Minha área de pesquisa trata
de questões enumerativas em Geometria Algébrica.
Tenho também me
interessado por singularidades e espaços de folheações
holomorfas.
Artigos +- recentes:
·
(com Daniel Leite)
Degrees of spaces of foliations of codimension one in CPn.
Journal of Pure and Applied Algebra
Resumo:
We consider the Zariski closure of
the set of foliations in CPn
defined by a differential 1-form of type
aFdG-bGdF, where F,G denote co-prime homogeneous
polynomials of degrees a, b.
Gomez-Mont and Lins-Neto have shown that
this
is an irreducible component of the space of
foliations of codimension 1.
Our main result gives a closed formula for the
degree of this component for a=2, b odd.
Foliations singular along a curve.
Transactions of the London Mathematical Society, v. 2, p. 80-92, 2015.
Resumo:
A general one-dimensional foliation in the complex projective space has finetely
many singularities. Given a family of positive dimensional subschemes in Pn, we study the loci
in the space of foliations defined by the requirement that the singularities contain a member of
the family. We show that the degrees of the loci are given by a polynomial. We compute it
explicitely in a few examples. We also include a formula for the number of isolated singularities
of a foliation in Pn general among those which are singular along a smooth curve.
·
(com
A. R. Silva),
Degree
of the variety of pairs of
nilpotent commuting matrices.
Resumo:
We compute the degrees of the projective varieties of pairs of
commuting nilpotent matrices of orders n = 2, 3, 4, to wit,
4, 72 and 5440.
(
Bulletin of the Brazilian Mathematical Society
, vol. dedicado a
Steven Kleiman
e
Aron Simis.)
v. 45, p. 837-863, 2014.
·
Hypersurfaces containing unexpected subvarieties.
Resumo:
Cubic 4-folds in P5 containing a general pair of 2-
planes are known to be rational. They form a family of codimension
2 in P55 . We find a polynomial which encodes, for all d>=3, the
degrees of the loci of hypersurfaces in P5 of degree d containing
some plane-pair.
In particular, we get the degree
(3,371,760) of that family of dimension 53
of rational cubic 4folds.
(dedicado a
Xavier Gomez-Mont)
Journal of Singularities, vol. 9, 219-225, 2014.
·
(com Angelo Felice Lopez e Fernando Cukierman)
Enumeration of Surfaces Containing an Elliptic Quartic Curve.
Resumo:
A very general surface of degree at least four in P3 contains no curves
other than intersections with surfaces. We find a formula for the degree of the
locus of surfaces in P3 of degree d >= 5 which contain some elliptic quartic curve. We also compute the degree of the locus of quartic surfaces containing an elliptic
quartic curve. Beware this is not a special case of the former, due to the fact that if a quartic surface contains an elliptic quartic curve it must contain a pencil of such curves.
Proc. Amer. Math. Soc. 142, 3305-3313, 2014.
·
Enumeration of tropes
Resumo: Singularities of the plane sections of a general surface
in three-space are well known, and counted; in particular, all plane
sections are reduced. Fixing integers d, k, we give formulas for the
degree of the locus of surfaces of degree d admitting a plane which
is tangent along some curve of degree k.
Dedicated to
Prof. Heisuke Hironaka on his 80th birthday.
(RACSAM
), v. 107, p. 213-220, (2013).
·
(com Viviana Ferrer)
Degenerate singularities of one dimensional
foliations
,
Comment. Math. Helv. 88, 305-321 (2013)
·
(com José Alberto Duarte Maia, Adriana Rodrigues Silva e Fernando Xavier)
Enumeration of surfaces containing a curve of low degree.
Resumo:
A very general surface of degree at least 4 in P3 contains no curves
besides intersection with another surface. Let W be some irreducible subvariety
of the Hilbert scheme of curves in P3 . Let NL(W,d) be the set of surfaces of fixed,
sufficiently high degree d which contain a member of W . It is a closed subvariety
of the linear system of such surfaces. We find polynomial formulas for the degrees
of NL(W,d) whenever W consists of curves of the following types: a union of up to three general lines; a conic or a twisted cubic curve.
The case of elliptic quartic curves is subtler and treated elsewhere.
J. Pure Appl. Algebra 217, no. 8, 1379-1394, (2013)
Livros:
· (com Joachim Kock)} A fórmula de Kontsevich para curvas racionais planas, 22^o Coloq. Bras.Mat. 1999. PDF
·
Introdução às curvas algébricas planas,
Coleção Matemática Universitária, IMPA
/ SBM, 160 p., 1996.
scripts para o calculo do grau de algumas componentes de espaços de folheações.
· Scripts em
maple para cálculos locais e aplicação da
fórmula de Bott para enumeração de cúbicas reversas: zip ,
txt
· Script para o
cálculo do número de curvas canônicas
em P3 incidentes 24 retas em
posição
geral.