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

TL;DR

This paper investigates the projective geometry of locally toric algebraic curves, proving that general tangent and osculating spaces typically meet the curve only at the point of contact, extending previous results and providing new applications.

Contribution

It extends classical results on tangent lines and osculating spaces of locally toric curves, offering simplified proofs and broader conditions for non-intersection in projective spaces.

Findings

General tangent lines to locally toric curves in P^3 meet only at the point of tangency.

Under mild conditions, the general osculating 2-space to certain locally toric curves in P^4 does not meet the curve again.

The results rely on an arithmetic approach and have applications in algebraic geometry.

Abstract

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