Semiampleness on Jacobian elliptic surfaces of Kodaira dimension one
Abstract
Let π: X→ ℙ1 be a semistable Jacobian elliptic surface over ℂ, and set χ=χ(𝒪X)≥3, so that κ(X)=1. Assume that the Mordell-Weil group of π is finite and that π has at least one reducible fiber, the reducible fibers being of types In1,⋯, Ins. Recently, Laface et al. proved that the zero section and the components of the reducible fibers generate NE̅(X) if and only if δ(π):=∑i=1s(⌊ ni2/4⌋)/ni≤χ. In particular, the Mori cone is rational polyhedral in this range. They also proved that N(π)=∑i=1s ni≤ 2χ+3 implies that X is a Mori dream surface. In this paper, we study the existence problem of Mori dream surfaces provided that N(π)≥ 2χ+4. Suppose δ(π)≤ χ. We first show that every nef isotropic divisor on X is semiample, and whenever each ni is even. Furthermore, when every ni is even, we obtain criteria for X to be a Mori dream surface: X is a Mori dream surface provided that either (i) all ni=2, or (ii) N(π)≤ 2χ+4; if some nj=4, then X is a Mori dream surface if and only if N(π)≤ 2χ+4. However, once some ni is odd, we construct a Jacobian elliptic surface π: Y→ℙ1 with δ(π)=χ=3, trivial Mordell-Weil group, and singular-fiber configuration I4+3I3+23I1 for which none of the eight non-vertical isotropic extremal rays is semiample. In particular, Y is not a Mori dream surface.
Related
- Bounded cohomology property on Jacobian elliptic surfaces with simplicial Mori cones
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- On irregular threefolds of Kodaira dimension two
- On the singularities and the Kodaira dimension of unitary Shimura varieties
BibTeX
@misc{laface2026semiampleness,
title = {{Semiampleness on Jacobian elliptic surfaces of Kodaira dimension one}},
author = {Antonio Laface and Sichen Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23306},
eprint = {2609.23306},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23306},
}
RIS
TY - UNPB
TI - Semiampleness on Jacobian elliptic surfaces of Kodaira dimension one
AU - Antonio Laface
AU - Sichen Li
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23306
UR - https://arxiv.org/abs/2609.23306
AB - Let π: X→ ℙ^1 be a semistable Jacobian elliptic surface over ℂ, and set χ=χ(𝒪_X)≥3, so that κ(X)=1. Assume that the Mordell-Weil group of π is finite and that π has at least one reducible fiber, the reducible fibers being of types I_n_1,⋯, I_n_s. Recently, Laface et al. proved that the zero section and the components of the reducible fibers generate NE̅(X) if and only if δ(π):=∑_i=1^s(⌊ n_i^2/4⌋)/n_i≤χ. In particular, the Mori cone is rational polyhedral in this range. They also proved that N(π)=∑_i=1^s n_i≤ 2χ+3 implies that X is a Mori dream surface. In this paper, we study the existence problem of Mori dream surfaces provided that N(π)≥ 2χ+4. Suppose δ(π)≤ χ. We first show that every nef isotropic divisor on X is semiample, and whenever each n_i is even. Furthermore, when every n_i is even, we obtain criteria for X to be a Mori dream surface: X is a Mori dream surface provided that either (i) all n_i=2, or (ii) N(π)≤ 2χ+4; if some n_j=4, then X is a Mori dream surface if and only if N(π)≤ 2χ+4. However, once some n_i is odd, we construct a Jacobian elliptic surface π: Y→ℙ^1 with δ(π)=χ=3, trivial Mordell-Weil group, and singular-fiber configuration I_4+3I_3+23I_1 for which none of the eight non-vertical isotropic extremal rays is semiample. In particular, Y is not a Mori dream surface.
ER -