MathOverflow

Let $f\colon X\to B$ be a locally trivial bundle of CW-complexes with fiber $F$. There are two standard ways to associate to it a spectral sequence converging to the cohomology $H^*(X,\mathbb{Q})$. I wonder whether these two spectral sequence are isomorphic...

I would like to clarify the details of a claim from Danilov's survey "Algebraic Varieties and Schemes" (contained as 2nd section in this book):
The thing is that one can try to encode non affine schemes in terms of "affine information" not only in terms of "classical" glueing datum, but more abstractly by means of apropriate schematic equivalence relation $R$ in same spirit as used in one possible...

Let $\Delta$ be a simplicial sphere, that is, a finite (abstract) simplicial complex whose canonical geometric realization $|\Delta|$ is homeomorphic to a sphere $\mathbf S^d\subset\Bbb R^{d+1}$.
Question: Can the homeomorphism $\phi :|\Delta|\to\mathbf S^d$ be chosen in a way, so that all combinatorial symmetries of $\Delta$ are realized geometrically?...

More percisely:
Question: Is there a (convex) polytope that has a unique realization up to, say, projective transformations?
I suppose I have to assume that it has more than $d+2$ vertices/facets if it is $d$-dimensional, to avoid some trivial cases such as simplices.By realization I mean just any other polytope that has the same combinatorics as the original one, that is...