Very stable extensions on arithmetic surfaces

TL;DR

This paper proves that general extensions of line bundles on smooth projective curves are very stable and derives new inequalities for lattice minima on arithmetic surfaces, advancing understanding in algebraic geometry and number theory.

Contribution

It introduces the concept of very stable extensions on arithmetic surfaces and establishes new inequalities for successive minima of related Euclidean lattices.

Findings

General extensions of line bundles are very stable.

New inequalities for successive minima of Euclidean lattices on arithmetic surfaces.

Enhanced understanding of stability and lattice properties in algebraic geometry.

Abstract

Given a line bundle L on a smooth projective curve over the complex numbers, we show that a general extension E of L by the trivial line bundle is very stable: line bundles contained in E have degree much less than half the degree of E. From this result we deduce new inequalities for the successive minima of the euclidean lattice H^1(X,L^{-1}), where L is an hermitian line bundle on the arithmetic surface X.