Bounded cohomology property on Jacobian elliptic surfaces with simplicial Mori cones

Abstract

Let $X$ be a Jacobian elliptic surface with finite Mordell-Weil group and exactly one reducible fiber. We prove that if $\chi ({\mathcal{O}}_X)\ge \rho (X)$, then the Mori cone of $X$ is simplicial. As an application, assuming additionally $q(X)=0$, we show that $X$ satisfies the bounded cohomology property (BCP): there exists a constant $c_X>0$ such that $h^1({\mathcal{O}}_X(C))\le c_X{h}^0({\mathcal{O}}_X(C))$ for every curve $C$ on $X$. We also establish a necessary and sufficient condition for the BCP to hold on minimal smooth projective surfaces $Y$ with $\kappa (Y)\ge 1$, $q(Y)=0$, and rational polyhedral Mori cones.

11 pages, comments are welcome!