[Paper Review] Five Lectures on Projective Invariants

Giorgio Ottaviani|arXiv (Cornell University)|May 13, 2013

Advanced Topics in Algebra47 references3 citations

TL;DR

This paper presents a comprehensive introduction to projective invariants using representation theory, focusing on invariants of forms and configurations of points in projective space. It develops computational tools such as graphical algebras, tableau functions, and Molien series to compute Hilbert series and generators of invariant rings, with key results including the Hilbert series for the $\mathrm{Alt}(6)$- and $\Sigma_6$-invariant rings of six unordered points on $\mathbb{P}^2$, and the identification of the Aronhold invariant as a Pfaffian and the Morley cubic as a geometric covariant.

ABSTRACT

We introduce invariant rings for forms (homogeneous polynomials) and for d points on the projective space, from the point of view of representation theory. We discuss several examples, addressing some computational issues. We introduce the graphical algebra for the invariants of d points on the line. This is an expanded version of the notes for the School on Invariant Theory and Projective Geometry, Trento, September 17-22, 2012.

Motivation & Objective

• To provide a modern, representation-theoretic framework for understanding classical projective invariants of forms and point configurations.
• To address computational challenges in invariant theory by introducing graphical algebras and tableau-based methods for computing invariants.
• To derive explicit Hilbert series and generator sets for invariant rings of six unordered points on $\mathbb{P}^1$ and $\mathbb{P}^2$, including relations among generators.
• To clarify the geometric significance of key invariants such as the Aronhold invariant and the Morley cubic in the context of Lüroth quartics and cubic surfaces.
• To demonstrate the failure of Kempe’s Lemma in higher-dimensional point configurations, showing the need for additional generators beyond linear invariants.

Proposed method

• Utilizes Schur-Weyl duality and representation theory of $GL(n+1)$, $SL(n+1)$, and $\Sigma_d$ to analyze invariants of forms and point sets.
• Applies Young diagrams and symmetrizers to compute dimensions of invariant subspaces and construct tableau functions.
• Employs the Molien formula to compute Hilbert series of invariant rings, particularly for $\mathrm{Alt}(6)$ and $\Sigma_6$ actions on six points.
• Introduces a graphical algebra for invariants of $d$ points on $\mathbb{P}^1$, enabling combinatorial computation of invariants via Kempe’s Lemma.
• Uses symbolic representation and the two Fundamental Theorems of invariants to construct all invariants from basic building blocks.
• Applies the Reynolds operator and plethysm techniques to decompose symmetric powers of symmetric tensors into irreducible representations.

Experimental results

Research questions

• RQ1 What is the structure of the invariant ring of $d$ unordered points on $\mathbb{P}^1$, and how can it be described using a graphical algebra?
• RQ2 How do the Hilbert series of the $\mathrm{Alt}(6)$- and $\Sigma_6$-invariant rings of six points on $\mathbb{P}^2$ differ, and what do their generators and relations reveal about the geometry of cubic surfaces?
• RQ3 What is the geometric meaning of the Aronhold invariant and the Morley cubic in the context of Lüroth quartics and the 6-point configuration on a cubic surface?
• RQ4 Why does Kempe’s Lemma fail for points on $\mathbb{P}^2$, and what new generators are required to describe the full invariant ring?
• RQ5 How can the Cremona hexahedral equations be derived from the invariants of six points on $\mathbb{P}^2$, and what do they reveal about the geometry of cubic surfaces?

Key findings

• The Hilbert series of the $\mathrm{Alt}(6)$-invariant ring of six unordered points on $\mathbb{P}^2$ is $\frac{1+t^{15}}{(1-t^2)^2(1-t^3)(1-t^5)(1-t^6)}$, with generators $a_2, d_2, a_3, a_5, a_6, \Delta$ and a relation expressing $\Delta^2$ as a polynomial in the others.
• The $\Sigma_6$-invariant ring of six unordered points on $\mathbb{P}^2$ has Hilbert series $\frac{1+t^{17}}{(1-t^2)(1-t^3)(1-t^4)(1-t^5)(1-t^6)}$, generated by $a_2, a_3, a_4, a_5, a_6, d_2\Delta$, with $(d_2\Delta)^2$ expressible in terms of the other generators.
• The Aronhold invariant of a plane quartic is realized as a Pfaffian of a skew-symmetric matrix, providing a geometric construction of this fundamental invariant.
• The Morley cubic, given by $\sum \overline{a}^2 a$, is a cubic covariant that detects whether the ramification quartic of a sextic of cubics through seven points is a Lüroth quartic.
• The Cremona hexahedral equations — $a+b+c+d+e+f=0$, $\sum \overline{a}a=0$, $\sum a^3=0$ — define the cubic surface as a hyperplane section of the Segre cubic primal, and the 15 lines of the form $a+b=c+d=e+f=0$ correspond to the 15 lines on a cubic surface.
• The invariant $d_2$ of six points on $\mathbb{P}^2$ is not expressible as a polynomial in the linear invariants $\overline{a},\ldots,\overline{f}$, showing that Kempe’s Lemma fails in this setting and requiring $d_2$ as an independent generator of degree 2.