An explicit expression of the Luroth invariant

TL;DR

This paper provides an explicit algorithm for expressing the Luroth invariant using Dixmier-Ohno invariants, offers a factorized form for Ciani quartics, and addresses open questions about singular Luroth quartics.

Contribution

It introduces a new explicit algorithm for the Luroth invariant and clarifies properties of specific quartic sub-loci, advancing invariant theory in algebraic geometry.

Findings

Explicit expression of the Luroth invariant in terms of Dixmier-Ohno invariants

Factorized form of the Luroth invariant on Ciani quartics

Answers to open questions on singular Luroth quartics

Abstract

In this short note, we give an algorithm to get an explicit expression of the Luroth invariant in terms of the Dixmier-Ohno invariants. We also get the explicit factorized expression on the locus of Ciani quartics in terms of the coefficients. Finally, we answer two open questions on sub-loci of singular Luroth quartics.