Number of singularities of a foliation on ${\mathbb P}^n$

Abstract

Abstract. Let D be a one dimensional foliation on a projective space, that is, an invertible subsheaf of the sheaf of sections of the tangent bundle. If the singularities of D are isolated, Baum-Bott formula states how many singularities, counted with multiplicity, appear. The isolated condition is removed here. Let m be the dimension of the singular locus of D. We give an upper bound of the number of singularities of dimension m, counted with multiplicity and degree, that D may have, in terms of the degree of the foliation. We give some examples where this bound is reached. We then generalize this result for a higher dimensional foliation on an arbitrary smooth and projective variety.

Codimension One Holomorphic Distributions on the Projective Three-space

“…Since Sing(F ) is 0-dimensional, it follows from [6, Corollary 3] that Sing(F ) must be contained on the 0-dimensional component S 0 (G ) of Sing(G ). However, Sing(F ) has length 5, while, following [47], the length of S 0 (G ) is at most 4.…”

“…The existence of distributions of type (1a) or (1b) is clear, and the characterization of their singular loci is an easy calculation together with the application of Theorem 3.8. Now let N be a null-correlation bundle; just setting k = 1 in Example 5.2, we obtain a distribution of the form (47) D : 0 → N → T P 3 → I Z/P 3 (4) → 0…”

Classification of foliations by curves of low degree on the three-dimensional projective space

“…Let us give a closer look into the formula expressed in display (26). As proven in the previous result, every Legendrian foliation can be expressed, up to a change of coordinates, as the wedge product of the canonical contact form with a 1-form ω ∈ H 0 (Ω 1 P 3 (d + 1))).…”

“…)) = 0 and Ext 2 (G * , G * ) = 0, being G * the direct sum of two lines bundles. Hence, we can apply the lemma and prove the statement, recalling that the dimension of the moduli space is given by (26) dim Ext 1 (G * , G * ) + dim Hom(G, Ω 1…”

On classification and moduli spaces of holomorphic distributions

“…We recall that length(U) is the number of isolated singularities of the distributions. We refer to [19,41,42,35,44] when the authors determine the number of isolated singularities of foliations by curves.…”