Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter February 28, 2023

Rational pullbacks of toric foliations

  • Javier Gargiulo Acea ORCID logo EMAIL logo , Ariel Molinuevo ORCID logo and Sebastián Velazquez ORCID logo
From the journal Forum Mathematicum


This article is dedicated to the study of singular codimension-one foliations on a simplicial complete toric variety X and their pullbacks by dominant rational maps φ : n X . First, we describe the singularities of and φ * for a generic pair ( φ , ) . Then we show that the first-order deformations of φ * arising from first-order unfoldings are the families of the form φ ε * , where φ ε is a perturbation of φ. We also prove that the deformations of the form φ * ε consist exactly of the families which are tangent to the fibers of φ. In order to do so, we state some results of independent interest regarding the Kupka singularities of these foliations.

Communicated by Jan Bruinier

Funding statement: The first author was fully supported by CNPq, Brazil, the project Print – Institutional Internationalization Program – CAPES, Brazil and Instituto de Matemática Pura e Aplicada, Brazil. The second author was fully supported by Universidade Federal do Rio de Janeiro, Brazil. The third author was fully supported by CNPq, Brazil and CONICET, Argentina.


We would like to thank Mariano Chehebar, Fernando Cukierman, Alicia Dickenstein, Alcides Lins Neto and Jorge Vitorio Pereira for fruitful conversations at various stages of this work. We are especially grateful to Federico Quallbrunn for his further valuable help. Gratitude is also due to Jarosław Buczyński for his suggestions and contributions regarding Section 2. Finally, we want to thank the anonymous referee for his/her constructive comments on the original manuscript.


[1] C. Araujo and S. Druel, On Fano foliations, Adv. Math. 238 (2013), 70–118. 10.1016/j.aim.2013.02.003Search in Google Scholar

[2] C. Araujo and S. Druel, On Fano foliations 2, Foliation Theory in Algebraic Geometry, Simons Symp., Springer, Cham (2016), 1–20. 10.1007/978-3-319-24460-0_1Search in Google Scholar

[3] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, 1969. Search in Google Scholar

[4] V. V. Batyrev and D. A. Cox, On the Hodge structure of projective hypersurfaces in toric varieties, Duke Math. J. 75 (1994), no. 2, 293–338. 10.1215/S0012-7094-94-07509-1Search in Google Scholar

[5] N. Bourbaki, Éléments de mathématique. Fascicule XXVII. Algèbre commutative. Chapitre 1: Modules plats. Chapitre 2: Localisation, Act. Sci. Indust. 1293, Hermann, Paris, 1961. Search in Google Scholar

[6] G. Brown and J. A. Buczyński, Maps of toric varieties in Cox coordinates, Fund. Math. 222 (2013), no. 3, 213–267. 10.4064/fm222-3-2Search in Google Scholar

[7] O. Calvo-Andrade, A. Molinuevo and F. Quallbrunn, On the geometry of the singular locus of a codimension one foliation in n , Rev. Mat. Iberoam. 35 (2019), no. 3, 857–876. 10.4171/rmi/1073Search in Google Scholar

[8] D. Cerveau, A. Lins Neto and S. J. Edixhoven, Pull-back components of the space of holomorphic foliations on ( n ) , n 3 , J. Algebraic Geom. 10 (2001), no. 4, 695–711. Search in Google Scholar

[9] D. Cerveau, A. Lins-Neto, F. Loray, J. V. Pereira and F. Touzet, Algebraic reduction theorem for complex codimension one singular foliations, Comment. Math. Helv. 81 (2006), no. 1, 157–169. 10.4171/CMH/47Search in Google Scholar

[10] S. C. Coutinho and J. V. Pereira, On the density of algebraic foliations without algebraic invariant sets, J. Reine Angew. Math. 594 (2006), 117–135. 10.1515/CRELLE.2006.037Search in Google Scholar

[11] D. A. Cox, J. B. Little and H. K. Schenck, Toric Varieties, Grad. Stud. Math. 124, American Mathematical Society, Providence, 2011. 10.1090/gsm/124Search in Google Scholar

