FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

1995, VOLUME 1, NUMBER 1, PAGES 263-280

On general elephant problem for three-dimensional Q-Fano fiber spaces over a surface

Yu. G. Prokhorov

We consider Q-Fano fiber spaces X/S over a surface, i. e., a three-dimensional variety X with terminal 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 $|-K_X + \varphi^* h|$. 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.

All articles are published in Russian.

Location: http://mech.math.msu.su/~fpm/eng/95/951/95115.htm
Last modified: October 3, 1997.