Quadratic forms and holomorphic foliations on singular surfaces

Annals of Mathematics, 2004

Journal of Singularities

It is well-known that a foliation by curves of degree greater than or equal to two, with isolated singularities, in the complex projective space of dimension greater than or equal to two, is uniquely determined by the scheme of its singular points. The main result in this paper is that the set of foliations which are uniquely determined by a subscheme (of the minimal possible degree) of its singular points, contains a nonempty Zariski-open subset. Our results hold in the projective space defined over any algebraically closed ground field.

Singularities – Sapporo 1998

Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 2013

Regular and Chaotic Dynamics, 2008

It is well known that changes in the Liouville foliations of the isoenergy surfaces of an integrable system imply that the bifurcation set has singularities at the corresponding energy level. We formulate certain genericity assumptions for two degrees of freedom integrable systems and we prove the opposite statement: the essential critical points of the bifurcation set appear only if the Liouville foliations of the isoenergy surfaces change at the corresponding energy levels. Along the proof, we give full classification of the structure of the isoenergy surfaces near the critical set under our genericity assumptions and we give their complete list using Fomenko graphs. This may be viewed as a step towards completing the Smale program for relating the energy surfaces foliation structure to singularities of the momentum mappings for non-degenerate integrable two degrees of freedom systems.

Bulletin of the London Mathematical Society

Consider a field k of characteristic 0, not necessarily algebraically closed, and a fixed algebraic curve f = 0 defined by a tame polynomial f ∈ k[x, y] with only quasihomogeneous singularities. We prove that the space of holomorphic foliations in the plane A 2 k having f = 0 as a fixed invariant curve is generated as k[x, y]-module by at most four elements, three of them are the trivial foliations f dx, f dy and df. Our proof is algorithmic and constructs the fourth foliation explicitly. Using Serre's GAGA and Quillen-Suslin theorem, we show that for a suitable field extension K of k such a module over K[x, y] is actually generated by two elements, and therefore, such curves are free divisors in the sense of K. Saito. After performing Groebner basis for this module, we observe that in many well-known examples, K = k.

2020

We characterize nondicrital generalized curve foliations with fixed reduced separatrix. Moreover, we give suficient conditions when a plane analytic curve is its reduced separatrix. For that, we introduce a distinguished expression for a given 1-form, called {\it Weierstrass form}. Then, using Weierstrass forms, we characterize the nondicritical generalized curve foliations: first, for foliations with monomial separatrix using toric resolution; second, for foliations with reduced separatrix, using the $GSV$-index. In this last case the characterization, which is our main result, could be interpreted in function of a polar of the foliation and a polar of its reduced separatrix.

Ergodic Theory and Dynamical Systems, 2016

In this paper we give complete analytic invariants for the set of germs of holomorphic foliations in $(\mathbb{C}^{2},0)$ that become regular after a single blow-up. Some of the invariants describe the holonomy pseudogroup of the germ and are called transverse invariants. The other invariants lie in a finite dimensional complex vector space. Such singularities admit separatrices tangentially to any direction at the origin. When enough separatrices are leaves of a radial foliaton (a condition that can always be attained if the multiplicity of the germ at the origin is at most four) we are able to describe and realize all the analytical invariants geometrically and provide analytic normal forms. As a consequence, we prove that any two such germs sharing the same transverse invariants are conjugated by a very particular type of birational transformation. We also provide explicit examples of universal equisingular unfoldings of foliations that cannot be produced by unfolding functions. ...

arXiv (Cornell University), 2023

In this work we describe dicritical foliations in (C 2 , 0) at a triple point of the resolution dual graph of an analytic plane branch C F using its semiroots. In particular, we obtain a constructive method to present a one-parameter family C Fu of separatrices for such foliations. As a by-product we relate the contact order between a special member of C Fu and C F with analytic discrete invariants of plane branches.

2013

Consider a complex one dimensional foliation on a complex surface near a singularity p. If I is a closed invariant set containing the singularity p, then I contains either a separatrix at p or an invariant real three dimensional manifold singular at p.