqueness of a specific nine-point configuration leading to an unexpected quartic.

Findings

Classified low degree unexpected curves.

Proved the uniqueness of the nine-point configuration for the unexpected quartic.

Abstract

The computation of the dimension of linear systems of plane curves through a bunch of given multiple points is one of the most classic issues in Algebraic Geometry. In general, it is still an open problem to understand when the points fail to impose independent conditions. Despite many partial results, a complete solution is not known, even if the fixed points are in general position. The answer in the case of general points in the projective plane is predicted by the famous Segre-Harbourne-Gimigliano-Hirschowitz conjecture. When we consider fixed points in special position, even more interesting situations may occur. Recently Di Gennaro, Ilardi and Vall\`{e}s discovered a special configuration $Z$ of nine points with a remarkable property: a general triple point always fails to impose independent conditions on the ideal of $Z$ in degree four. The peculiar structure and properties of this kind of \textit{unexpected curves} were studied by Cook II, Harbourne, Migliore and Nagel. By using both explicit geometric constructions and more abstract algebraic arguments, we classify low degree unexpected curves. In particular, we prove that the aforementioned configuration $Z$ is the unique one giving rise to an unexpected quartic.