On Mori chamber and stable base locus decompositions

Laface, Antonio; Massarenti, Alex; Rischter, Rick

doi:10.1090/tran/7985

On Mori chamber and stable base locus decompositions
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by Antonio Laface, Alex Massarenti and Rick Rischter PDF

Trans. Amer. Math. Soc. 373 (2020), 1667-1700 Request permission

Abstract:

The effective cone of a Mori dream space admits two wall-and-chamber decompositions called Mori chamber and stable base locus decompositions. In general the former is a nontrivial refinement of the latter. We investigate, from both the geometrical and combinatorial viewpoints, the differences between these decompositions. Furthermore, we provide a criterion to establish whether the two decompositions coincide for a Mori dream space of Picard rank two, and we construct an explicit example of a Mori dream space of Picard rank two for which the decompositions are different, showing that our criterion is sharp. Finally, we classify the smooth toric $3$-folds of Picard rank three for which the two decompositions are different.

References

Daniele Arcara, Aaron Bertram, Izzet Coskun, and Jack Huizenga, The minimal model program for the Hilbert scheme of points on $\Bbb {P}^2$ and Bridgeland stability, Adv. Math. 235 (2013), 580–626. MR 3010070, DOI 10.1016/j.aim.2012.11.018
Ivan Arzhantsev, Ulrich Derenthal, Jürgen Hausen, and Antonio Laface, Cox rings, Cambridge Studies in Advanced Mathematics, vol. 144, Cambridge University Press, Cambridge, 2015. MR 3307753
Hamid Ahmadinezhad, On pliability of del Pezzo fibrations and Cox rings, J. Reine Angew. Math. 723 (2017), 101–125. MR 3604978, DOI 10.1515/crelle-2014-0095
Carolina Araujo and Alex Massarenti, Explicit log Fano structures on blow-ups of projective spaces, Proc. Lond. Math. Soc. (3) 113 (2016), no. 4, 445–473. MR 3556488, DOI 10.1112/plms/pdw034
Hamid Ahmadinezhad and Francesco Zucconi, Mori dream spaces and birational rigidity of Fano 3-folds, Adv. Math. 292 (2016), 410–445. MR 3464026, DOI 10.1016/j.aim.2016.01.008
Aaron Bertram and Izzet Coskun, The birational geometry of the Hilbert scheme of points on surfaces, Birational geometry, rational curves, and arithmetic, Simons Symp., Springer, Cham, 2013, pp. 15–55. MR 3114922, DOI 10.1007/978-1-4614-6482-2_{2}
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, DOI 10.1090/S0894-0347-09-00649-3
Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
Michel Brion, Variétés sphériques et théorie de Mori, Duke Math. J. 72 (1993), no. 2, 369–404 (French). MR 1248677, DOI 10.1215/S0012-7094-93-07213-4
Michel Brion, Introduction to actions of algebraic groups, Les cours du CIRM, 1, no. 1, Hamiltonian actions: Invariants et classification, 2010.
Dawei Chen and Izzet Coskun, Stable base locus decompositions of Kontsevich moduli spaces, Michigan Math. J. 59 (2010), no. 2, 435–466. MR 2677631, DOI 10.1307/mmj/1281531466
Dawei Chen and Izzet Coskun, Towards Mori’s program for the moduli space of stable maps, Amer. J. Math. 133 (2011), no. 5, 1389–1419. With an appendix by Charley Crissman. MR 2843103, DOI 10.1353/ajm.2011.0040
Izzet Coskun, Tommaso de Fernex, and Angela Gibney (eds.), Surveys on recent developments in algebraic geometry, Proceedings of Symposia in Pure Mathematics, vol. 95, American Mathematical Society, Providence, RI, 2017. Papers from the Bootcamp for the 2015 Summer Research Institute on Algebraic Geometry held at the University of Utah, Salt Lake City, UT, July 6–10, 2015. MR 3701416, DOI 10.1090/pspum/095
