Free curves, Eigenschemes and Pencils of curves

Abstract

Let $R=K[x,y,z]$. A reduced plane curve $C=V(f)\subset {\mathbf{P}}^2$ is $free$ if its associated module of tangent derivations $\mathrm{Der}(f)$ is a free $R$-module, or equivalently if the corresponding sheaf $T_{{\mathbf{P}}^2}(-\log C)$ of vector fields tangent to $C$ splits as a direct sum of line bundles on ${\mathbf{P}}^2$. In general, free curves are difficult to find, and in this note, we describe a new method for constructing free curves in ${\mathbf{P}}^2$. The key tools in our approach are eigenschemes and pencils of curves, combined with an interpretation of Saito's criterion in this context. Previous constructions typically applied only to curves with quasihomogeneous singularities, which is not necessary in our approach. We illustrate our method by constructing large families of free curves.