Plücker formula

Abstract: Given a point S ∈ ℙ2: = ℙ2(ℂ) and an irreducible algebraic curve 𝒞 of ℙ2 (with any type of singularities), we consider the lines ℛ m obtained by reflection of the lines (S m) on 𝒞 (for m ∈ 𝒞). The caustic by reflection Σ S (𝒞) is classically defined as the Zariski closure of the envelope of the reflected lines ℛ m . We identify this caustic with the Zariski closure of Φ(𝒞), where Φ is some rational map. We use this approach to give general and explicit formulas for the degree (with multiplicity) of caustics by reflection. Our formulas are expressed in terms of intersection numbers of the initial curve 𝒞 (or of its branches). Our method is based on a fundamental lemma for rational map thanks to the notion of Φ-polar and on the computation of intersection numbers. In particular, we use precise estimates related to the intersection numbers of 𝒞 with its polar at any point and to the intersection numbers of 𝒞 with its Hessian curve. These computations are linked with generalized Plucker formulas for the class...

Abstract: The classical Plucker formula for a plane curve was generalized by Teissier to the case of a hypersurface with isolated singularities and further by Kleiman to the case of an arbitrary n-dimensional projective variety V with isolated singularities. The formula relates the zeroth rank of V (the degree of the dual variety) to the Segre numbers of the conormal module and certain Buchsbaum—Rim multiplicities associated to the singular points of V. A second generalization was obtained by Pohl. It relates the (n-1)th rank of V to the first Chern class of a desingularization of V and the degree of the cuspidal divisor. We describe, for a projective variety V with arbitrary singularities, a natural class in the Chow group of the singular locus whose top dimensional part is given by Buchsbaum—Rim multiplicities, and we obtain generalizations of both formulas. The formulas are equations in the Chow group of V. They imply numerical formulas for all the ranks of V.

Abstract: We prove a formula which compares intersection numbers of conormal varieties of two projective varieties and their dual varieties. When one of them is linear, we can recover the usual Plucker formula for the degree of the dual variety. The basic strategy of the proof is to study a category of Lagrangian subvarieties in the cotangent bundle of a projective space under a birational transformation.

Abstract: A generalized Frenet formula for holomorphic 2-spheresS 2 in Grassmann manifoldsG(k,n) is introduced and a generalized Pliicker formula is achieved. By use of these formulae some pinching theorems for Gaussian curvature are obtained which generalize the corresponding results about holomorphic and minimalS 2 inCP n-1. Einsteinian holomorphic subbundle of the trivial bundleS 2 ×C n is investigated.