Some conic bundles (Q1280614)

The main result of the paper is the following theorem: Let \(X\) be a conic bundle on a normal surface \(S\). Assume that \(X\) has log-terminal singularities and \(-K_X\equiv Q\), \(Q\) being a Cartier divisor. Let \(A\) be a sufficiently ample divisor on \(S\). Let \(p : X \rightarrow S\) denote the projective morphism onto \(S\). Then the linear system \(|p^*A - Q |\) contains a reduced irreducible divisor with only log-terminal singularities. This generalises the results of \textit{V. V. Shokurov} [Math. USSR, Izv. 14, 395-405 (1980); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 43, 430-441 (1979; Zbl 0407.14017)] and \textit{Yu. G. Prokhorov} [Sb. Math. 186, No. 9, 1341-1352 (1995); translation from Mat. Sb. 186, No. 9, 113-124 (1995; Zbl 0868.14020)].