Osculating spaces and diophantine equations (with an appendix by Pietro Corvaja and Umberto Zannier)
| dc.creator | Bolognesi, Michele | |
| dc.creator | Pirola, Gian Pietro | |
| dc.date | 2007-11-09 | |
| dc.date | 2007-11-13 | |
| dc.date.accessioned | 2026-07-07T08:42:13Z | |
| dc.date.available | 2026-07-07T08:42:13Z | |
| dc.description | 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. | |
| dc.description | 23 pages | |
| dc.identifier | https://arxiv.org/abs/0711.1487 | |
| dc.identifier | http://arxiv.org/abs/0711.1487 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/141885 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Number Theory | |
| dc.subject | 14H50,14H45 | |
| dc.title | Osculating spaces and diophantine equations (with an appendix by Pietro Corvaja and Umberto Zannier) | |
| dc.type | text |