Variations of geometric invariant quotients for pairs, a computational approach
HTML articles powered by AMS MathViewer
- by Patricio Gallardo and Jesus Martinez-Garcia PDF
- Proc. Amer. Math. Soc. 146 (2018), 2395-2408 Request permission
Abstract:
We study GIT compactifications of pairs formed by a hypersurface and a hyperplane. We provide a general setting to characterize all polarizations which give rise to different GIT quotients. Furthermore, we describe a finite set of one-parameter subgroups sufficient to determine the stability of any GIT quotient. We characterize all maximal orbits of non-stable and strictly semistable pairs, as well as minimal closed orbits of strictly semistable pairs. Our construction gives natural compactifications of the space of log smooth pairs for Fano and Calabi-Yau hypersurfaces.References
- Daniel Allcock, The moduli space of cubic threefolds, J. Algebraic Geom. 12 (2003), no. 2, 201–223. MR 1949641, DOI 10.1090/S1056-3911-02-00313-2
- Igor Dolgachev, Lectures on invariant theory, London Mathematical Society Lecture Note Series, vol. 296, Cambridge University Press, Cambridge, 2003. MR 2004511, DOI 10.1017/CBO9780511615436
- Igor V. Dolgachev and Yi Hu, Variation of geometric invariant theory quotients, Inst. Hautes Études Sci. Publ. Math. 87 (1998), 5–56. With an appendix by Nicolas Ressayre. MR 1659282
- Patricio Gallardo, On the GIT quotient of quintic surfaces, arXiv preprint arXiv:1310.3534 (2013), 1–28.
- Patricio Gallardo and Jesus Martinez-Garcia, GIT stability of pairs and hypersurface singularities, to appear.
- Patricio Gallardo and Jesus Martinez-Garcia, Moduli of cubic surfaces and their anticanonical divisors, arXiv:1607.03697 (2016), 19 pages.
- Patricio Gallardo and Jesus Martinez-Garcia, VGIT package for python, available at https://doi.org/10.15125/BATH-00458 (2017).
- Radu Laza, Deformations of singularities and variation of GIT quotients, Trans. Amer. Math. Soc. 361 (2009), no. 4, 2109–2161. MR 2465831, DOI 10.1090/S0002-9947-08-04660-6
- Radu Laza, The moduli space of cubic fourfolds, J. Algebraic Geom. 18 (2009), no. 3, 511–545. MR 2496456, DOI 10.1090/S1056-3911-08-00506-7
- Shigeru Mukai, An introduction to invariants and moduli, Cambridge Studies in Advanced Mathematics, vol. 81, Cambridge University Press, Cambridge, 2003. Translated from the 1998 and 2000 Japanese editions by W. M. Oxbury. MR 2004218
- D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906, DOI 10.1007/978-3-642-57916-5
- Yuji Odaka, Cristiano Spotti, and Song Sun, Compact moduli spaces of del Pezzo surfaces and Kähler-Einstein metrics, J. Differential Geom. 102 (2016), no. 1, 127–172. MR 3447088
- Peter Orlik and Louis Solomon, Singularities. II. Automorphisms of forms, Math. Ann. 231 (1977/78), no. 3, 229–240. MR 476735, DOI 10.1007/BF01420243
- Michael Thaddeus, Geometric invariant theory and flips, J. Amer. Math. Soc. 9 (1996), no. 3, 691–723. MR 1333296, DOI 10.1090/S0894-0347-96-00204-4
Additional Information
- Patricio Gallardo
- Affiliation: Department of Mathematics, Washington University, Campus Box 1146, One Brookings Drive, St. Louis, Missouri 63130-4899
- MR Author ID: 1228133
- Email: pgallardocandela@wustl.edu
- Jesus Martinez-Garcia
- Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
- MR Author ID: 1073616
- Email: J.Martinez.Garcia@bath.ac.uk
- Received by editor(s): March 29, 2016
- Received by editor(s) in revised form: September 7, 2017
- Published electronically: February 16, 2018
- Communicated by: Lev Borisor
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 2395-2408
- MSC (2010): Primary 14L24, 14H10, 14Q10; Secondary 14J45, 14J32
- DOI: https://doi.org/10.1090/proc/13950
- MathSciNet review: 3778143