On general elephant problem for three-dimensional $\mathbf{Q}$-Fano fiber spaces over a surface

Yu. G. Prokhorov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 1, pp. 263-280

Résumé

We consider $\mathbf{Q}$-Fano fiber spaces $X/S$ over a surface, i. e., a three-dimensional variety $X$ with terminal $\mathbf{Q}$-factorial singularities and a projective morphism $\varphi:X\to S$ onto a normal surface $S$ such that $\varphi_*\mathcal{O}_X=\mathcal{O}_S$, $\rho(X/S)=1$ and $-K_X$ $\varphi$-ample. In this situation we discuss Reid's conjecture on general elephants, i. e. on general members of the linear system $\left|-K_X+\varphi^*h\right|$. We prove that the surface $S$ has only cyclic quotient singularities, besides if for $X/S$ the elephants conjecture is true, then singularities of $S$ are Du Val singularities of the type $A_n$. In the last case some conditions on singularities of $X$ and $S$ are obtained.