Linear projections and successive minima

Abstract

Let $X$ be an arithmetic surface, and let $L$ be a line bundle on $X$. Choose a metric $h$ on the lattice $\Lambda$ of sections of $L$ over $X$. When the degree of the generic fiber of $X$ is large enough, we get lower bounds for the successive minima of $(\Lambda ,h)$ in terms of the normalized height of $X$. The proof uses an effective version (due to C. Voisin) of a theorem of Segre on linear projections and Morrison's proof that smooth projective curves of high degree are Chow semistable.