Smooth projective surfaces with bounded cohomology property

In this paper, we first prove that every Mori dream surface $X$ satisfies the bounded cohomology property (BCP for short). Namely, there exists a constant $c_X>0$ such that $h^1(\mathcal O_X(C))\le c_Xh^0(\mathcal O_X(C))$ for every curve $C$ on $X$. We then prove that there is a positive constant $m(Y)$ such that $l_C:=(K_Y\cdot C)(C^2)^{-1}\le m(Y)$ for every ample curve $C$ on a geometrically ruled surface $Y$ over a curve of genus $g$, and $Y$ satisfies the BCP if $g\le1$.

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