I am reading a text in which one speaks of quadric hypersurfaces $\mathcal{Q}$ (in $\mathbb{P}^n(k)$ with $k$ a finite field) which are "everywhere nonreduced."
What does this mean ? (And is there a difference with general fields ?)
What would be the classification of everywhere nonreduced quadric hypersurfaces $\mathcal{Q}$ in $\mathbb{P}^n(k)$ ($k$ still a finite field) ?
This might mean two different things.
First, among quadric hypersurfaces there are hypersurfaces which are everywhere non-reduced. These are double planes (their equations are squares of linear functions).
Second, one could mean a subscheme, whose associated reduced scheme is a (reduced) quadric hypersurface, but whose scheme structure is everywhere non-reduced.