Conic bundle ($f:X\to B$)

A flat proper morphism of algebraic varieties whose geometric fibres are plane conics, usually presented as a quadratic equation in a projective bundle of relative dimension two. The generic fibre is normally required to be smooth; singular conics can occur in special fibres. A conic bundle from a line on a smooth cubic surface has five reducible fibres over the projective line.

In characteristic different from two, a local equation for a conic bundle is $z^{\mathsf{T}}Mz=0$ with $M$ symmetric. A fibre is singular exactly when the determinant vanishes. Changing the fibre coordinates multiplies that determinant by a nonzero square, so its zero locus is intrinsically defined even when the quadratic coefficients belong to twisting line bundles.

Let the base be a smooth algebraic curve, the total surface be smooth, and the field be algebraically closed of characteristic different from two. At a rank-two singular fibre with kernel vector $v$, the fibre derivatives vanish at $[v]$. Smoothness requires $v^{\mathsf{T}}{M}^{\prime }v\ne 0$. The adjugate matrix is a nonzero multiple of $v{v}^{\mathsf{T}}$, so the determinant has a simple zero. At rank at most one, the projective kernel contains a line and the quadratic $v^{\mathsf{T}}{M}^{\prime }v$ has a zero on it; that point would make the total surface singular. Thus every singular fibre is a pair of distinct projective lines and its discriminant zero is simple.