Some remarks on infinitesimal deformations of a conic bundle. II. (Q2719780)

Let \(f:X\to Y\) be a conic bundle over a compact complex surface \(Y\) and \(\Delta\subset Y\) its discriminant locus. In this case, \(E:=f_* \omega_X^{-1}\) is a rank 3 vector bundle on \(Y\) and the \(Y\)-morphism: \(X\to \mathbb{P}(E)\) defined by the canonical morphism: \(f^*f_* \omega_X^{-1} \to \omega_X^{-1}\) is an embedding. Using results from part I [\textit{M. Ebihara}, Saitama Math. J. 18, 1-21 (2000; see the preceding review Zbl 0978.14008)], the author proves two types of results:NEWLINENEWLINENEWLINE(1) If \(Y=\mathbb{P}^2\), \(\Delta\) is smooth and \(E\) is a direct sum of line bundles, then there is no nontrivial small deformation of \(f\) which is again a conic bundle over \(Y\) with the same discriminant locus \(\Delta\);NEWLINENEWLINENEWLINE(2) Under the same assumptions, any small deformation of \(X\) is realized as an embedded deformation of \(X\) in \(\mathbb{P}(E)\).