Restricted successive minima

Successive Minima and Radii

In this note we present inequalities relating the successive minima of a o-symmetric convex body and the successive inner and outer radii of the body. These inequalities build a bridge between known inequalities involving only either the successive minima or the successive radii.

Successive-Minima-Type Inequalities

We show analogues of Minkowski’s theorem on successive minima, where the volume is replaced by the lattice point enumerator. We further give analogous results to some recent theorems by Kannan and Lovász on covering minima.

Linear projections and successive minima

In arithmetic geometry, cohomology groups are not vector spaces as in classical algebraic geometry but rather euclidean lattices. As a consequence, to understand these groups we need to evaluate not only their rank, but also their successive minima, which are fundamental invariants in the geometry of numbers. The goal of this article is to perform this task for line bundles on projective curves...

Secant Varieties and Successive Minima

Let X be a semi-stable arithmetic surface over the spectrum S of the ring of integers in a number field K. We assume that the generic fiber XK is geometrically irreducible and has positive genus. Consider a line bundle L on X , which is non-negative on every vertical fiber and equipped with an admissible metric (in the sense of Arakelov [A]) of positive curvature. In [S] we have shown that any ...

On successive minima of indefinite quadratic forms