[Paper Review] An affine open covering of $\mathcal{M}_g$ for $g \le 5$

Claudio Fontanari, Stefano Pascolutti|arXiv (Cornell University)|Mar 14, 2011

Algebraic Geometry and Number Theory10 references4 citations

TL;DR

This paper proves that the moduli space $σ\mathcal{M}_g$ of smooth curves of genus $g$ is the union of $g-1$ affine open subsets for all $g$ with $2 \leq g \leq 5$, confirming a conjecture of Looijenga. The proof uses modular forms and theta characteristics, constructing explicit affine open covers via vanishing loci of modular forms associated to theta constants and level-2 structures on the Satake compactification of $\mathcal{A}_g$. The key result is the construction of such covers for $g=4$ and $g=5$, extending known results for $g=2,3$. The method relies on showing that certain divisors in the Satake compactification contain products of periods of lower-genus curves, ensuring the complements are affine.

ABSTRACT

We prove that the moduli space $\mathcal{M}_g$ of smooth curves of genus $g$ is the union of $g-1$ affine open subsets for every $g$ with $2 \le g \le 5$, as predicted by an intriguing conjecture of Eduard Looijenga.

Motivation & Objective

• To prove Looijenga's conjecture that $\mathcal{M}_g$ is the union of $g-1$ affine open subsets for $2 \leq g \leq 5$.
• To extend the known results for $g=2,3$ to $g=4,5$ using modular forms and theta characteristics.
• To construct explicit affine open covers of $\mathcal{M}_g$ via vanishing loci of modular forms on the Satake compactification $\mathcal{A}_g^*$.
• To verify that the complements of certain divisors in $\mathcal{M}_g$ are affine by showing their closures in $\mathcal{A}_g^*$ contain products of periods of lower-genus curves.
• To establish that the intersection of key divisors is empty, ensuring the open cover condition holds.

Proposed method

• Use the Satake compactification $\mathcal{A}_g^* = \operatorname{Proj}(A(\Gamma))$ of the moduli space of principally polarized abelian varieties with level-2 structure.
• Define three key divisors in $\mathcal{M}_4$: $\Theta_{\text{null}}$, $D_1$, and $D_H$, whose complements are candidates for affine open sets.
• For $g=5$, include an additional divisor $D_T$ to form the open cover, with $D_H$ and $D_T$ defined via vanishing of modular forms $F_H$ and $F_T$.
• Apply Corollary 1 to verify that the complement of each divisor is affine by showing its closure in $\mathcal{A}_g^*$ contains products of periods of smooth and nodal curves.
• Use the splitting property of theta constants under block-diagonal period matrices to compute the number of vanishing theta constants on such products.
• Leverage Lemma 3 to show that if more than $v(g)$ even theta constants vanish on a product period, then the modular form $F_H$ or $F_T$ vanishes, implying the period lies in the closure of the divisor.

Experimental results

Research questions

• RQ1 Can the moduli space $\mathcal{M}_g$ be covered by $g-1$ affine open subsets for $g=4$ and $g=5$, as predicted by Looijenga's conjecture?
• RQ2 Do the vanishing loci of specific modular forms $F_H$ and $F_T$ on $\mathcal{A}_g[2]$ define divisors in $\mathcal{M}_g$ whose complements are affine?
• RQ3 Is the intersection of $\Theta_{\text{null}}$, $D_1$, and $D_H$ (or $D_T$) empty in $\mathcal{M}_g$, ensuring the open cover condition?
• RQ4 Does the closure of the divisor $D_H$ in $\mathcal{A}_g^*$ contain all products of periods of smooth and nodal curves of lower genus?
• RQ5 Can the number of vanishing even theta constants on a product of period matrices be used to verify that a modular form vanishes on such points?

Key findings

• For $g=4$, $\mathcal{M}_4$ is the union of three affine open subsets: $\mathcal{M}_4 \setminus \Theta_{\text{null}}$, $\mathcal{M}_4 \setminus D_1$, and $\mathcal{M}_4 \setminus D_H$, with $D_H$ defined by the vanishing of a modular form $F_H$.
• For $g=5$, $\mathcal{M}_5$ is the union of four affine open subsets: $\mathcal{M}_5 \setminus \Theta_{\text{null}}$, $\mathcal{M}_5 \setminus D_1$, $\mathcal{M}_5 \setminus D_H$, and $\mathcal{M}_5 \setminus D_T$, with $D_T$ defined by the vanishing of a modular form $F_T$.
• The closure of $D_H$ in $\mathcal{A}_4^*$ contains all products of periods of smooth and nodal curves of genus 2 and 2, or genus 1 and 3, due to more than $v(4)=10$ even theta constants vanishing on such points.
• The closure of $D_H$ in $\mathcal{A}_5^*$ contains all such products because more than $v(5)=66$ even theta constants vanish on period matrices of the form $\operatorname{diag}(\tau_1, \tau_2)$ with $g_1+g_2=5$.
• The intersection $\Theta_{\text{null}} \cap D_1 \cap D_H$ is empty in $\mathcal{M}_4$, and $\Theta_{\text{null}} \cap D_1 \cap D_T \cap D_H$ is empty in $\mathcal{M}_5$, ensuring the open cover condition.
• The complements $\mathcal{M}_g \setminus D_H$ and $\mathcal{M}_g \setminus D_T$ are affine because the closures of $D_H$ and $D_T$ in $\mathcal{A}_g^*$ contain the relevant boundary strata of products of periods.