The moduli space of stable quotients

Abstract

A moduli space of stable quotients of the rank n trivial sheaf on stable curves is introduced. Over nonsingular curves, the moduli space is Grothendieck's Quot scheme. Over nodal curves, a relative construction is made to keep the torsion of the quotient away from the singularities. New compactifications of classical spaces arise naturally: a nonsingular and irreducible compactification of the moduli of maps from genus 1 curves to projective space is obtained. Localization on the moduli of stable quotients leads to new relations in the tautological ring generalizing Brill-Noether constructions. The moduli space of stable quotients is proven to carry a canonical 2-term obstruction theory and thus a virtual class. The resulting system of descendent invariants is proven to equal the Gromov-Witten theory of the Grassmannian in all genera. Stable quotients can also be used to study Calabi-Yau geometries. The conifold is calculated to agree with stable maps. Several questions about the behavior of stable quotients for arbitrary targets are raised.

50 pages

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Cited by in corpus (56)

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• A Mathematical Theory of the Gauged Linear Sigma Model
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• Moduli of stable maps in genus one and logarithmic geometry I
• A mirror theorem for genus two Gromov-Witten invariants of quintic threefolds
• Relations in the tautological ring
• Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics
• Relations on ${\overline{M}}_{g,n}$ via equivariant Gromov-Witten theory of ${\mathbb{P}}^1$
• Quasimap Wall-crossings and Mirror Symmetry
• Gromov-Witten invariants of stable maps with fields
• Higher-genus wall-crossing in the gauged linear sigma model
• Tautological relations in moduli spaces of weighted pointed curves
• Virtual push-forwards
• The hypergeometric functions of the Faber-Zagier and Pixton relations
• The kappa ring of the moduli of curves of compact type: II
• Genus-One Mirror Symmetry in the Landau-Ginzburg Model
• Differential Rings from Special Kähler Geometry
• Effective divisors on moduli spaces of curves and abelian varieties
• Localized Chern Characters for 2-periodic complexes
• Relative quasimaps and mirror formulae
• Chow Rings of Heavy/Light Hassett Spaces via Tropical Geometry
• Computing top intersections in the tautological ring of $M_g$
• Multi-Point Virtual Structure Constants and Mirror Computation of CP^2-model
• Quasimap wall-crossing for GIT quotients
• A Brief Survey of FJRW Theory
• Deformation of quotients on a product
• Scattering amplitudes of stable curves
• Stable quotients and the holomorphic anomaly equation
• A new method toward the Landau-Ginzburg/Calabi-Yau correspondence via quasi-maps
• The Geometry of Stable Quotients in Genus One
• A mathematical theory of D-string world-sheet instantons, II: Moduli stack of $Z$-(semi)stable morphisms from Azumaya nodal curves with a fundamental module to a projective Calabi-Yau 3-fold
• Quasimaps to moduli spaces of sheaves
• Stable maps and stable quotients
• Hori's Equation for Gravitational Virtual Structure Constants of Calabi-Yau Hypersurface in CP^{N-1}
• The Moduli Space in the Gauged Linear Sigma Model
• On the asymptotic enumerativity property for Fano manifolds
• Enumerativity of virtual Tevelev degrees
• Equivariant holomorphic anomaly equation
• Tropical double ramification loci
• Higher-genus wall-crossing in Landau-Ginzburg theory
• Quasimaps to GIT fiber bundles and applications
• The local/logarithmic correspondence and the degeneration formula for quasimaps
• Double and Triple Givental's J-functions for Stable Quotients Invariants
• Vanishing of Gromov-Witten invariants of product of P1
• Crepant resolution conjecture for ${\mathbb{C}}^5/{\mathbb{Z}}_5$
• Double $J$-function of stable quasimap invariants for complete intersection in Grassmannian
• Alternate compactifications of the moduli space of genus one maps
• Comparison of two desingularization of the Kontsevich's moduli space of elliptic stable maps
• A survey on mixed spin P-fields
• On the cohomology of the Losev-Manin moduli space
• A wall-crossing formula and the invariance of GLSM correlation functions
• $K$-theoretic quasimap wall-crossing
• Automorphisms of tropical Hassett spaces
• Period Integrals (Givental's I-function) of Calabi-Yau Hypersurface in $C{P}^{N-1}$ and Intersection Numbers of Moduli Space of Quasimaps from $C{P}^1$ with Two Marked Points to $C{P}^{N-1}$