The Picard number of fibred Mori dream surfaces
Abstract
Let S be a smooth complex projective surface endowed with a fibration f : S → C onto a smooth projective curve C. We prove that, if S is a Mori dream space (or, more generally, if its pseudo-effective cone is polyhedral) then the Picard number ρ(S) can be effectively computed by counting the irreducible components of the reducible fibres of f. A first simple consequence is that, given an elliptic fibration f : S → ℙ1 with a section and such that S is a Mori dream space, the Mordell-Weil group of the general fibre of f is finite. The main application is a simple criterion for proving that a surface fibred over a curve is not a Mori dream space. We show that certain Horikawa surfaces, Fermat surfaces in ℙ3 of every degree ≥ 4, particular product-quotient surfaces, and the minimal simply connected numerical Godeaux surface constructed by Craighero and Gattazzo are not Mori dream spaces.
Related
- Mori dream fibers and the geometric generic fiber
- Bound of automorphisms of fibred surfaces in positive characteristic
- Bounded cohomology property on Jacobian elliptic surfaces with simplicial Mori cones
- The derived Picard group of Fukaya categories of closed surfaces
- Residual finiteness and cuspidal cohomology of Picard modular surfaces
BibTeX
@misc{fallucca2026picard,
title = {{The Picard number of fibred Mori dream surfaces}},
author = {Federico Fallucca and Roberto Pignatelli and Francesco Polizzi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.29485},
eprint = {2608.29485},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29485},
}
RIS
TY - UNPB
TI - The Picard number of fibred Mori dream surfaces
AU - Federico Fallucca
AU - Roberto Pignatelli
AU - Francesco Polizzi
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2608.29485
UR - https://arxiv.org/abs/2608.29485
AB - Let S be a smooth complex projective surface endowed with a fibration f : S → C onto a smooth projective curve C. We prove that, if S is a Mori dream space (or, more generally, if its pseudo-effective cone is polyhedral) then the Picard number ρ(S) can be effectively computed by counting the irreducible components of the reducible fibres of f. A first simple consequence is that, given an elliptic fibration f : S → ℙ^1 with a section and such that S is a Mori dream space, the Mordell-Weil group of the general fibre of f is finite. The main application is a simple criterion for proving that a surface fibred over a curve is not a Mori dream space. We show that certain Horikawa surfaces, Fermat surfaces in ℙ^3 of every degree ≥ 4, particular product-quotient surfaces, and the minimal simply connected numerical Godeaux surface constructed by Craighero and Gattazzo are not Mori dream spaces.
ER -