Some remarks on infinitesimal deformations of a conic bundle. I (Q2719779)

Let \(f:X\to Y\) be a conic bundle over a compact complex surface \(Y\) and \(\Delta\subset Y\) its discriminant locus. Any small deformation of \(f\) (with \(Y\) fixed) is again a conic bundle. The author studies the relationship between the infinitesimal deformations of \(f\) and the infinitesimal embedded deformations of \(\Delta\) in \(Y\).NEWLINENEWLINENEWLINEMore precisely, according to the general theory of \textit{E. Horikawa} [J. Math. Soc. Japan 25, 372-396 (1973; Zbl 0253.32022)], the first order infinitesimal deformations of \(f\) are parametrized by \(D_{X/Y}:= \text{H}^1 (\text{T}_X \to f^* \text{T}_Y)\) (here H\(^1\) denotes the hypercohomology). On the other hand, the first order infinitesimal embedded deformations of \(\Delta\) in \(Y\) are parametrized by \(\text{H}^0 (\Delta, N_{\Delta \mid Y})\). For \(\Delta\) smooth, the author gives an explicit description of the ``tangent map'': \(D_{X/Y}\to \text{H}^0 (\Delta, N_{\Delta \mid Y})\).