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We recall that the Brill–Noether Theorem gives necessary and sufficient conditions for the existence of a g d r . Here we consider a general n-fold, etale, cyclic cover p : C ˜ → C of a curve C of genus g and investigate for which numbers r , d a g d r exists on C ˜ . For r = 1 this means computing the gonality of C ˜ . Using degeneration to a special singular example (containing a Castelnuovo canonical curve) and the theory of limit linear series for tree-like curves we show that the Plucker formula yields a necessary condition for the existence of a g d r which is only slightly weaker than the sufficient condition given by the results of Laksov and Kleimann [24] , for all n , r , d .

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...

Inthispaperwestudygermsofpolynomialsformedbytheproductofsemi-weighted homogeneous polynomials of the same type, which we call semi-weighted homogeneous arrangements. It is shown how the Le numbers of such polynomials are computed using only their weights and degree of homogeneity. A key point of the main theorem is to find the number called polar ratio of this polynomial class. An important consequence is the description of the Euler characteristic of the Milnor fibre of such arrangements only depending on their weights and degree of homogeneity.

We study the process of resolution of linear systems on 2-dimensional regular local rings by quadratic transformations and we give an effective characterization of the Zariski open set of equisingular elements in the linear system. We give a description of the behaviour at the first infinitesimal neighborhood by means of the Plucker Formulas.

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.

We prove a Plucker formula,for a projective variety X with arbitrary singularities, which expresses the class of X, the degree of the dual variety, in terms of Euler characteristics of X and of two linear sections of X. Moreover, we show that there is no formula whatsoever expressing this degree as a difference of two terms, a deformation invariant and a correction for singularities.

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.