= Reduced singular fibres of a smooth conic bundle
{title2=$\operatorname{rank}M=2,\quad\Delta'(t)\ne0$}
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 $v^{\mathsf T}M'v\ne0$. The <adjugate matrix> is a nonzero multiple of $vv^{\mathsf T}$, so the determinant has a simple zero. At rank at most one, the projective kernel contains a line and the quadratic $v^{\mathsf T}M'v$ 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.
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