Birationally rigid del Pezzo fibrations
The main object of study of the present paper are the del Pezzo fibrations of degree 1 or 2, i.e. the smooth projective complex threefolds \(X\) with \(\text{rk\,Pic}(X) = 2\) admitting a fibration \(\tau: X \rightarrow {\mathbb P}^1\) with a general fiber -- a del Pezzo surface of degree \(1\) or \(2\). By a result of \textit{A. Pukhlikov} [Izv. Math. 62, No. 1, 115--155 (1998; Zbl 0948.14008)] a del Pezzo fibration \(X\) of degree 1 or 2, for which \(K_X^2\) lies outside the interiour of the cone \(\overline{\text{NE}}(X)\) of numerically effective 1-cycles on \(X\), can't be birational: (a) to a conic bundle, (b) to a non-equivalent fibration of rational surfaces, (c) to a Fano 3-fold with terminal \({\mathbb Q}\)-factorial singularities. Theorem 2, the main result of this paper, states that if in addition \(K_X^2\) does not lie on the boundary \(\delta \overline{\text{NE}}(X)\) then \(X\) can't be birational to a fibration into surfaces with Kodaira dimension \(0\), or to a Fano 3-fold with canonical singularities. In example 3 is constructed a del Pezzo fibration with \(K_X^2 \in \delta \overline{\text{NE}}(X)\) which objects some of the conclusions of theorem 2 (see also theorem 27). By remark 25 and theorem 26, there exist del Pezzo fibrations of degree \(\leq 2\) with \(K_X^2 \in \overline{\text{NE}}(X)\) for which all conclusions of theorem 2 take place (see also theorem 28).
scientific article; zbMATH DE number 2168078