Nef cone and successive minima: an example

TL;DR

We compute the nef and pseudo-effective cones of $C\times J$ for a genus $g>1$ curve $C$ and its Jacobian $J$ under the minimal Picard number condition $\rho(C\times J)=3$, obtaining explicit descriptions and showing the nef and psef cones coincide. Using these cone descriptions, we derive the successive minima for the relative height in the fibration $C\times J\to J$ and exhibit a counterexample to Zhang's geometric theorem on successive minima for $\dim B>1$. The boundary of the nef cone is generated by pull-backs $L_{m,n}=g m^2\alpha_1+n^2\theta_2+mnQ$, with every nef class approximable by such, leading to a semi-ample boundary. The computations produce explicit numerical invariants: $e_1(h^{\theta}_{\mathcal{L}})=e_2(h^{\theta}_{\mathcal{L}})=(g-\frac{1}{g})(g-1)!$, $h^{\theta}_{\mathcal{L}}(C_K)=(g-1)(g-1)!$, and a sequence of algebraic points with height attaining $e_1$, illustrating the failure of the conjectured inequality in higher-dimensional bases.

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.

Theorems & Definitions (12)

• Theorem 1.1: Theorem \ref{['cone thm']}
• Theorem 1.2: Theorem \ref{['second main']}
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Theorem 2.3: Theorem \ref{['first main thm']}
• proof
• Lemma 3.1
• proof