320 research outputs found
Hyperelliptic -osculating covers and rational surfaces
Let and denote, respectively, the projective line and
a fixed elliptic curve marked at its origin, both defined over an algebraically
closed field of arbitrary characteristic \emph{\textbf{p}}
\neq2. We will consider all finite separable marked morphisms , such that is a degree- cover of , ramified at the smooth point . Canonically
associated to there is the Abel (rational) embedding of into its
\emph{generalized Jacobian}, , and , the flag of
hyperosculating planes to at (cf.
\textbf{2.1. & 2.2.}). On the other hand, we also have the homomorphism
\iota_\pi: X \to \Jac\,\Gamma, obtained by dualizing . There is a
smallest positive integer such that the tangent line to is
contained in . We call it \emph{the osculating order} of .
Studying, characterizing and constructing those with given \textit{osculating
order} but maximal possible arithmetic genus, is one of the main issues.
The other one, to which the first issue reduces, is the construction of all
rational curves in a particular anticanonical rational surface associated to
(i.e.: a rational surface with an effective anticanonical divisor)
Remarks on endomorphisms and rational points
Let X be a variety over a number field and let f: X --> X be an "interesting"
rational self-map with a fixed point q. We make some general remarks concerning
the possibility of using the behaviour of f near q to produce many rational
points on X. As an application, we give a simplified proof of the potential
density of rational points on the variety of lines of a cubic fourfold
(originally obtained by Claire Voisin and the first author in 2007).Comment: LaTeX, 22 pages. v2: some minor observations added, misprints
corrected, appendix modified
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