It is well known that any plane cubic curve can be obtained as the discriminant locus of a conic bundle (actually even just of a net of conics). Does this hold true also for all nodal cubics (with double lines over the nodes)? How does one see this?



plane cubics and conic bundles
Yes, at least if we're not in characteristic 2.
Since all nodal cubics are projectively equivalent, it is enough to find one example. Trying a few symmetric $3 \times 3$ determinants soon turns up the matrix
$$
M = \left[ \begin{array}{ccc}
 x & x & y \cr 
 x & z & 0 \cr
 y & 0 & x
\end{array} \right]
$$
with determinant $x^2(z-x)-zy^2$. So the discriminant locus of
the associated net of conics is $zy^2 = x^2(z-x)$, which has a
node at $(x:y:z) = (0:0:1)$ [set $z=1$ to get the more familiar
affine model $y^2 = x^2 - x^3$ with a node at the origin].
At that point $M$ becomes
$$
\left[ \begin{array}{ccc}
 0 & 0 & 0 \cr
 0 & 1 & 0 \cr
 0 & 0 & 0
\end{array} \right],
$$
where the conic degenerates to a double line as desired.


