31 research outputs found
Divisors on graphs, Connected flags, and Syzygies
We study the binomial and monomial ideals arising from linear equivalence of
divisors on graphs from the point of view of Gr\"obner theory. We give an
explicit description of a minimal Gr\"obner bases for each higher syzygy
module. In each case the given minimal Gr\"obner bases is also a minimal
generating set. The Betti numbers of the binomial ideal and its natural initial
ideal coincide and they correspond to the number of 'connected flags' in the
graph. In particular the Betti numbers are independent of the characteristic of
the base field. For complete graphs the problem was previously studied by
Postnikov and Shapiro and by Manjunath and Sturmfels. The case of a general
graph was stated as an open problem.Comment: to appear in International Mathematics Research Notices (IMRN
The monodromy pairing and discrete logarithm on the Jacobian of finite graphs
Every graph has a canonical finite abelian group attached to it. This group
has appeared in the literature under a variety of names including the sandpile
group, critical group, Jacobian group, and Picard group. The construction of
this group closely mirrors the construction of the Jacobian variety of an
algebraic curve. Motivated by this analogy, it was recently suggested by Norman
Biggs that the critical group of a finite graph is a good candidate for doing
discrete logarithm based cryptography. In this paper, we study a bilinear
pairing on this group and show how to compute it. Then we use this pairing to
find the discrete logarithm efficiently, thus showing that the associated
cryptographic schemes are not secure. Our approach resembles the MOV attack on
elliptic curves
Faltings height and N\'eron-Tate height of a theta divisor
We prove a formula which, given a principally polarized abelian variety
over the field of algebraic numbers, relates the stable Faltings
height of with the N\'eron-Tate height of a symmetric theta divisor on .
Our formula completes earlier results due to Bost, Hindry, Autissier and
Wagener. We introduce the notion of a tautological model of a principally
polarized abelian variety over a complete discretely valued field, and we
express the non-archimedean N\'eron functions for a symmetric theta divisor on
in terms of tautological models and the tropical Riemann theta function.Comment: Referee's remarks taken into account, errors fixed, main result
unchanged, added examples dealing with elliptic curve