Regular zero locus
= Regular zero locus
{title2=$\operatorname{rank}(ds|_V)=r$}
The zero locus of a holomorphic <section of a vector bundle> of a rank-$r$ bundle is regular when its derivative has rank $r$ along the locus. Its kernel is then the tangent bundle of the smooth zero set. Smoothness of the underlying set alone is insufficient: the square of a reduced defining function has the same set of zeros but zero derivative there.