= Conic bundle from a line on a smooth cubic surface
{title2=$f:S\to\mathbb P^1,\quad[f^{-1}(t)]=H-L$}
Write a line as $L=\{x_2=x_3=0\}$ and the surface equation as $F=x_2Q_2+x_3Q_3$. Smoothness says $A=Q_2|_L$, $B=Q_3|_L$ have no common zero. Projection $[x_2:x_3]$ extends over $L$ by $[-B:A]$. Its fibres are the residual <plane conics> in the planes through $L$, since substituting $x_2=sw$, $x_3=tw$ gives $F=w(sQ_2+tQ_3)$. Different residual conics are disjoint everywhere, including their points on $L$, because they are fibres of one <morphism of algebraic varieties>. The family is flat: over the smooth base curve its local rings are torsion-free over the corresponding <discrete valuation ring>; smoothness of the total surface supplies the reduced-fibre conclusions.
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