In the paper of Eckl/Puhklikov (http://arxiv.org/abs/1210.3715) the following terminology is introduced:
" Let $X \subset Y$ be a subvariety of codimension 1 in a smooth quasiprojective complex variety $Y$ of dimension $n$. A point $P ∈ X$ is called a quadratic point of rank $r$ if there are analytic coordinates $z = (z_1, . . . , z_n)$ of $Y$ around $P$ and a quadratic form $q_2(z)$ of rank r such that the germ of $X$ in $P$ is given by $(P ∈ X) ∼= {q_2(z) + higherorderterms = 0} ⊂ Y$. "
This is the first I've seen this terminology, but I would be interested to see if there are any other situations in which it features. Furthermore, is there some geometric interpretation of the rank of a quadratic singularity? Or some commutative algebra/algebraic group interpretation, like associating to it a symmetric $n \times n$ matrix?
