Osculating degeneration of curves

Abstract

The main objects of this paper are osculating spaces of order m to 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 m are curves in Pm+1 Whose degree n is 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 Pm+2. The case m = 1 of it is a classical formula proved with modern techniques. by Le Barz (Le Barz, P. (1982). Formules multisecantes pour les courbes gauches quelconques. In: Enumerative Geometry and Classical Algebraic Geometry. Prog. in Mathematics 24, Birkhauser, pp. 165-197).