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Osculating spaces and diophantine equations (with an appendix by Pietro Corvaja and Umberto Zannier) | | Michele Bolognesi
; Gian Pietro Pirola
; | | Date: |
9 Nov 2007 | | Abstract: | This paper deals with some classical problems about the projective geometry
of complex algebraic curves. We call extit{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 $P^3$ 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 $P^3$ 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 $P^4$ 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. | | Source: | , 0711.1487 | | Services: | Forum | Review | PDF | Favorites | |
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