On singular Luroth quartics

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 $¶^{14}$. We compute the degree of them and we discuss the consequences of this computation on the explicit form of the Lüroth 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 Lüroth quartics.

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