Nef cone and successive minima: an example

TL;DR

This paper calculates the nef and pseudo-effective cones of a specific algebraic variety and examines the behavior of successive minima, revealing a counterexample to Zhang's theorem in this context.

Contribution

It provides explicit computations of cones and successive minima for a minimal Picard number variety, challenging existing theorems.

Findings

Computed nef and pseudo-effective cones of C×J for minimal Picard number

Calculated successive minima of a height function in the relative setting

Demonstrated that Zhang's theorem of successive minima does not hold in this case

Abstract

In this paper, we compute the nef cone and the pseudo-effective cone of $C\times J$ for a smooth projective curve $C$ and its Jacobian variety $J$ such that $C\times J$ has the minimal Picard number. As a consequence, we also compute the successive minima of a height function for the relative setting $C\times J\to J$, and our result shows that Zhang's theorem of successive minima does not hold in this case.