On rigid plane curves

Abstract

We exhibit a series of new examples of rigid plane curves, that is, curves, whose collection of singularities determines them almost uniquely up to a projective transformation of the plane.

Deformations of plane curves and Jacobian syzygies

“…Example Consider the weighted homogeneous singularity with . We will denote this singularity by following . Then the equianalytic ideal is generated by and , while the equisingular ideal is generated by and all the monomial with .…”

“…Example In this example we list the free and nearly free curves of degree d such that and . For details we refer to . All of them are strongly ‐rigid except for the curve discussed below.…”

“…The equianalytic and the equisingular deformations of a plane curve singularity are rather well understood, and the corresponding theory is briefly recalled in the second section following . On the other hand, the equianalytic and the equisingular deformations of the plane curve C , encoded in the (possibly non‐reduced) analytic subspaces of the projective space , are well understood when C is nodal, see , but much less in the general case, see . Here d is the degree of the curve C , is the list of all the singularities of C , or denotes the equianalytic or the equisingular deformations, and , such that parametrizes all the degree d plane curves.…”