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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

1 vote
1 answer
169 views

Is the vector bundle over a vector bundle, a vector bundle over the base scheme?

Suppose we have $E\to X$, a vector bundle over $X$ and $E'\to E$ a vector bundle over $E$. Composing the structure maps gives a smooth scheme $E'\to X$ over $X$. My question is, when is this a vector …
0 votes
1 answer
142 views

Are projective bundles corresponding to non-isomorphic vector bundles always non-isomorphic?

Suppose we are given a scheme $S$ and two vector bundles $V$ and $W$ over $S$. Is it always true that $\mathbb{P}(V)\cong \mathbb{P}(W)$ implies that $V\cong W$ as $S$-schemes? If the statement is fal …
1 vote
1 answer
238 views

Are there nonaffine schemes over which every exact sequence of vector bundles is split?

Is there an example of a non-affine scheme $X$ such that every short exact sequence of vector bundles over $X$ splits? If there are such examples then what if we ask it to be true of all (not necessar …