Abstract
We determine the effective cone of the Quot scheme parametrizing all rank r, degree d quotient sheaves of the trivial bundle of rank n on \({\mathbb{P}^1}\). More specifically, we explicitly construct two effective divisors which span the effective cone, and we also express their classes in the Picard group in terms of a known basis.
Similar content being viewed by others
References
Bertram A.: Towards a Schubert calculus for maps from a Riemann surface to a Grassmannian. Internat. J. Math. 5, 811–825 (1994)
Bertram A.: Quantum Schubert calculus. Adv. Math. 128, 289–305 (1997)
Bertram A., Daskalopoulos G., Wentworth R.: Gromov invariants for holomorphic maps from Riemann surfaces to Grasmannians. J. Am. Math. Soc. 9, 529–571 (1996)
Coskun, I., Starr, J.: Divisors on the space of maps to Grassmannians. Int. Math. Res. Notices 2006, 25, Article ID 35273 (2006)
Ramirez C.: The degree of the variety of rational ruled surfaces and Gromov–Witten invariants. Trans. Am. Math. Soc. 358, 11–24 (2006)
Ramirez C.: On a stratification of the Kontsevich moduli space \({\overline{M}_{0,n}(G(2,4),d)}\) and enumerative geometry. J. Pure Appl. Algebr. 213, 857–868 (2009)
Ravi M., Rosenthal J., Wang X.: Degree of the generalized Plücker embedding of a Quot scheme and quantum cohomology. Math. Ann. 311, 11–26 (1998)
Shao, Y.: A compactification of the space of algebraic maps from \({\mathbb{P}^1}\) to a Grassmannian. Ph.D. Thesis, University of Arizona
Strømme, S.: On parametrized rational curves in Grassmann varieties. In: Lecture Notes in Mathematics, vol. 1266, pp. 251–272. Springer, Berlin (1987)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jow, SY. The effective cone of the space of parametrized rational curves in a Grassmannian. Math. Z. 272, 947–960 (2012). https://doi.org/10.1007/s00209-011-0966-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-011-0966-8