# References on the dual frame of a projective frame.

A projective frame $\mathcal{F}$ of a projective space $P=P(E)$ defines a basis of the vector space $E$, defined up to multiplication by a scalar. The corresponding dual basis of $E^*$ is well defined up to a scalar, thus projectivizes to a unique projective frame of the dual $P^*$ of $P$, which shall be called the dual frame $\mathcal{F}^*$ of $\mathcal{F}$.

I woundered some time ago how one can construct $\mathcal{F}^*$ from the dual hyperplanes of the elements of $\mathcal{F}$. This is not a difficult question, but surprisingly it is neither completely trivial nor could I find a reference on this problem.

To see why it is not trivial, remember that a projective frame of dimension $n$ has $n+2$ elements; it is trivial to find the $n+1$ first elements of $\mathcal{F}^*$, but the last one is a little bit more difficult to construct.

I have a solution that seems nice to me, in part because it relates several different kind of projective polarities. I think it would be surprising that it is not already well-known, though.

You are right; do you know some reference for this? By the way, there exist several other interpretation of this polarity (for example, by considering the faces of the simplex defined by the $n+1$ first points of the frame as an algebraic hypersurface of degree $n+1$). – Benoît Kloeckner Jan 11 '11 at 14:49