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 -KX
$\varphi$-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
An.
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.