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

Abstract

AbstractThis 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 \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$(t^{a_1},\dots ,t^{a_n})$\end{document}, with a1, …, an relatively prime positive integers. In this paper we prove that the general tangent line to a locally toric curve in P3 meets the curve only at the point of tangency. More generally, under mild hypotesis, up to a finite number of anomalous parametrizations \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$(t^{a_1},\dots ,t^{a_n})$\end{document}, the general osculating 2‐space to a locally toric curve of genus g < 2 in P4 does not meet the curve again. The arithmetic part of the proof of this result relies on the results contained in the Appendix. By means of the same methods we give some applications and we propose possible further developments. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim