Abstract. In the 80's D. Eisenbud and J. Harris considered the following problem: "What are the limits of Weierstrass points in families of curves degenerating to stable curves?" But for the case of stable curves of compact type, treated by them, this problem remained wide open since then. In the present article, we propose a concrete approach to this problem, and give a quite explicit solution for stable curves with just two irreducible components meeting at points in general position.

Given a hypersurface in the complex projective n-space we prove several known formulas for the degree of its polar map by purely algebrogeometric methods. Furthermore, we give formulas for the degree of its polar map in terms of the degrees of the polar maps of its components. As an application, we classify the plane curves with polar map of low degree, including a very simple proof of I. Dolgachev's classification of homaloidal plane curves. 26102.769/2008.

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We develop a method to compute limits of dual plane curves in Zeuthen families of any kind. More precisely, we compute the limit 0-cycle of the ramification scheme of a general linear system on the generic fiber, only assumed geometrically reduced, of a Zeuthen family of any kind.

Given a hypersurface in the complex projective space, we prove that the degree of its toric polar map is given by the signed topological Euler characteristic of a distinguished open set, namely the complement of the union of the hypersurface and the coordinate hyperplanes. In addition, we prove that if the hypersurface is in general position or is nondegenerate with respect to its Newton polytope, then the coefficients of the Chern-Schwartz-MacPherson class of the distinguished open set agree, up to sign, with the multidegrees of the toric polar map. In the latter case, we also recover the multidegrees from mixed volumes.For plane curves, a precise formula for the degree of the toric polar map is obtained in terms of local invariants. Finally, we construct families, in arbitrary dimension, of irreducible hypersurfaces whose toric polar map is birational.