Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On birational geometry of the space of parametrized rational curves in Grassmannians


Author: Atsushi Ito
Journal: Trans. Amer. Math. Soc. 369 (2017), 6279-6301
MSC (2010): Primary 14C20, 14M99
DOI: https://doi.org/10.1090/tran/6840
Published electronically: March 1, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study the birational geometry of the Quot
schemes of trivial bundles on $ \mathbb{P}^1$ by constructing small $ \mathbb{Q}$-factorial modifications of the Quot schemes as suitable moduli spaces. We determine all the models which appear in the minimal model program on the Quot schemes. As a corollary, we show that the Quot schemes are Mori dream spaces and log Fano.


References [Enhancements On Off] (What's this?)

  • [ACGH] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932
  • [ABCH] Daniele Arcara, Aaron Bertram, Izzet Coskun, and Jack Huizenga, The minimal model program for the Hilbert scheme of points on $ \mathbb{P}^2$ and Bridgeland stability, Adv. Math. 235 (2013), 580-626. MR 3010070, https://doi.org/10.1016/j.aim.2012.11.018
  • [BM] Arend Bayer and Emanuele Macrì, Projectivity and birational geometry of Bridgeland moduli spaces, J. Amer. Math. Soc. 27 (2014), no. 3, 707-752. MR 3194493, https://doi.org/10.1090/S0894-0347-2014-00790-6
  • [Bi] Caucher Birkar, Singularities on the base of a Fano type fibration, J. Reine Angew. Math. 715 (2016), 125-142. MR 3507921, https://doi.org/10.1515/crelle-2014-0033
  • [BCHM] Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405-468. MR 2601039, https://doi.org/10.1090/S0894-0347-09-00649-3
  • [Ch] Dawei Chen, Mori's program for the Kontsevich moduli space $ \overline {\mathcal {M}}_{0,0}(\mathbb{P}^3,3)$, Int. Math. Res. Not. IMRN 2008, Art. ID rnn 067, 17 pp.. MR 2439572, https://doi.org/10.1093/imrn/rnn016
  • [Gr] Alexander Grothendieck, Techniques de construction et théorèmes d'existence en géométrie algébrique. IV. Les schémas de Hilbert, Séminaire Bourbaki, Vol. 6, Exp. No. 221, 249-276, Soc. Math. France, Paris, 1995 (French). MR 1611822
  • [Ha] Brendan Hassett, Classical and minimal models of the moduli space of curves of genus two, Geometric methods in algebra and number theory, Progr. Math., vol. 235, Birkhäuser Boston, Boston, MA, 2005, pp. 169-192. MR 2166084, https://doi.org/10.1007/0-8176-4417-2_8
  • [HK] Yi Hu and Sean Keel, Mori dream spaces and GIT, Dedicated to William Fulton on the occasion of his 60th birthday, Michigan Math. J. 48 (2000), 331-348. MR 1786494, https://doi.org/10.1307/mmj/1030132722
  • [Jo] Shin-Yao Jow, The effective cone of the space of parametrized rational curves in a Grassmannian, Math. Z. 272 (2012), no. 3-4, 947-960. MR 2995148, https://doi.org/10.1007/s00209-011-0966-8
  • [Ma] Cristina Martínez Ramirez, On a stratification of the Kontsevich moduli space $ \overline M_{0,n}(G(2,4),d)$ and enumerative geometry, J. Pure Appl. Algebra 213 (2009), no. 5, 857-868. MR 2494376, https://doi.org/10.1016/j.jpaa.2008.10.009
  • [Ni] Nitin Nitsure, Construction of Hilbert and Quot schemes, Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Amer. Math. Soc., Providence, RI, 2005, pp. 105-137. MR 2223407
  • [Ro] Joachim Rosenthal, On dynamic feedback compensation and compactification of systems, SIAM J. Control Optim. 32 (1994), no. 1, 279-296. MR 1255971, https://doi.org/10.1137/S036301299122133X
  • [Sh] Y. Shao, A compactification of the space of parametrized rational curves in Grassmannians, arXiv:1108.2299.
  • [SS] Frank Sottile and Bernd Sturmfels, A sagbi basis for the quantum Grassmannian, J. Pure Appl. Algebra 158 (2001), no. 2-3, 347-366. MR 1822848, https://doi.org/10.1016/S0022-4049(00)00053-0
  • [St] Stein Arild Strømme, On parametrized rational curves in Grassmann varieties, Space curves (Rocca di Papa, 1985) Lecture Notes in Math., vol. 1266, Springer, Berlin, 1987, pp. 251-272. MR 908717, https://doi.org/10.1007/BFb0078187
  • [Ve] Kartik Venkatram, Birational geometry of the space of rational curves in homogeneous varieties, ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)-Massachusetts Institute of Technology, 2011. MR 2992651

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 14C20, 14M99

Retrieve articles in all journals with MSC (2010): 14C20, 14M99


Additional Information

Atsushi Ito
Affiliation: Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
Email: aito@math.kyoto-u.ac.jp

DOI: https://doi.org/10.1090/tran/6840
Keywords: Quot scheme, small $\mathbb{Q}$-factorial modification, Mori dream space
Received by editor(s): August 12, 2015
Received by editor(s) in revised form: September 19, 2015
Published electronically: March 1, 2017
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society