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 . First, we describe the singularities of and for a generic pair . Then we show that the first-order deformations of arising from first-order unfoldings are the families of the form , where is a perturbation of φ. We also prove that the deformations of the form 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.
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.
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