Raquel Mallavibarrena

Higher order dual varieties of projective surfaces

We investigate higher order dual varieties of projective manifolds whose osculatory behavior is the best possible. In particular, for a k-jet ample surface we prove the nondegeneratedness of the k-th dual variety and for 2-regular surfaces we investigate the degree of the second dual variety.

Osculating Degeneration of Curves

Abstract The main objects of this paper are osculating spaces of order mto smooth algebraic curves, with the property of meeting the curve again. We prove that the only irreducible curves with an infinite number of this type of osculating spaces of order mare curves in P m+1whose degree nis greater than m + 1. This is a generalization of the result and proof of Kaji (Kaji, H. (1986). On the tangentially degenerate curves. J. London Math. Soc.33(2):430–440) that corresponds to the case m = 1. We also obtain an enumerative formula for the number of those osculating spaces to curves in P m+2. The case m = 1 of it is a classical formula proved with modern techniques by Le Barz (Le Barz, P. (1982). Formules multisécantes pour les courbes gauches quelconques. In: Enumerative Geometry and Classical Algebraic Geometry. Prog. in Mathematics 24, Birkhäuser, pp. 165–197). Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.