Polarity with respect ot a foliation and Cayley-Bacharach Theorems

Abstract

We prove that a foliation of degree di¨erent from one on the projective plane (de®ned over any algebraically closed ground ®eld), is uniquely determined by its subscheme of singular points (which we call the singular subscheme of the foliation). Conversely, we provide three di¨erent characterizations of those subschemes that are the singular subscheme of some foliation. Introduction and statement of resultsLet k be an algebraically closed ®eld and let P 2 be the projective plane over k. This paper deals with the geometry of foliations F on P 2 .Foliations over the (complex) projective plane have been recently studied in several papers, for instance in connection with the so-called Poincare  Problem (see [7], [6], [3], [1], and [12]), or with the problem of complex-cycles (see [14]), or even, with the study of polynomial foliations on the a½ne plane (see [2]), to mention a few. An overview to this literature (and to the references contained therein) makes it clear that the geometry of plane foliations is far from being well-understood. This paper studies the geometry of F through the algebro-geometric properties of its subscheme Sing SF of singular points, which will be referred throughout as the singular subscheme of F. We focus on the following two natural problems: First, to describe the geometry of the singular subscheme of a foliation and apply it to understand the foliation from such subscheme. Second, to give characterizations of those subschemes of P 2 that are the singular one of some foliation.We regard a foliation F as a rational map p p F : P 2 3 P 2 (called its polarity map), together with the two-dimensional linear system h h F of plane curves giving rise

Sections with isolated singularities

“…Let F ∈ Fol r (P n , L) be a holomorphic foliation by curves of degree r ≥ 1, with isolated singularities, and let F ∈ Fol r (P n , L) be another foliation such that SingS(F ) ⊇ SingS(F). Then F = F. This result generalizes both Theorem 2.6 in [8], where SingS(F) is assumed to be reduced, and Theorem 3.5 in [5], where n = 2 and the degree r used therein was taken to be that of a homogeneous polynomial vector field X in C n+1 giving rise to F, so that r = r − 1.…”

“…For such foliations in projective spaces, we derive a precise generalization of the above-mentioned results from [5] and [8] in Theorem 3.5 below.…”

“…In the previous paper [5], the authors have shown that an algebraic foliation of degree at least 2 in the projective plane is determined by the subscheme of its singular points (or its singular subscheme, after Definition 2.1 below). This result was known to hold for foliations by curves in projective spaces having a reduced singular subscheme (see [8]).…”

Special subschemes of the scheme of singularities of a plane foliation

“…These computations together with[2, Proposition 4.3] provide our first result:Corollary 3.3. Let F be a foliation of degree r 2 on P 2 , with isolated singularities.…”

GEOMETRIC INVARIANT THEORY FOR HOLOMORPHIC FOLIATIONS ON ℂℙ2 OF DEGREE 2

“…A set of seven points in ‫ސރ‬ 2 is said to be in general position if no three of these lie on one line and no six of them lie on one conic. Moreover, in the case of degree 2, the corollary 4.10 in [3] says that seven different points of ‫ސރ‬ 2 are the singular set of a unique non-degenerate holomorphic foliation of degree 2 if and only if there are not present six points in a conic. On the other hand, in Remark 2 we prove that a non-degenerate foliation of degree d has an invariant line if and only if this line has d + 1 singular points.…”