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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 .

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.

Given any light position S in the complex projective plane P^2 and any algebraic curve C of P^2 (with any kind of singularities), we consider the incident lines coming from S (i.e. the lines containing S) and their reflected lines after reflection off the mirror curve C. The caustic by reflection is the Zariski clusure of the envelope of these reflected lines. We introduce the notion of reflected polar curve and express the class of the caustic by reflection in terms of intersection numbers of C with the reflected polar curve, thanks to a fundamental lemma established in [14]. This approach enables us to get an explicit formula for the class of the caustic by reflection in every case in terms of intersection numbers of the initial curve C., Comment: 21 pages, 1 figure

Given a point S and any irreducible algebraic curve C in P^2 (with any type of singularities), we consider the caustic of reflection defined as the Zariski closure of the envelope of the reflected lines from the point S on the curve C. We identify this caustic with the Zariski closure of the image of C by a rational map. Thanks to a general fundamental lemma, we give a formula for the degree of the caustic of reflection in terms of multiplicity numbers of C (or of its branches). Our formula holds in every case. We also give some precisions about Pl\"ucker formulas., Comment: 35 pages, 1 figure

Let $R$ be a standard graded Noetherian algebra over an Artinian local ring. Motivated by the work of Achilles and Manaresi in intersection theory, we first express the multiplicity of $R$ by means of local $j$-multiplicities of various hyperplane sections. When applied to a homogeneous inclusion $A\subseteq B$ of standard graded Noetherian algebras over an Artinian local ring, this formula yields the multiplicity of $A$ in terms of that of $B$ and of local $j$-multiplicities of hyperplane sections along ${\rm Proj}\,(B)$. Our formulas can be used to find the multiplicity of special fiber rings and to obtain the degree of dual varieties for any hypersurface. In particular, it gives a generalization of Teissier's Pl\"{u}cker formula to hypersurfaces with non-isolated singularities. Our work generalizes results by Simis, Ulrich and Vasconcelos on homogeneous embeddings of graded algebras., Comment: To appear in Transactions of American Mathematical Society

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.