The Hesse Pencil Variety

TL;DR

The paper studies the Hesse pencil variety for plane cubics, defining $H_8$ as the $8$-dimensional locus in $G(1,9)$ of pencils generated by a smooth cubic and its Hessian. A key tool is a cubic skew-invariant $R$ in $∧^3(Sym^3(C^3))$ that detects pencils arising from a cubic and its Hessian, leading to a linear section $N$ of $G(1,9)$; the authors prove $H_8=N$ by matching dimension and the Schubert multidegree $(1,3,9,12,6)$, obtained via a Hesse-configuration count yielding degree $622$. They show $H_8$ coincides with the $SL(3)$-orbit closure of the Fermat-like pencil $⟨x^3+y^3+z^3,xyz⟩$ and contains eight additional $SL(3)$-orbits, while its singular locus is the union of two specific orbits. The work also develops the binary quartic baseline, where the analogous Hesse pencil variety $H_3$ is a smooth Fano $3$-fold realized as the intersection of $G(1,4)$ with three hyperplanes, with degree $5$ and multidegree $(1,2)$. Overall, the paper gives explicit equations, orbit structure, and a complete geometric description of Hesse pencils for plane cubics in terms of invariant theory and Schubert calculus.

Abstract

We introduce and study the Hesse pencil variety $H_8$, obtained as the Zariski closure in the Grassmannian $G(1,9)$ of the set of pencils generated by a smooth plane cubic and its Hessian. We prove that $H_8$ has dimension $8$ and can be realized as the intersection of $G(1,9)$ with ten hyperplanes corresponding to the Schur module $\mathbb{S}_{(5,1)}\mathbb{C}^3$. Moreover, $H_8$ coincides with the closure of the $SL(3)$-orbit of the pencil $\langle x^3+y^3+z^3,\ xyz\rangle$ and contains eight additional orbits. The variety is singular, and its singular locus is precisely the union of two orbits, $O(\langle x^3,x^2y\rangle)$ and $O(\langle x^2y,x^2z\rangle)$. A key ingredient in our study is a cubic skew-invariant $R\in \bigwedge^3(\mathrm{Sym}^3\mathbb{C}^3)$ defined by $R(l^3,m^3,n^3)=(l\wedge m\wedge n)^3$, whose vanishing characterizes pencils generated by a cubic and its Hessian. This invariant allows us to write explicit equations defining $H_8$. A crucial geometric step in our argument is the fact that through four general points of $\mathbb{P}^2$ there pass exactly six Hesse configurations, which enables us to compute the multidegree of $H_8$ and conclude that it coincides with the variety defined by the invariant $R$.

Figures (9)

• Figure 1: This figure represents nine points in Hesse configuration and the twelve lines that characterize them. The eight black lines are clearly visible, while the remaining four could not be explicitly drawn. Instead, four different colours were used to connect the three points on each of these lines.
• Figure 1: $\{y=z\}\cap\{x=0\}=(0,1,1)$
• Figure 2: $\{x=z\}\cap\{y=0\}=(1,0,1)$
• Figure 3: $\{x=y\}\cap\{z=0\}=(1,1,0)$
• Figure :

Theorems & Definitions (43)

• Definition 1.0
• Definition 1.0
• Theorem 1.1
• Corollary 1.1
• Proposition 1.1
• Definition 2.1: The Hesse Pencil
• Proposition 2.2
• proof
• Definition 2.2
• Proposition 2.3