[Paper Review] The Moduli Space of Stables Maps with Divisible Ramification

Oliver Leigh|arXiv (Cornell University)|Dec 17, 2018

Algebraic Geometry and Number Theory16 references4 citations

TL;DR

This paper constructs a natural compactification of the moduli space of stable maps with divisible ramification, where every ramification divisor is r-fold. It establishes a perfect obstruction theory and virtual fundamental class for genus zero maps, using rth roots of canonical sections to define the ramification condition, enabling applications to r-spin Hurwitz theory and the r-spin ELSV formula via virtual localization.

ABSTRACT

We develop a theory for stable maps to curves with divisible ramification. For a fixed integer $r>0$, we show that the condition of every ramification locus being divisible by $r$ is equivalent to the existence of an $r$th root of a canonical section. We consider this condition in regards to both absolute and relative stable maps and construct natural moduli spaces in these situations. We construct an analogue of the Fantechi-Pandharipande branch morphism and when the domain curves are genus zero we construct a virtual fundamental class. This theory is anticipated to have applications to $r$-spin Hurwitz theory. In particular it is expected to provide a proof of the $r$-spin ELSV formula [SSZ'15, Conj. 1.4] when used with virtual localisation.

Motivation & Objective

• To develop a compact moduli space for stable maps with ramification divisible by a fixed integer r.
• To reframe the ramification condition using rth roots of canonical sections, enabling compactification in the singular and nodal curve setting.
• To construct a virtual fundamental class for genus zero maps with divisible ramification.
• To generalize the Fantechi-Pandharipande branch morphism to this setting.
• To provide a foundation for proving the r-spin ELSV formula using virtual localization techniques.

Proposed method

• Define the moduli space via the existence of a line bundle L on the domain curve C, a section σ of L, and an isomorphism e:L^r → ω_C ⊗ f^*ω_X^∨ such that e(σ^r) = δ, where δ is the dual of the differential map.
• Use r-prestable curves—stacks with balanced r-orbifold points over nodes of the coarse space—to handle singular domains and ensure proper compactification.
• Construct a branch morphism analogous to the Fantechi-Pandharipande morphism, mapping to a space of branched covers.
• Establish a perfect obstruction theory by pulling back via a flat morphism from the moduli space of stable maps to a target space of degenerate targets.
• Use cone constructions and derived categories to define the virtual fundamental class in genus zero.
• Leverage étale base change in genus zero to transfer the perfect obstruction theory from a partial compactification to the full moduli space.

Experimental results

Research questions

• RQ1 How can the condition of divisible ramification (i.e., ramification divisor divisible by r) be naturally compactified in the moduli space of stable maps?
• RQ2 What is the correct geometric formulation of divisible ramification that extends to singular and nodal curves?
• RQ3 Can a virtual fundamental class be constructed for the moduli space of stable maps with divisible ramification when the domain has genus zero?
• RQ4 How does the branch morphism generalize in this context of r-divisible ramification?
• RQ5 What is the relationship between this moduli space and r-spin Hurwitz theory, particularly in proving the r-spin ELSV formula?

Key findings

• The moduli space of stable maps with r-divisible ramification, denoted $\overline{\mathcal{M}}^{1/r}_{g}(X,d)$, is a proper Deligne-Mumford stack.
• The space is non-empty only when r divides $2g - 2 - d(2g_X - 2)$, ensuring a topological obstruction condition is satisfied.
• The forgetful map $\chi: \overline{\mathcal{M}}^{1/r}_{g}(X,d) \to \overline{\mathcal{M}}_{g}(X,d)$ is flat and of relative dimension zero, and is an immersion over the open locus of maps with r-divisible ramification.
• For genus zero, a perfect obstruction theory exists, leading to a virtual fundamental class of dimension m, where m is the expected dimension.
• The virtual fundamental class is constructed via a cone construction on a pullback of the relative dualizing complex, using the rth root data.
• The étale base change from a partial compactification ensures the virtual fundamental class descends to the full moduli space in genus zero.