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We show that the set of singular holomorphic foliations on projective spaces with split tangent sheaf and good singular set is open in the space of holomorphic foliations. We also give a cohomological criterion for the rigidity of holomorphic foliations induced by group actions and prove the existence of rigid codimension one foliations of degree n − 1 on P n for every n ≥ 3.1991 Mathematics Subject Classification. 32J18,32Q55,37F75.

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Abstract. We show that the singular holomorphic foliations induced by dominant quasi-homogeneous rational maps fill out irreducible components of the space Fq(r, d) of singular foliations of codimension q and degree d on the complex projective space P r , when 1 ≤ q ≤ r − 2. We study the geometry of these irreducible components. In particular we prove that they are all rational varieties and we compute their projective degrees in several cases.

We compute the multiplicity of the discriminant of a line bundle L over a nonsingular variety S at a given section X, in terms of the Chern classes of L and of the cotangent bundle of S, and the Segre classes of the jacobian scheme of X in S. For S a surface, we obtain a precise formula that expresses the multiplicity as a sum of a term due to the non-reduced components of the section, and a term that depends on the Milnor numbers of the singularities of X red . Also, under certain hypotheses, we provide formulas for the "higher discriminants" that parametrize sections with a singular point of prescribed multiplicity. As an application, we obtain criteria for the various discriminants to be "small". Introduction.Let S ⊂ P N be a smooth variety over an algebraically closed field of characteristic 0. Denote by D ⊂P N the dual variety of S, consisting of all the hyperplanes in P N that are tangent to S. The basic question that we consider in this note is:Our answer is in the style of the well known formula for the degree of the dual in terms of the Chern classes of S and of the hyperplane bundle L (see for example [3], p. 63). We associate with each hyperplane section X a zero-dimensional class in the Chow group of S, obtained by capping the Segre class in S of the singular scheme of X by the Chern classes of the bundle P 1 L of principal parts of order one (which are easily expressible in terms of the classes of L and of the cotangent bundle of S). Assuming that D is a hypersurface, the multiplicity of D at X is then essentially the degree of this class. For example, if S = P 2 , and one writes the section X as i m i X i with m i natural numbers and X i irreducible divisors, then m X (D) has a simple expression in terms of the integers m i , the degrees of the X i , and the singularities of X red = i X i (see 3.1).The main general formula is stated in §1, in a more general setting also addressing the same question for the 'higher discriminants' D (r) ⊂P N , consisting of sections of S that have a point of multiplicity at least r + 1. The formula gives m X D (r) under a hypothesis on the dimension of the osculating planes of S (automatic in the case of dual varieties) and assuming that the discriminants have the expected dimension. In these hypotheses, the formula also specializes easily to compute the degree of the discriminants (Corollary 1.1).In fact the class introduced in §1 can be used to detect whether the dual variety is not a hypersurface-that is, whether it is 'small': we show (Proposition 1.3) that the class vanishes if and only if the dual is small-again, we prove the corresponding result for all discriminants, under the same hypothesis on the osculating planes.The main difficulty in applying the theorem to specific situations lies in the computation of the Segre class s(J r X, S) of the Jacobian schemes of a hyperplane section. One important case occurs when S is a surface, or more generally when the 1 Supported in part by the