= Discriminant quintic of a cubic surface conic bundle
{title2=$\deg\Delta=5$}
The matrix of the residual <plane conic> in coordinates $(u,v,w)$ has coefficient degrees $\left(\begin{smallmatrix}1&1&2\\1&1&2\\2&2&3\end{smallmatrix}\right)$ in the plane parameter $(s,t)$. Every term of its <determinant> has degree five. The <reduced singular fibres of a smooth conic bundle> lemma rules out an identically zero determinant and makes each zero simple. Over an <algebraically closed field>, there are therefore exactly five singular fibres, each a pair of distinct coplanar <projective lines>. No residual conic contains the original line, because $sA+tB\equiv0$ would make $A,B$ proportional and force a common zero. Pairs from different fibres are disjoint.
Back to article page