[12] F. Cukierman, J. Gargiulo Acea and C. Massri, Stability of logarithmic differential one-forms, Trans. Amer. Math. Soc. 371 (2019), no. 9, 6289–6308. 10.1090/tran/7443Search in Google Scholar

[13] F. Cukierman and J. V. Pereira, Stability of holomorphic foliations with split tangent sheaf, Amer. J. Math. 130 (2008), no. 2, 413–439. 10.1353/ajm.2008.0011Search in Google Scholar

[14] F. Cukierman, J. V. Pereira and I. Vainsencher, Stability of foliations induced by rational maps, Ann. Fac. Sci. Toulouse Math. (6) 18 (2009), no. 4, 685–715. 10.5802/afst.1221Search in Google Scholar

[15] A. S. de Medeiros, Structural stability of integrable differential forms, Geometry and Topology, Lecture Notes in Math. 597, Springer, Berlin (1977), 395–428. 10.1007/BFb0085369Search in Google Scholar

[16] I. Dolgachev, Weighted projective varieties, Group Actions and Vector Fields, Lecture Notes in Math. 956, Springer, Berlin (1982), 34–71. 10.1007/BFb0101508Search in Google Scholar

[17] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Publ. Math. Inst. Hautes Études Sci. 28 (1966), 1–255. 10.1007/BF02684343Search in Google Scholar

[18] R. Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), no. 2, 121–176. 10.1007/BF01467074Search in Google Scholar

[19] I. Kupka, The singularities of integrable structurally stable Pfaffian forms, Proc. Natl. Acad. Sci. USA 52 (1964), 1431–1432. 10.1073/pnas.52.6.1431Search in Google Scholar PubMed PubMed Central

[20] B. Malgrange, Frobenius avec singularités. II. Le cas général, Invent. Math. 39 (1977), no. 1, 67–89. 10.1007/BF01695953Search in Google Scholar

[21] C. Massri, A. Molinuevo and F. Quallbrunn, The Kupka scheme and unfoldings, Asian J. Math. 22 (2018), no. 6, 1025–1045. 10.4310/AJM.2018.v22.n6.a3Search in Google Scholar

[22] C. Massri, A. Molinuevo and F. Quallbrunn, Foliations with persistent singularities, J. Pure Appl. Algebra 225 (2021), no. 6, Paper No. 106630. 10.1016/j.jpaa.2020.106630Search in Google Scholar

[23] A. Molinuevo, Unfoldings and deformations of rational and logarithmic foliations, Ann. Inst. Fourier (Grenoble) 66 (2016), no. 4, 1583–1613. 10.5802/aif.3044Search in Google Scholar

[24] D. Mumford, The Red Book of Varieties and Schemes, Lecture Notes in Math. 1358, Springer, Berlin, 1999. 10.1007/b62130Search in Google Scholar

[25] J. V. Pereira and C. Spicer, Hypersurfaces quasi-invariant by codimension one foliations, Math. Ann. 378 (2020), no. 1–2, 613–635. 10.1007/s00208-019-01833-4Search in Google Scholar

[26] R. P. Stanley, Combinatorics and Commutative Algebra, 2nd ed., Progr. Math. 41, Birkhäuser, Boston, 1996. Search in Google Scholar

[27] T. Suwa, Unfoldings of complex analytic foliations with singularities, Japan. J. Math. (N. S.) 9 (1983), no. 1, 181–206. 10.4099/math1924.9.181Search in Google Scholar

[28] T. Suwa, Unfoldings of foliations with multiform first integrals, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 3, 99–112. 10.5802/aif.932Search in Google Scholar

[29] T. Suwa, Unfoldings of meromorphic functions, Math. Ann. 262 (1983), no. 2, 215–224. 10.1007/BF01455312Search in Google Scholar

[30] S. Velazquez, Toric foliations with split tangent sheaf, Bull. Sci. Math. 175 (2022), Paper No. 103099. 10.1016/j.bulsci.2022.103099Search in Google Scholar

Received: 2022-09-09
Revised: 2023-01-19
Published Online: 2023-02-28
Published in Print: 2023-05-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 30.11.2023 from
Scroll to top button