Nef cone and successive minima: an example

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.

💡 Research Summary

The paper studies the product of a smooth projective curve C of genus g > 1 and its Jacobian J, focusing on the case where the Picard number of C×J is minimal, i.e. ρ(C×J)=3. In this “minimal” situation the Néron‑Severi group decomposes as NS(C×J)=NS(C)⊕NS(J)⊕End(J). The authors construct three explicit generators of NS(C×J)⊗ℝ:

* α₁ = p₁⁎α, where α∈Pic(C) satisfies (2g‑2)α≅ω_C;
* θ₂ = p₂⁎θ, where θ⊂J is the principal polarization obtained from the (g‑1)-fold symmetric product of C;
* Q = (i_α×id)⁎P, the pull‑back of the Poincaré bundle on J×Ĵ via the Abel‑Jacobi map i_α:C→J.

The line bundle Q corresponds to the identity element in End(J).

For any pair of integers (m,n) the morphism f_{m,n}:C×J→J, (x,y)↦m(x‑α)+ny has pull‑back

 f_{m,n}⁎θ = g m²·α₁ + n²·θ₂ + mn·Q.

Since θ is ample on J, every such pull‑back is nef. By allowing real coefficients (m,n)∈ℝ² and adding non‑negative multiples of α₁ and θ₂, the authors obtain the full nef cone. They prove:

 Nef(C×J) = { a·α₁ + b·θ₂ + c·Q | a≥0, b≥0, ab ≥ g c² }

 Amp(C×J) = Big(C×J) = { a·α₁ + b·θ₂ + c·Q | a>0, b>0, ab > g c² }.

Thus the pseudo‑effective cone coincides with the nef cone, and the ample cone coincides with the big cone. Moreover, every boundary divisor of the nef cone is semi‑ample, being a pull‑back of an ample divisor on J.

The second part of the paper applies this cone description to a height problem. Let π:p₂:C×J→J be the projection, M=θ∈Pic(J) the ample line bundle on the base, and consider the line bundle

 L = g·α₁ + θ₂ + Q ∈ Pic(C×J).

For the function field K = k(J) (the field of rational functions on J) the generic fiber is X = C_K. The relative height associated to (L, M) is

 h_{θ}^{L}(x) = (1/deg x)·\overline{x}·L·(π⁎θ)^{d‑1},

where \overline{x} is the Zariski closure of the point x∈X(K) as an effective divisor on C×J. Using the explicit cone description, the authors write \overline{x}=a·α₁ + b·θ₂ + c·Q and compute intersection numbers directly. They obtain:

 e₁(h_{θ}^{L}) = e₂(h_{θ}^{L}) =