Osculating spaces and diophantine equations (with an appendix by Pietro Corvaja and Umberto Zannier)

2007/11/09 by Michele Bolognesi, Gian Pietro Pirola, Bolognesi, Michele +1
Mathematics · #14H45 #14H50 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:14H45 #msc:14H50

paper · pdf · doi:10.48550/arxiv.0711.1487

23 pages

arxiv created 2007/11/13 · arxiv updated 2009/12/01

Abstract

This paper deals with some classical problems about the projective geometry of complex algebraic curves. We call locally toric a projective curve that in a neighbourhood of every point has a local analytical parametrization of type (t^a₁,...,t^a_n), with a₁,..., a_n relatively prime positive integers. In this paper we prove that the general tangent line to a locally toric curve in \bP³ meets the curve only at the point of tangency. This result extends and simplifies those of the paper \citekaji by H.Kaji where the same result is proven for any curve in \bP³ such that every branch is smooth. More generally, under mild hypotesis, up to a finite number of anomalous parametrizations (t^a₁,...,t^a_n), the general osculating 2-space to a locally toric curve of genus g<2 in \bP⁴ does not meet the curve again. The arithmetic part of the proof of this result relies on the Appendix \citecz:rk to this paper. By means of the same methods we give some applications and we propose possible further developments.

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