Dawei Chen, Mori’s program for the Kontsevich moduli space $\overline {\scr M}_{0,0}(\Bbb P^3,3)$, Int. Math. Res. Not. IMRN , posted on (2008), Art. ID rnn 067, 17. MR 2439572, DOI 10.1093/imrn/rnn016
Alessio Corti, Factoring birational maps of threefolds after Sarkisov, J. Algebraic Geom. 4 (1995), no. 2, 223–254. MR 1311348
Olivier Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001. MR 1841091, DOI 10.1007/978-1-4757-5406-3
A. Fahrner and J. Hausen, On intrinsic quadrics, Canadian Journal of Mathematics, 2019, 1-37. DOI 10.4153/CJM-2018-037-5.
Anne Fahrner, Jürgen Hausen, and Michele Nicolussi, Smooth projective varieties with a torus action of complexity 1 and Picard number 2, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018), no. 2, 611–651. MR 3801292
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, DOI 10.1007/0-8176-4417-2_{8}
Brendan Hassett and Donghoon Hyeon, Log canonical models for the moduli space of curves: the first divisorial contraction, Trans. Amer. Math. Soc. 361 (2009), no. 8, 4471–4489. MR 2500894, DOI 10.1090/S0002-9947-09-04819-3
Brendan Hassett and Donghoon Hyeon, Log minimal model program for the moduli space of stable curves: the first flip, Ann. of Math. (2) 177 (2013), no. 3, 911–968. MR 3034291, DOI 10.4007/annals.2013.177.3.3
Yi Hu and Sean Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331–348. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786494, DOI 10.1307/mmj/1030132722
Jürgen Hausen and Simon Keicher, A software package for Mori dream spaces, LMS J. Comput. Math. 18 (2015), no. 1, 647–659. MR 3418031, DOI 10.1112/S1461157015000212
M. Hering, A. Küronya, and S. Payne, Asymptotic cohomological functions of toric divisors, Adv. Math. 207 (2006), no. 2, 634–645. MR 2271020, DOI 10.1016/j.aim.2005.12.007
Christopher D. Hacon and James McKernan, The Sarkisov program, J. Algebraic Geom. 22 (2013), no. 2, 389–405. MR 3019454, DOI 10.1090/S1056-3911-2012-00599-2
César Lozano Huerta, Birational geometry of the space of complete quadrics, Int. Math. Res. Not. IMRN 23 (2015), 12563–12589. MR 3431630, DOI 10.1093/imrn/rnv043
Simon Keicher, Computing the GIT-fan, Internat. J. Algebra Comput. 22 (2012), no. 7, 1250064, 11. MR 2999370, DOI 10.1142/S0218196712500646
Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR 2095472, DOI 10.1007/978-3-642-18808-4
A. Massarenti, On the birational geometry of spaces of complete forms I: Collineations and quadrics, Journal of Algebra, to appear. https://arxiv.org/abs/1803.09161, 2018.
A. Massarenti, On the birational geometry of spaces of complete forms II: Skew-forms, https://arxiv.org/abs/1803.09011, 2018.
Alex Massarenti and Rick Rischter, Spherical blow-ups of Grassmannians and Mori dream spaces, J. Algebra 494 (2018), 188–219. MR 3723177, DOI 10.1016/j.jalgebra.2017.09.037
S. Mukai, Finite generation of the Nagata invariant rings in a-d-e cases, RIMS preprint 1502, 2005.
Shinnosuke Okawa, On images of Mori dream spaces, Math. Ann. 364 (2016), no. 3-4, 1315–1342. MR 3466868, DOI 10.1007/s00208-015-1245-5
Nicolas Perrin, On the geometry of spherical varieties, Transform. Groups 19 (2014), no. 1, 171–223. MR 3177371, DOI 10.1007/s00031-014-9254-0
R. Rischter, Projective and birational geometry of Grassmannians and other special varieties, PhD thesis, https://arxiv.org/abs/1705.05673, 2017.
Igor R. Shafarevich, Basic algebraic geometry. 1, Translated from the 2007 third Russian edition, Springer, Heidelberg, 2013. Varieties in projective space. MR 3100243