[Paper Review] Heisenberg invariant quartics and SU_C(2) for a curve of genus four

W. M. Oxbury, Christian Pauly|ArXiv.org|Mar 21, 1997

Algebraic Geometry and Number Theory7 references16 citations

TL;DR

This paper investigates the geometry of the moduli space $Σ_C(2)$ of semistable rank 2 vector bundles with trivial determinant on a genus 4 curve $C$, showing that its image under the morphism $φ: \Sigma_C(2) \to |2\Theta|$ lies in the singular locus of a unique irreducible Heisenberg-invariant quartic $Q_C \subset \mathbb{P}^{15}$. The key result establishes the existence and uniqueness of $Q_C$ when $C$ has no vanishing theta-nulls, using cubic normality and Verlinde formula arguments, with implications for the structure of higher secant varieties and Prym varieties via restriction to $|2\Xi|$ subspaces.

ABSTRACT

If C is a curve of genus 4 without vanishing theta-nulls then there exists a unique (irreducible) Heisenberg-invariant quartic Q_C in |2Θ| = P^{15} such that Sing Q_C contains the image of SU_C(2), the moduli space of rank 2 vector bundles with trivial determinant. Moreover, in each eigen-P^7 of the Heisenberg action on |2Θ|, Q_C restricts to the classical Coble quartic of the corresponding Prym-Kummer variety. We compare Q_C with the hypersurface G_3 in |2Θ| of divisors containing a translate of C in J(C), and show that in the eigen-P^7s G_3 recovers Beauville--Debarre's quadrisecant planes of the Prym-Kummers (this works for any genus). Using the Recillas construction this enables us to deduce, contrary to the analogous result for genus 3, that Q_C and G_3 are distinct.

Motivation & Objective

• To extend the Narasimhan–Ramanan results on $\mathcal{SU}_C(2)$ for genus 2 and 3 curves to genus 4.
• To characterize the image $\phi(\mathcal{SU}_C(2))$ in $|2\Theta| \cong \mathbb{P}^{15}$ via Heisenberg-invariant quartics.
• To investigate whether $\phi(\mathcal{SU}_C(2))$ coincides with the singular locus of a canonical quartic $Q_C$, analogous to the Coble quartic in genus 3.
• To analyze the relationship between $\mathcal{SU}_C(2)$ and the Heisenberg-invariant varieties $G_d \subset |2\Theta|$ defined by translates of $W_{g-d}$.
• To explore the restriction of $Q_C$ and $G_3$ to Prym varieties $P_\eta$ via the Recillas correspondence, and compare them to known secant plane structures.

Proposed method

• Prove cubic normality of $\phi(\mathcal{SU}_C(2))$ in $\mathbb{P}^{15}$ using geometric and cohomological techniques.
• Apply the Verlinde formula to show that the ideal of $\phi(\mathcal{SU}_C(2))$ contains exactly sixteen independent cubics.
• Use symmetry of the Heisenberg action to deduce that these cubics are partial derivatives of a single irreducible quartic $Q_C$, establishing its uniqueness.
• Restrict the ambient space $|2\Theta| \cong \mathbb{P}^{15}$ to eigen-planes $|2\Xi| \subset \mathbb{P}^7$ under the action of $J_C[2]$, identifying them with $|2\Theta|$ of Prym varieties.
• Use Beauville–Debarre results on quadrisecant planes of Prym Kummer varieties to compare the restriction of $G_3$ and $Q_C$ to $|2\Xi|$, showing they differ.
• Construct a filtration $\mathrm{Kum}(J) \subset G_2 \subset \phi(\mathcal{SU}_C(2)) \subset G_3 \subset \mathbb{P}^{15}$, where $G_2$ is a divisor ruled by 4-planes and $G_3$ by 10-planes, polar with respect to theta characteristics.

Experimental results

Research questions

• RQ1 Does there exist a unique irreducible Heisenberg-invariant quartic $Q_C \subset \mathbb{P}^{15}$ containing $\phi(\mathcal{SU}_C(2))$ in its singular locus for a genus 4 curve without vanishing theta-nulls?
• RQ2 Is $\phi(\mathcal{SU}_C(2))$ equal to the singular locus of $Q_C$, i.e., is its ideal generated by cubics?
• RQ3 How do the varieties $G_d \subset |2\Theta|$ defined by translates of $W_{g-d}$ relate to $\mathcal{SU}_C(2)$ and $Q_C$ in genus 4?
• RQ4 What is the behavior of $Q_C$ and $G_3$ upon restriction to $|2\Xi|$ for Prym varieties $P_\eta$ associated to 2-torsion points $\eta \in J_C[2]$?
• RQ5 Do the quadrisecant planes of $\mathrm{Kum}(P_\eta)$, which are known to exist for genus 3 Pryms, coincide with the restriction of $G_3$ to $|2\Xi|$?

Key findings

• For a nonhyperelliptic genus 4 curve $C$ without vanishing theta-nulls, there exists a unique irreducible Heisenberg-invariant quartic $Q_C \subset \mathbb{P}^{15}$ such that $\phi(\mathcal{SU}_C(2)) \subset \mathrm{Sing}(Q_C)$.
• The image $\phi(\mathcal{SU}_C(2))$ is cubically normal in $\mathbb{P}^{15}$, and the Verlinde formula implies exactly sixteen independent cubics vanish on it.
• These cubics are the partial derivatives of a single quartic $Q_C$, which is therefore unique and irreducible.
• The restriction of $Q_C$ to each $|2\Xi| \subset \mathbb{P}^{15}$ (fixed by $\eta \in J_C[2]$) is the Coble quartic of the Prym Kummer $\mathrm{Kum}(P_\eta)$.
• The restriction of $G_3$ to $|2\Xi|$ is the hypersurface ruled by the 4-parameter family of quadrisecant planes of $\mathrm{Kum}(P_\eta)$, which is distinct from the Coble quartic.
• The filtration $\mathrm{Kum}(J) \subset G_2 \subset \phi(\mathcal{SU}_C(2)) \subset G_3 \subset \mathbb{P}^{15}$ holds, with $G_2$ a divisor ruled by 4-planes and $G_3$ by 10-planes polar with respect to any theta characteristic.