Plane conic
= Plane conic
{title2=$q(X,Y,Z)=0,\quad\deg q=2$}
A degree-two <projective plane curve>, allowing singular and nonreduced cases. In characteristic different from two, write its equation as $z^{\mathsf T}Mz$ with $M$ a symmetric <matrix>. It is smooth exactly when $M$ has <matrix rank> three. Over an <algebraically closed field>, rank two gives two distinct <projective lines>, while rank one gives a double line.