[Paper Review] A compactification of the space of maps from curves

Bumsig Kim, Andrew Kresch+1 more|arXiv (Cornell University)|May 31, 2011

Algebraic Geometry and Number Theory20 references4 citations

TL;DR

This paper constructs a new compactification, denoted $\overline{\mathfrak{U}}_{g,\mu}(X,\beta)$, of the moduli space of genus $g$ stable ramified maps from $n$-pointed curves to a nonsingular projective variety $X$, with prescribed ramification indices $\mu_i$ at marked points. The compactification is a proper Deligne-Mumford stack equipped with a natural virtual fundamental class, enabling the definition of ramified Gromov-Witten invariants via integration against this class.

ABSTRACT

We construct a new compactification of the moduli space of maps from pointed nonsingular projective stable curves to a nonsingular projective variety with prescribed ramification indices at the points. It is shown to be a proper Deligne-Mumford stack equipped with a natural virtual fundamental class.

Motivation & Objective

• To construct a proper compactification of the moduli space of maps from genus $g$ curves to a nonsingular projective variety $X$, with prescribed ramification at $n$ marked points.
• To unify four known compactifications: Fulton-MacPherson configuration spaces, Kontsevich's stable maps, Harris-Mumford's admissible covers, and Li's stable relative maps.
• To define ramified Gromov-Witten invariants using a virtual fundamental class on the new moduli space.
• To establish a potential link between these invariants and BPS counts in algebraic geometry, as conjectured by Pandharipande.

Proposed method

• Utilizes Fulton-MacPherson degeneration spaces as target geometries, constructed via iterated blowups of $X[n] \times X$ along smooth centers.
• Employs log structures and the notion of (log) admissible maps to control ramification and ensure properness.
• Applies deformation theory for log maps and constructs a perfect obstruction theory via the dual of $R\pi_*(f^*T^\dagger_{\mathcal{W}}(-\sum \mu_i p_i))$.
• Uses the virtual fundamental class construction from [4] and [21], relying on the algebraicity and pure dimensionality of the base stack $\mathfrak{B}$ of prestable curves with FM spaces.
• Introduces a natural morphism $Te_i: \overline{\mathfrak{U}}_{g,\mu}(X,\beta) \to \mathbb{P}TX$ to define invariants via integration against $\psi$-classes and pullbacks of cohomology classes.
• Applies results from [23] and [21] to ensure the existence of the virtual class and to verify properness and Deligne-Mumford structure.

Experimental results

Research questions

• RQ1 Can a compactification of the moduli space of stable ramified maps be constructed that incorporates known compactifications like stable maps and admissible covers?
• RQ2 Does the resulting moduli space carry a natural virtual fundamental class enabling the definition of Gromov-Witten invariants in the ramified setting?
• RQ3 Is there a conjectural link between ramified Gromov-Witten invariants and BPS counts, particularly in the locally Fano case?
• RQ4 Can the invariants defined via this compactification recover known integer-valued BPS invariants as conjectured by Pandharipande?

Key findings

• The moduli space $\overline{\mathfrak{U}}_{g,\mu}(X,\beta)$ is a proper Deligne-Mumford stack over an algebraically closed field of characteristic zero.
• It admits a natural virtual fundamental class, making it suitable for defining ramified Gromov-Witten invariants.
• The construction unifies four key compactifications: Fulton-MacPherson configuration spaces, Kontsevich's stable maps, Harris-Mumford's admissible covers, and Li's stable relative maps.
• When $\mu = (1,\dots,1)$, the space reduces to the standard unramified Gromov-Witten moduli space $\overline{\mathfrak{U}}_{g,n}(X,\beta)$, recovering classical invariants.
• For locally Fano curve classes on a threefold, the ramified invariants are conjectured to coincide with BPS counts $n_{g,\beta}(\gamma_1,\dots,\gamma_n)$, and this is supported by Zinger's result on integrality in the unramified case.
• The virtual fundamental class is constructed via a perfect obstruction theory using log tangent sheaves and deformation-theoretic tools from [23] and [21].