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On page 3 of [this paper][1], the authors give a Bruhat cell decomposition of a quadric hypersurface $Q$ of complex dimension $n$. This may be a stupid question, but it doesn't seem clear to me what exactly the decomposition is. They only explicitly describe the real dimension $2n-2$, $n$, and $n+2$ cells, but say that there is exactly one cell for each even real dimension. Particularly, I would like to know what the complex codimension 2 cells are. [1]: http://www.imsc.res.in/~kapil/papers/topol.pdf

On page 3 of this paper, the authors give a Bruhat cell decomposition of a quadric hypersurface $Q$ of complex dimension $n$. This may be a stupid question, but it doesn't seem clear to me what exactly the decomposition is. They only explicitly describe the real dimension $2n-2$, $n$, and $n+2$ cells, but say that there is exactly one cell for each even real dimension. Particularly, I would like to know what the complex codimension 2 cells are.