Conic discriminant
= Conic discriminant
{title2=$\Delta=\det M$}
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>.