On singular Luroth quartics

TL;DR

This paper investigates the geometric properties of singular L"uroth quartics, computes their degrees, and explores implications for the L"uroth invariant using Cremona transformations and moduli space analysis.

Contribution

It provides the first degree computations for the irreducible components of singular L"uroth quartics and analyzes their impact on understanding the L"uroth invariant.

Findings

The locus of singular L"uroth quartics has two irreducible components of codimension two.

The degrees of these components are explicitly computed.

The class of the closure of nonsingular L"uroth quartics in  3 is determined.

Abstract

Plane quartics containing the ten vertices of a complete pentalateral and limits of them are called L\"uroth quartics. The locus of singular L\"uroth quartics has two irreducible components, both of codimension two in $\P^{14}$. We compute the degree of them and we discuss the consequences of this computation on the explicit form of the L\"uroth invariant. One important tool are the Cremona hexahedral equations of the cubic surface. We also compute the class in $\overline{M}_3$ of the closure of the locus of nonsingular L\"uroth quartics.