Stable maps and Quot schemes

Abstract

In this paper we study the relationship between two different compactifications of the space of vector bundle quotients of an arbitrary vector bundle on a curve. One is Grothendieck's Quot scheme, while the other is a moduli space of stable maps to the relative Grassmannian. We establish an essentially optimal upper bound on the dimension of the two compactifications. Based on that, we prove that for an arbitrary vector bundle, the Quot schemes of quotients of large degree are irreducible and generically smooth. We precisely describe all the vector bundles for which the same thing holds in the case of the moduli spaces of stable maps. We show that there are in general no natural morphisms between the two compactifications. Finally, as an application, we obtain new cases of a conjecture on effective base point freeness for pluritheta linear series on moduli spaces of vector bundles.

39 pages, 1 figure; final version with a few expository changes suggested by the referee

References in corpus (5)

• Geography of Brill-Noether loci for small slopes
• Vector bundles on curves and generalized theta functions: recent results and open problems
• Stable maps and Hurwitz schemes in mixed characteristic
• Quantum cohomology of flag varieties
• Quantum cohomology of flag manifolds

Cited by in corpus (20)

• The moduli space of stable quotients
• Generalized theta linear series on moduli spaces of vector bundles on curves
• Quantum Witten localization and abelianization for qde solutions
• Moduli spaces of stable quotients and wall-crossing phenomena
• Virtual push-forwards
• Parabolic reductions of principal bundles
• Relative quasimaps and mirror formulae
• Vector bundles on curves and theta functions
• Hyperquot schemes on curves: virtual class and motivic invariants
• Coherent Sheaves on Ribbons and their Moduli
• 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
• Maximal subbundles and Gromov invariants
• Stable maps and stable quotients
• A weighted topological quantum field theory for Quot schemes on curves
• Ulrich bundles on intersections of two 4-dimensional quadrics
• Maximal subbundles, quot schemes, and curve counting
• On the intersection theory of Quot schemes and moduli of bundles with sections
• Some examples of vector bundles in the base locus of the generalized theta divisor
• Effective very ampleness for generalized theta divisors
• Theta functions on the moduli space of parabolic bundles