On singular Lüroth quartics

Abstract

Plane quartics containing the ten vertices of a complete pentalateral and limits of them are called Lüroth quartics. The locus of singular Lüroth quartics has two irreducible components, both of codimension two in ℙ14. We compute the degree of them and discuss the consequences of this computation on the explicit form of the Lüroth invariant. One important tool is the Cremona hexahedral equations of the cubic surface. We also compute the class in $\bar M_3 $ of the closure of the locus of nonsingular Lüroth quartics.