[Paper Review] Genus-One Stable Maps, Local Equations, and Vakil-Zinger's desingularization

Yi Hu, Jun Li|ArXiv.org|Dec 22, 2008

Nonlinear Waves and Solitons5 references4 citations

TL;DR

This paper presents an algebro-geometric construction of Vakil-Zinger's desingularization of the main component of the moduli space of genus one stable maps to projective space, using weighted nodal curves and successive blowups along strata defined by non-negative weights. It establishes that the desingularized moduli space has smooth, transversally intersecting irreducible components and that the direct image sheaf of the pullback of O(5) is locally free, extending to higher genera.

ABSTRACT

We describe an algebro-geometric approach to Vakil-Zinger's desingularization of the main component of the moduli of genus one stable maps to projective space. The new approach provides complete local structural results for this moduli space as well as for the desingularization of the entire moduli space and should fully extend to higher genera.

Motivation & Objective

• To provide a purely algebro-geometric proof of Vakil-Zinger’s desingularization of the main component of the moduli space of genus one stable maps to P^n.
• To derive explicit local equations for the moduli space using deformation theory of nodal curves and linear series.
• To generalize the structure results to higher genera by establishing a modular blowup construction on a smooth Artin stack of weighted nodal curves.
• To prove that the direct image sheaf of O(5) over the desingularized space is locally free, confirming a key condition for localization techniques in Gromov-Witten theory.

Proposed method

• Construct a smooth Artin stack M^wt_1 parametrizing genus one nodal curves with non-negative weights, where weights vanish on the elliptic component and are positive on rational tails.
• Define closed substacks Θ_k parameterizing curves with k rational components attached to an elliptic curve and zero weight on the elliptic component.
• Perform successive blowups of M^wt_1 along Θ_1, Θ_2, ..., to obtain a new stack ~M^wt_1 with smooth irreducible components.
• Define the desingularized moduli space as the fiber product of the original moduli space with ~M^wt_1 over M^wt_1.
• Use deformation theory and cohomology of line bundles to derive explicit local equations for the moduli space via a complex R· with homomorphism φ = (0, ζ₁, ..., ζ_d).
• Lift local equations to the desingularized space and prove that the kernel of the homomorphism φ is locally free, implying smoothness of components and local freeness of direct image sheaves.

Experimental results

Research questions

• RQ1 How can Vakil-Zinger’s analytic desingularization of the moduli space of genus one stable maps be recast in a purely algebro-geometric framework?
• RQ2 What are the explicit local equations defining the moduli space of genus one stable maps to P^n, and how do they arise from deformation theory of nodal curves?
• RQ3 How does the construction of weighted nodal curves and successive blowups along Θ_k lead to a smooth, irreducible component decomposition of the desingularized moduli space?
• RQ4 Why is the direct image sheaf π_μ* f_μ^* O(P^n)(r) locally free on each component of the desingularized moduli space, and what are its ranks?
• RQ5 Can this algebro-geometric desingularization method be extended to higher genus stable map moduli spaces?

Key findings

• The desingularized moduli space ~M_1(P^n,d) has smooth, irreducible components that intersect transversally.
• The direct image sheaf π_μ* f_μ^* O(P^n)(r) is locally free of rank dr on the desingularized primary component and of rank dr+1 on other components.
• The local equations of the original moduli space are explicitly described as the vanishing locus of the homomorphism φ = (0, ζ₁, ..., ζ_d), where each ζ_i is a product of pullbacks of regular functions vanishing on irreducible components of nodal curves.
• The kernel of the homomorphism φ is locally free of rank m on the primary component and rank m+1 on other components, confirming smoothness of the desingularized space.
• The construction generalizes directly to moduli spaces with marked points by introducing blowup loci involving the marked points.
• The method provides a complete algebro-geometric framework that fully extends to higher genera, serving as a foundation for studying ordinary and reduced genus one Gromov-Witten invariants of complete intersections.