Stable maps and Quot schemes

By Mihnea Cristian Popa, Mike Roth. 2000

59 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0c9b1afdc87a476a59a2200129c46ec11bc3ee5d

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

QUANTUM KIRWAN MORPHISM AND GROMOV-WITTEN INVARIANTS OF QUOTIENTS I (2012, background): “6], [15, Section 8], [8], [42] for any smooth projective variety X embedded in a projective space P(V ), there exists a proper morphism Mg,0(C ×X) → QuotF/C where QuotF/C is the quot scheme of the trivial bundle F = C × V ∨ compactifying Hom(C,P(V )).”
Clifford Indices for Vector Bundles on Curves (2008, unknown): “Semistable vector bundle, Clifford index, gonality, Brill-Noether theory.”
The moduli space of stable quotients (2009, methodology): “We have described the morphism c pointwise – we refer the reader to Lian, Liu and Yau [23], Manolache [26] and Popa and Roth [38] for a scheme theoretic construction.”