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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the differential geometry of holomorphic plane curves

Authors: Jorge Luiz Deolindo-Silva and Farid Tari
Journal: Trans. Amer. Math. Soc. 373 (2020), 6817-6833
MSC (2010): Primary 57R45, 14H50, 53A15
Published electronically: July 3, 2020
MathSciNet review: 4155192
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Abstract: We consider the geometry of regular holomorphic curves in $\mathbb C^2$ viewed as surfaces in the affine space $\mathbb R^4$. We study the $\mathcal A$-singularities of parallel projections of generic such surfaces along planes to transverse planes. We show that at any point on the surface which is not an inflection point of the curve there are two tangent directions determining two planes along which the projection has singularities of type butterfly or worse. The integral curves of these directions form a pair of foliations on the surface defined by a binary differential equation (BDE). The singularities of this BDE are the inflection points of the curve together with other points that we call butterfly umbilic points. We determine the configurations of the solution curves of the BDE at its singularities. Finally, we prove that an affine view of an algebraic curve of degree $d\ge 2$ in $\mathbb CP^2$ has $8d(d-2)$ butterfly umbilic points.

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Additional Information

Jorge Luiz Deolindo-Silva
Affiliation: Departamento de Matemática - UFSC - Campus Blumenau, Rua João Pessoa, 2514 - Velha, CEP: 89036-004 - Blumenau - SC, Brazil
MR Author ID: 1244547

Farid Tari
Affiliation: Instituto de Ciências Matemáticas e de Computação - USP, Avenida Trabalhador são-carlense, 400 - Centro, CEP: 13566-590 - São Carlos - SP, Brazil
MR Author ID: 313337

Received by editor(s): March 14, 2019
Published electronically: July 3, 2020
Additional Notes: The second author was partially supported by the grants FAPESP 2014/00304-2 and CNPq 303772/2018-2.
Article copyright: © Copyright 2020 American Mathematical Society