[Paper Review] The Schottky problem in genus five

Charles Siegel|arXiv (Cornell University)|Feb 24, 2013

Algebraic Geometry and Number Theory29 references3 citations

TL;DR

This paper solves the Schottky problem in genus five by analyzing the Prym map for genus six curves and using incidence structures on fibers of the map $\beta_5$. It proves that the small Schottky locus $\mathcal{S}_5^{\text{small}}$ is precisely the Jacobian locus $\mathcal{J}_5$, resolving a long-standing conjecture by Donagi and confirming the irreducibility of the small Schottky locus in genus five.

ABSTRACT

In this paper, we present a solution to the Schottky problem in the spirit of Schottky and Jung for genus five curves. To do so, we exploit natural incidence structures on the fibers of several maps to reduce all questions to statements about the Prym map for genus six curves. This allows us to find all components of the big Schottky locus and thus, to show that the small Schottky locus introduced by Donagi is irreducible.

Motivation & Objective

• To resolve the Schottky problem in genus five by characterizing the Jacobian locus within the moduli space of principally polarized abelian varieties.
• To prove that the small Schottky locus $\mathcal{S}_5^{\text{small}}$ is irreducible and equal to the Jacobian locus $\mathcal{J}_5$.
• To extend the Schottky-Jung program by analyzing degenerations and contracted loci of the map $\beta_5$ on the moduli space of twisted Prym varieties.
• To complete the classification of components of the big Schottky locus in genus five using incidence structures and local degree computations.

Proposed method

• The paper uses the map $\beta_5$ from the moduli space of twisted Prym varieties to $\mathcal{A}_4$, analyzing its fibers via $Q_6^-$-configurations over $\mathbb{F}_2$.
• It applies incidence geometry on point-line configurations over $\mathbb{F}_2$ to compute the degree of $\beta_5$, which is found to be 119.
• The study of degenerations of abelian varieties in genus five, based on Izadi’s work, enables the classification of components in the preimage of $\alpha_4(\mathcal{A}_4)$.
• Contracted loci of $\beta_5$ are analyzed to identify which components are not of positive local degree, isolating the relevant components of the Schottky locus.
• Local degree computations are performed on $\overline{\mathcal{RJ}}_5$, $\overline{\mathcal{RC}^0}$, and $\partial^I\overline{\mathcal{RA}}_5^t$, yielding degrees 54, 1, and 64 respectively.
• The sum of local degrees (1 + 54 + 64 = 119) confirms that no other components contribute to the Schottky locus, completing the classification.

Experimental results

Research questions

• RQ1 Is the small Schottky locus $\mathcal{S}_5^{\text{small}}$ irreducible and equal to the Jacobian locus $\mathcal{J}_5$ in genus five?
• RQ2 Which components of the preimage $\beta_5^{-1}(\alpha_4(\mathcal{A}_4))$ contribute to the Schottky locus with positive local degree?
• RQ3 What is the total degree of the map $\beta_5$, and how does it constrain the structure of the Schottky locus?
• RQ4 Can the Schottky-Jung conjecture be corrected and verified in genus five using Prym map geometry and degeneration techniques?
• RQ5 Which loci in the moduli space of twisted Prym varieties are contracted by $\beta_5$, and how do they affect the Schottky locus?

Key findings

• The map $\beta_5$ has degree 119, computed via incidence structures on $Q_6^-$-configurations over $\mathbb{F}_2$.
• The only component of $\beta_5^{-1}(\alpha_4(\mathcal{A}_4))$ that is blown down is $\mathcal{A}_4 \times \mathcal{RA}_1$, while all others have positive local degree.
• The local degree of $\beta_5$ is 1 on $\overline{\mathcal{RC}^0}$, 54 on $\overline{\mathcal{RJ}}_5$, and 64 on $\partial^I\overline{\mathcal{RA}}_5^t$, summing to 119.
• The sum of local degrees confirms that no other components of $\mathcal{RS}_5$ contribute, so $\mathcal{RS}_5 = \overline{\mathcal{RJ}}_5 \cup \overline{\mathcal{RC}^0} \cup \partial^I\overline{\mathcal{RA}}_5^t \cup \mathcal{A}_4 \times \mathcal{RA}_1$.
• Since only Jacobians have all nonzero 2-torsion points in $\mathcal{RS}_5$, it follows that $\mathcal{S}_5^{\text{small}} = \mathcal{J}_5$.
• The small Schottky locus $\mathcal{S}_5^{\text{small}}$ is irreducible, as it is equal to the irreducible Jacobian locus $\mathcal{J}_5$.