2,733 research outputs found
Computing canonical heights using arithmetic intersection theory
For several applications in the arithmetic of abelian varieties it is
important to compute canonical heights. Following Faltings and Hriljac, we show
how the canonical height on the Jacobian of a smooth projective curve can be
computed using arithmetic intersection theory on a regular model of the curve
in practice. In the case of hyperelliptic curves we present a complete
algorithm that has been implemented in Magma. Several examples are computed and
the behavior of the running time is discussed.Comment: 29 pages. Fixed typos and minor errors, restructured some sections.
Added new Example
Toric Intersection Theory for Affine Root Counting
Given any polynomial system with fixed monomial term structure, we give
explicit formulae for the generic number of roots with specified coordinate
vanishing restrictions. For the case of affine space minus an arbitrary union
of coordinate hyperplanes, these formulae are also the tightest possible upper
bounds on the number of isolated roots. We also characterize, in terms of
sparse resultants, precisely when these upper bounds are attained. Finally, we
reformulate and extend some of the prior combinatorial results of the author on
which subsets of coefficients must be chosen generically for our formulae to be
exact.
Our underlying framework provides a new toric variety setting for
computational intersection theory in affine space minus an arbitrary union of
coordinate hyperplanes. We thus show that, at least for root counting, it is
better to work in a naturally associated toric compactification instead of
always resorting to products of projective spaces
Partial Degree Formulae for Plane Offset Curves
In this paper we present several formulae for computing the partial degrees
of the defining polynomial of the offset curve to an irreducible affine plane
curve given implicitly, and we see how these formulae particularize to the case
of rational curves. In addition, we present a formula for computing the degree
w.r.t the distance variable.Comment: 24 pages, no figure
Curves having one place at infinity and linear systems on rational surfaces
Denoting by the linear system of plane
curves passing through generic points of the projective
plane with multiplicity (or larger) at each , we prove the
Harbourne-Hirschowitz Conjecture for linear systems determined by a wide family of systems of multiplicities
and arbitrary degree . Moreover, we provide an
algorithm for computing a bound of the regularity of an arbitrary system
and we give its exact value when is in the above family.
To do that, we prove an -vanishing theorem for line bundles on surfaces
associated with some pencils ``at infinity''.Comment: This is a revised version of a preprint of 200
Computing functions on Jacobians and their quotients
We show how to efficiently compute functions on jacobian varieties and their
quotients. We deduce a quasi-optimal algorithm to compute isogenies
between jacobians of genus two curves
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