Free curves, Eigenschemes and Pencils of curves
Roberta Di Gennaro, Giovanna Ilardi, Rosa Maria Mirò-Roig, Hal Schenck, Jean Vallès
TL;DR
The paper develops a geometric framework to construct free plane curves by combining eigenschemes with pencils of curves, extending Saito's freeness criterion beyond quasihomogeneous singularities. It introduces the canonical derivation $\delta_{f,g}=[\nabla f \wedge \nabla g]$ and its eigenscheme $\Gamma$, showing that freeness of a union from a pencil is characterized by membership of the defining polynomial in the saturated ideal $I_{\Gamma}$. A key theorem states that for a union $F_k$ of pencil members with appropriate degrees, a curve $V(F)$ is free with exponents $(n+m-2, N-n-m+1)$ iff $F\in(I_{\Gamma})_N$, enabling constructive freeness results. The paper illustrates the method with detailed examples (Hesse, conic/sextic pencils, osculating conics) and extends the approach to nets and reflection arrangements, revealing how freeness behaves under adding pencil or net members and highlighting non-quasihomogeneous cases. Overall, the work broadens the toolkit for generating free divisors in $\mathbb P^2$ and connects determinantal and eigenscheme techniques to classical problems in plane curve freeness.
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.