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This paper deals with some classical problems about the projective geometryof complex algebraic curves. We call \textit{locally toric} a projective curvethat in a neighbourhood of every point has a local analytical parametrizationof type $(t^{a_1},...,t^{a_n})$, with $a_1,..., a_n$ relatively prime positiveintegers. In this paper we prove that the general tangent line to a locallytoric curve in $\bP^3$ meets the curve only at the point of tangency. Thisresult extends and simplifies those of the paper \cite{kaji} by H.Kaji wherethe same result is proven for any curve in $\bP^3$ such that every branch issmooth. More generally, under mild hypotesis, up to a finite number ofanomalous parametrizations $(t^{a_1},...,t^{a_n})$, the general osculating2-space to a locally toric curve of genus $g<2$ in $\bP^4$ does not meet thecurve again. The arithmetic part of the proof of this result relies on theAppendix \cite{cz:rk} to this paper. By means of the same methods we give someapplications and we propose possible further developments.

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