[论文解读] Some Classical Invariants, from Harmonic Quadruples to Triangle Groups

Giorgio Ottaviani, Vincenzo Galgano|arXiv (Cornell University)|Mar 3, 2026

Mathematics and Applications被引用 0

一句话总结

The notes survey harmonic and equianharmonic invariants for binary quartics and ternary cubics, relate them to SL(2) and SL(3) actions, and connect finite polyhedral groups and triangle groups with moduli and covariants, including an appendix on Pfaffians and exercises.

ABSTRACT

These notes are an expanded version of the lectures held in Tromso, in May 2025 at the "Lie-Stormer Summer School : Invariant Theory from classics to modern developments", in the framework of TiME events. We emphasize the analogy between binary quartics and ternary cubics (and subsequently modular forms) based on their harmonic and equianharmonic invariants. Triangle groups are presented in both the elliptic and the hyperbolic setting with their associated tilings. The topics include the discussion of a short Hilbert paper on polynomials which are powers, that was proposed to the participants. The appendix contains some exercises, with sketches of solutions, and a section devoted to Pfaffians edited by Vincenzo Galgano.

研究动机与目标

• Motivate invariant theory through harmonic quadruples and cross-ratio as a projective invariant framework.
• Develop the analogy between binary quartics and ternary cubics via their harmonic and equianharmonic invariants.
• Explore finite polyhedral groups, ADE classification, and the role of covariants in classifying SL(2) orbits.
• Present connections to hyperbolic and elliptic triangle groups and modular forms.
• Provide an appendix with exercises and Pfaffian techniques related to the invariants.

提出的方法

• Define and study SL(2) invariants of four ordered points on P1 using the cross-ratio and its harmonic/equianharmonic specializations.
• Introduce binary quartics via Sym^4(C^2) and express I and J as classical invariants linked to transvectants.
• Use transvectants (f,g)_n to generate the invariant ring and express I and J in terms of covariants.
• Relate ternary cubics to binary quartics through Salmon’s theorem, obtaining Aronhold and T invariants as pullbacks and describing equianharmonic/harmonic cases.
• Analyze the nullcone and semistability in the SL(2) and SL(3) settings, including stabilizers and orbit closures.
• Discuss hyperbolic triangle groups, modular forms, and Pfaffian descriptions of invariants in appendix.

Figure 2: AD is the harmonic mean between AB and AC. Recall that the sound frequencies are reciprocal to the lengths.

实验结果

研究问题

• RQ1 How do harmonic and equianharmonic configurations of points in P1 reflect classical invariant theory for binary forms?
• RQ2 What is the structure of the SL(2)-invariant ring for binary quartics and how do I and J generate it?
• RQ3 How does Salmon’s theorem connect ternary cubics to binary quartics via invariants A (Aronhold) and T, and what are the equianharmonic/harmonic conditions?
• RQ4 What are the role and geometry of semistability, nullcones, and orbit stabilizers in the SL(2) and SL(3) settings?
• RQ5 How do triangle groups and modular forms arise in the study of these invariants and their covariants?

主要发现

• The SL(2)-invariant ring of Sym^4(C^2) is generated by the two invariants I and J, i.e., C[Sym^4(C^2)]^{SL(2)} = C[I,J].
• A binary quartic is harmonic iff a certain determinant J vanishes, and equianharmonic iff a certain quadratic form I vanishes; the discriminant is D = I^3 − 27J^2.
• The Hessian map on quartics has Jacobian proportional to I·J, and its polar relation to J links Hessians to the geometry of the invariant ring.
• Salmon’s theorem yields a bridge from ternary cubics to binary quartics, with the Aronhold invariant A and the invariant T for ternary cubics pulling back I and J, encoding equianharmonic and harmonic cases respectively.
• The Enriques–Fano and Aluffi–Faber results classify smooth SL(2)-orbit closures in projective spaces of binary forms, relating to polyhedral groups and yielding four distinguished smooth Fano 3-folds (genus 12 case U22) with rich geometric and representation-theoretic structure.

Figure 3: Apollonius construction: D is the fourth harmonic after A, B, C. The construction does not depend on the conic. The figure is from the introduction of [ DCG ] .