Multiplicities of discriminants

Abstract

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

On singular Lüroth quartics

“…Since D contains the double conics and the cuspidal curves, D contains H, Δ 0 and Δ 1 . In [1] it is proved that D vanishes with multiplicity 14 on double conics and with multiplicity 2 on cuspidal quartics. If we consider any test curve intersecting double conics and cuspidal quartics transversely, we will have to perform a base change in order to get stable reduction.…”

New and old results in resultant theory

“…The simplest example of such a synthesis corresponds to n = 3 and r = 2: the system of equations has form (6). It can be written using a 3×6 matrix, and one more 3×6 matrix is needed to complete a 6×6 square matrix.…”

Discriminants and Artin conductors

“…Remark 2.3.1. In particular, if P 1 is a general line through P ∈ ∆ X , the above multiplicity is then the definition of the multiplicity of the point P in ∆ X which was studied over the complex numbers in [Dim86], [Par91], [Nem88] where formulas in terms of (generalized) Milnor numbers was given, and general formulas in terms of Segre classes was given in [AC93]. In characteristic p, if we assume that the singularities are isolated, a general line through P will define a smooth total space over a pencil, and the formula of Deligne [SGA 7-II], Exposé XVI, Proposition 2.1 can be used together with general geometry of pencils to prove that the multiplicity of the discriminant is the total Milnor number in the sense of idem.…”