Birational automorphisms of algebraic threefolds with a~pencil of Del Pezzo surfaces

A. V. Pukhlikov

Izvestiya. Mathematics , Tome 62 (1998) no. 1, pp. 115-155

Résumé

In this paper it is proved that there is only one pencil of rational surfaces on a smooth three-dimensional variety $V$ fibred into Del Pezzo surfaces of degree 1, 2 or 3 by a morphism $\pi\colon V\to\mathbb P^1$ (Iskovskikh's conjecture), provided that the class of 1-cycles $(MK^2_V-f)$ (where $K_V$ is the canonical class and $f$ is the class of a line in a fibre) is not effective for any $M\in\mathbb Z$. This condition is satisfied if $V$ is sufficiently “twisted” over the base of the pencil. This implies that these varieties admit no conic bundle structures, and, in particular, that they are nonrational. Certain higher-dimensional generalizations of this theorem are considered: similar statements are true for varieties with a pencil of thee-dimensional quartics of general position, and analogous results are obtained in these cases.