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.Comment: 39 pages, 1 figure; final version with a few expository changes suggested by the refere

Virtual push-forwards

“…S M g;n .G.k; r /; d/ be the universal curve over S M g;n .G.k; r /; d/. Let Proof This has been proved by Marian, Oprea and Pandharipande [19] and a similar situation appears in Popa-Roth [21]. Let us shortly sketch the proof.…”

“…Let O S M g;nC1 .G.1;r /;d/ .E/ D M O M g;nC1 .D/. The line bundle O S M g;nC1 .G.1;r /;d/ .E/ has degree 1 when restricted to the general fiber of the induced map from E to S M g;n .G.1; r /; d/ (see Popa-Roth [21]). We attach the weight ı to such an E if the degree of S restricted to the general fiber in D is ı .…”

“…The relation between moduli spaces of stable maps to projective bundles on some variety X and moduli spaces of stable maps to X have been studied by Gathmann under very restrictive hypothesis. The relation between moduli spaces of stable quotients and moduli spaces of stable maps has been studied by Popa [21], Marian, Oprea and Pandharipande [19] and Toda [22]. Relations between virtual classes have been proved in [19] and [22] by localization techniques.…”

Stable maps and stable quotients

“…Then R 1 p * S((a i + 1)D i ) = 0 and from Lemma 7.1. in [19] we obtain that p * S((a i + 1)D i ) is a vector bundle.…”

“…I am particularly grateful to I Ciocan-Fontanine for pointing out several delicate issues and for very inspiring discussions. Many thanks to M Popa for explaining to me some aspects of his paper [19] and to B Fantechi to whom I owe most of Sections 2.1 and 2.2. I was supported by SFB-647 and by a Marie Curie Intra-European Fellowship: FP7-PEOPLE-2011-IEF.…”