Open loci of ideals with applications to birational maps

Hassanzadeh, S.; Mostafazadehfard, Maral

doi:10.1090/proc/16665

Open loci of ideals with applications to birational maps
HTML articles powered by AMS MathViewer

by S. Hamid Hassanzadeh and Maral Mostafazadehfard;

Proc. Amer. Math. Soc. 152 (2024), 1841-1856

DOI: https://doi.org/10.1090/proc/16665

Published electronically: March 1, 2024

HTML | PDF | Request permission

Abstract:

In this work we show that the loci of ideals in principal class, ideals of grade at least two, and ideals of maximal analytic spread are Zariski open sets in the parameter space. As an application, we show that the set of birational maps of clear polynomial degree $d$ over an arbitrary projective variety $X$, denoted by $Bir(X)_{d}$, is a constructible set. This extends a previous result by Blanc and Furter.

References

Cinzia Bisi, Alberto Calabri, and Massimiliano Mella, On plane Cremona transformations of fixed degree, J. Geom. Anal. 25 (2015), no. 2, 1108–1131. MR 3319964, DOI 10.1007/s12220-013-9459-9
Jérémy Blanc and Jean-Philippe Furter, Topologies and structures of the Cremona groups, Ann. of Math. (2) 178 (2013), no. 3, 1173–1198. MR 3092478, DOI 10.4007/annals.2013.178.3.8
Jérémy Blanc, Algebraic structures of groups of birational transformations, Algebraic groups: structure and actions, Proc. Sympos. Pure Math., vol. 94, Amer. Math. Soc., Providence, RI, 2017, pp. 17–30. MR 3645066, DOI 10.1090/pspum/094/02
M. P. Brodmann and R. Y. Sharp, Local cohomology, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 136, Cambridge University Press, Cambridge, 2013. An algebraic introduction with geometric applications. MR 3014449
Frederick W. Call, A theorem of Grothendieck using Picard groups for the algebraist, Math. Scand. 74 (1994), no. 2, 161–183. MR 1298359, DOI 10.7146/math.scand.a-12487
Serge Cantat and Stéphane Lamy, Normal subgroups in the Cremona group, Acta Math. 210 (2013), no. 1, 31–94. With an appendix by Yves de Cornulier. MR 3037611, DOI 10.1007/s11511-013-0090-1
A. V. Doria, S. H. Hassanzadeh, and A. Simis, A characteristic-free criterion of birationality, Adv. Math. 230 (2012), no. 1, 390–413. MR 2900548, DOI 10.1016/j.aim.2011.12.005
D. Grayson and M. Stillman, Macaulay 2–a system for computation in algebraic geometry and commutative algebra, 1997, Available via anonymous ftp from math.uiuc.edu, 1998.
Craig Huneke and Irena Swanson, Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006. MR 2266432
S. H. Hassanzadeh and A. Simis, Bounds on degrees of birational maps with arithmetically Cohen-Macaulay graphs, J. Algebra 478 (2017), 220–236. MR 3621670, DOI 10.1016/j.jalgebra.2017.01.010
Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
Ivan Pan and Alvaro Rittatore, Some remarks about the Zariski topology of the Cremona group, Preprint, arXiv:1212.5698, 2012.
Jean-Pierre Serre, Le groupe de Cremona et ses sous-groupes finis, Astérisque 332 (2010), Exp. No. 1000, vii, 75–100 (French, with French summary). Séminaire Bourbaki. Volume 2008/2009. Exposés 997–1011. MR 2648675
Aron Simis, Cremona transformations and some related algebras, J. Algebra 280 (2004), no. 1, 162–179. MR 2081926, DOI 10.1016/j.jalgebra.2004.03.025
Aron Simis, Bernd Ulrich, and Wolmer V. Vasconcelos, Codimension, multiplicity and integral extensions, Math. Proc. Cambridge Philos. Soc. 130 (2001), no. 2, 237–257. MR 1806775, DOI 10.1017/S0305004100004667