Plus-one generated curves, Brian\c{c}on-type polynomials and eigenscheme ideals

TL;DR

This paper introduces minimal plus-one generated curves, explores their relation to free and nearly free curves, and characterizes them via eigenscheme ideals, providing new examples and counterexamples in algebraic geometry.

Contribution

It defines minimal plus-one generated curves, relates them to free curves, and characterizes them using eigenscheme ideals, expanding understanding of curve classifications.

Findings

New examples of free, nearly free, and minimal plus-one generated curves.

Counter-examples to the conjecture linking supersolvability and freeness.

Characterization of plus-one generated curves via eigenscheme ideals.

Abstract

We define the minimal plus-one generated curves and prove a result explaining why they are the closest relatives of the free curves, after the nearly free curves. Then we look at the projective closures of the general and of the special fibers of some Brian\c{c}on-type polynomials constructed by E. Artal Bartolo, Pi. Cassou-Nogu\`es and I. Luengo Velasco. They yield new examples of free, nearly free or minimal plus-one generated curves, as well as counter-examples to the conjecture saying that a supersolvable curve is free. In the final section we give a characterization of plus-one generated curves in terms of eigenscheme ideals, similar to the characterization of free curves given by R. Di Gennaro, G. Ilardi, R.M. Mir\'o-Roig, H. Schenck and J. Vall\`es in a recent paper. Then we apply this result to the construction of minimal plus-one generated curves obtained by putting together at least two members in a pencil of curves related to Brian\c{c}on-type polynomials.