[Paper Review] On the hypersurface of Luroth quartics

Giorgio Ottaviani, Edoardo Sernesi|ArXiv.org|Mar 30, 2009

Algebraic Geometry and Number Theory13 references3 citations

TL;DR

This paper provides a rigorous reconstruction of Morley's 1919 proof that the hypersurface of Lüroth quartic plane curves in ℙ¹⁴ has degree 54. Using Geiser involutions and the Morley invariant—a skew-symmetric, multihomogeneous polynomial of degree three in seven points—it shows that Lüroth quartics correspond precisely to configurations where this invariant vanishes. The degree 54 is confirmed via geometric control of the 36 Cremona planes associated to cubic surfaces, linking to moduli spaces of vector bundles and the Barth map's degree via vector bundle techniques.

ABSTRACT

The hypersurface of Luroth quartic curves inside the projective space of plane quartics has degree 54. We give a proof of this fact along the lines outlined in a paper by Morley, published in 1919. Another proof has been given by Le Potier and Tikhomirov in 2001, in the setting of moduli spaces of vector bundles on the projective plane. Morley's proof uses the description of plane quartics as branch curves of Geiser involutions and gives new geometrical interpretations of the 36 planes associated to the Cremona hexahedral representations of a nonsingular cubic surface.

Motivation & Objective

• To provide a rigorous foundation for Morley's 1919 proof of the degree of the Lüroth hypersurface, which had been considered obscure or incomplete.
• To reconstruct and clarify Morley’s method using Geiser involutions and the geometric interpretation of the Morley invariant as a skew-symmetric, multihomogeneous polynomial of degree three in seven points.
• To establish the precise connection between the vanishing of the Morley invariant and the Lüroth property of quartic curves.
• To link the geometric construction to moduli spaces of vector bundles on ℙ², particularly via the Barth map, and confirm the degree 54 result through vector bundle techniques.
• To demonstrate that the Barth map from the moduli space M(0,4) to the Lüroth hypersurface is generically injective, implying degree 54 for the image.

Proposed method

• The paper uses Geiser involutions—degree-two rational maps on ℙ²—defined by linear systems of cubics with seven distinct base points, whose branch curves are plane quartics.
• It introduces the Morley invariant Ψ, a skew-symmetric, multihomogeneous polynomial of degree three in the coordinates of seven points, vanishing precisely when the associated quartic is Lüroth.
• The proof relies on Bateman’s 13-dimensional family of configurations where the Morley invariant vanishes, showing these correspond exactly to Lüroth quartics.
• It constructs a rational map from the product ℙ(𝑆²𝑉∨) × ℙ(𝑆³𝑉∨) to the space of quartics, showing dominance and generic finiteness via apolarity and Veronese-type embeddings.
• It identifies the Lüroth hypersurface with the image of the Barth map from the moduli space M(0,4), using vector bundle invariants and Donaldson theory to confirm the degree.
• It employs Cremona hexahedral representations of cubic surfaces and their 36 associated Cremona planes to control the locus of Lüroth quartics, linking to the 54-degree result.

Experimental results

Research questions

• RQ1 What is the precise degree of the hypersurface of Lüroth quartic plane curves in ℙ¹⁴?
• RQ2 How can Morley’s 1919 proof, which was considered incomplete, be reconstructed with full geometric and algebraic rigor?
• RQ3 What is the geometric significance of the Morley invariant in characterizing Lüroth quartics?
• RQ4 How do the 36 Cremona planes associated to a cubic surface relate to the degree 54 of the Lüroth hypersurface?
• RQ5 What is the relationship between the Barth map from M(0,4) to the Lüroth hypersurface and the degree of the image variety?

Key findings

• The Lüroth hypersurface in ℙ¹⁴ has degree exactly 54, confirming Morley’s original claim.
• The Morley invariant Ψ is a skew-symmetric, multihomogeneous polynomial of degree three in seven points, vanishing if and only if the associated quartic is Lüroth.
• The locus of configurations where the Morley invariant vanishes contains Bateman’s 13-dimensional family of Lüroth configurations as a component.
• The Barth map from the moduli space M(0,4) of stable rank-two vector bundles with (c₁,c₂)=(0,4) to the Lüroth hypersurface is generically injective, implying deg(Im(b)) = 54.
• The 36 Cremona planes associated to a cubic surface correspond to the 36 rational projections from points on the surface whose branch curves are Lüroth quartics, and this number is key to the degree 54 count.
• The rational map from ℙ(𝑆²𝑉∨) × ℙ(𝑆³𝑉∨) to the space of quartics is dominant and generically finite, confirming the irreducibility and dimension of the Lüroth hypersurface.