Foliations with isolated singularities on Hirzebruch surfaces

Abstract

We study foliations ℱ {\mathcal{F}} on Hirzebruch surfaces S δ {S_{\delta}} and prove that, similarly to those on the projective plane, any ℱ {\mathcal{F}} can be represented by a bi-homogeneous polynomial affine 1-form. In case ℱ {\mathcal{F}} has isolated singularities, we show that, for δ = 1 {\delta=1} , the singular scheme of ℱ {\mathcal{F}} does determine the foliation, with some exceptions that we describe, as is the case of foliations in the projective plane. For δ ≠ 1 {\delta\neq 1} , we prove that the singular scheme of ℱ {\mathcal{F}} does not determine the foliation. However, we prove that, in most cases, two foliations ℱ {\mathcal{F}} and ℱ ′ {\mathcal{F}^{\prime}} given by sections s and s ′ {s^{\prime}} have the same singular scheme if and only if s ′ = Φ ⁢ ( s ) {s^{\prime}=\Phi(s)} , for some global endomorphism Φ of the tangent bundle of S δ {S_{\delta}} .

Algebraic Integrability of Planar Polynomial Vector Fields by Extension to Hirzebruch Surfaces

“…and thus, we can identify the points on U 00 with the points (x, y) on C 2 through the isomorphism (1, x; 1, y) → (x, y). A similar identification holds for any other open set U ij giving rise to change of coordinates maps in the overlap of two open sets U ij (see, for instance, [25] for more details).…”

“…Let F δ be a foliation on a Hirzebruch surface F δ . By [25], F δ is determined by a differential 1-form…”

“…are the affine coordinates in U 00 ∼ = C 2 considered in Subsection 2.4, then the foliation on U 00 defined by the restriction of F δ is given by the differential 1-form A δ,1 (1, x, 1, y)dx + B δ,1 (1, x, 1, y)dy. There are analogous expressions for the restriction of F δ to the remaining affine open subsets U ij , see [25] for details.…”

“…As said before, any foliation on the complex projective plane P 2 is algebraic in the sense that it is the extension of the field of directions induced by a planar polynomial vector field; in fact, such a foliation can be described by a homogeneous polynomial vector field on C 3 , or by a homogeneous polynomial differential 1-form on C 3 satisfying certain condition (see [28,6]). In [27], it is shown that foliations on a Hirzebruch surface behave similarly. We explain it in the next section.…”

Algebraic integrability of planar polynomial vector fields by extension to Hirzebruch surfaces

“…Let F = [f] be a singular foliation (with isolated singularities) on a Hirzebruch surface. Then it is given by f [27]).…”