[Paper Review] On the geometry of the compactification of the universal Picard variety

Claudio Fontanari|ArXiv.org|Feb 18, 2002

Algebraic Geometry and Number Theory11 references5 citations

TL;DR

This paper investigates the geometry of the compactified universal Picard variety $\overline{P}_{d,g}$, constructed via geometric invariant theory by Caporaso. It establishes a natural injective morphism from Cornalba's moduli space of spin curves $\overline{S}_g$ into $\overline{P}_{d,g}$, and provides a complete description of the rational divisor class group of $\overline{P}_{d,g}$, showing it is generated by the tautological line bundle $\mathcal{L}_{d,g}$, the pullback of the Hodge class $\phi_d^*(\lambda)$, and boundary divisors $\mathcal{D}_i$, with no nontrivial relations under certain conditions.

ABSTRACT

Here we focus on the geometry of $\pdgbar$, the compactification of the universal Picard variety constructed by L. Caporaso. In particular, we show that the moduli space of spin curves constructed by M. Cornalba naturally injects into $\pdgbar$ and we give generators and relations of the rational divisor class group of $\pdgbar$, extending previous work by A. Kouvidakis.

Motivation & Objective

• To study the geometric structure of the compactified universal Picard variety $\overline{P}_{d,g}$, constructed via geometric invariant theory by Caporaso.
• To investigate the relationship between $\overline{P}_{d,g}$ and the moduli space of spin curves $\overline{S}_g$, particularly whether $\overline{S}_g$ embeds naturally into $\overline{P}_{d,g}$.
• To compute the rational divisor class group of $\overline{P}_{d,g}$, extending Kouvidakis' work on the Picard group of the non-compactified $P_{d,g}$.
• To determine conditions under which the rational Picard group of $\overline{P}_{d,g}$ is freely generated by explicit classes.

Proposed method

• Leveraging the geometric invariant theory construction of $\overline{P}_{d,g}$, the paper analyzes its boundary components and their codimensions.
• Establishing a natural morphism from $\overline{S}_g$ to $\overline{P}_{d,g}$ by associating to each spin curve a line bundle of degree $d$ on a quasistable curve.
• Using the known Picard group of $P_{d,g}$, as established by Kouvidakis, to deduce generators for the rational divisor class group of $\overline{P}_{d,g}$.
• Applying a lifting argument from Arbarello and Cornalba's families of curves to show the absence of nontrivial relations in the divisor class group.
• Employing the fact that $\overline{P}_{d,g}$ is a geometric quotient and has finite quotient singularities under the coprime condition $(d-g+1,2g-2)=1$ to relate Weil divisors to Cartier divisors.
• Proving injectivity of the pullback map $\phi_d^*$ on rational Chow groups to establish linear independence of the generators.

Experimental results

Research questions

• RQ1 Does the moduli space of spin curves $\overline{S}_g$ naturally embed into the compactified universal Picard variety $\overline{P}_{d,g}$?
• RQ2 What are the generators of the rational divisor class group of $\overline{P}_{d,g}$, and what relations exist among them?
• RQ3 Under what conditions is the rational Picard group of $\overline{P}_{d,g}$ freely generated by explicit classes?
• RQ4 How does the rational divisor class group of $\overline{P}_{d,g}$ relate to the tautological classes on $\overline{\mathcal{M}}_g$?

Key findings

• There exists a natural injective morphism from the moduli space of spin curves $\overline{S}_g$ to the compactified universal Picard variety $\overline{P}_{d,g}$, establishing a precise geometric link between the two spaces.
• The rational divisor class group of $\overline{P}_{d,g}$ is generated by the tautological line bundle $\mathcal{L}_{d,g}$, the pullback of the Hodge class $\phi_d^*(\lambda)$, and the boundary divisors $\mathcal{D}_i$ for $i=0,\ldots,\lfloor g/2\rfloor$.
• When $g \geq 4$, the rational divisor class group is generated by these classes, and the relations are trivial, meaning no nontrivial linear relations exist among them.
• For $g=3$, the hyperelliptic locus $H$ in $\mathcal{M}_3$ satisfies $[H] = 18\lambda$, so $[\phi_d^{-1}(H)] = 18\phi_d^*(\lambda)$, which is consistent with the generator set.
• The pullback map $\phi_d^*:\mathrm{Pic}(\overline{\mathcal{M}}_g) \otimes \mathbb{Q} \to \mathrm{Pic}(\overline{P}_{d,g}) \otimes \mathbb{Q}$ is injective when $d \geq 20(g-1)$ and $(d-g+1,2g-2)=1$, ensuring the generators are linearly independent.
• Under the coprime condition $(d-g+1,2g-2)=1$, the rational Picard group of $\overline{P}_{d,g}$ is freely generated by $\mathcal{L}_{d,g}$, $\phi_d^*(\lambda)$, and the $\mathcal{D}_i$ classes.