Speaker: Hal Schenck (Auburn)

 

Title: Free curves, eigenschemes, and pencils of curves

 

 

Abstract: Let \(R=K[x,y,z]\). A reduced plane curve \(C = V(f)\) in \(P^2\) is free if its associated module of tangent derivations Der\((f)\) is a free \(R\)-module, or equivalently if the corresponding sheaf \(T_{P^2}(−log C)\) of vector fields tangent to \(C\) splits as a direct sum of line bundles on \(P^2\). In general, free curves are difficult to find, and in this note, we describe a new method for constructing free curves in \(P^2\). 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.

 

This is joint work with Roberta Di Gennaro, Giovanna Ilardi, Rosa Maria Mirò-Roig, Hal Schenck, and Jean Vallès.