4
votes
1answer
824 views

How to resolve a wedge product of vector bundles

Let $X$ be an algebraic variety. Consider an exact sequence $$0\to A\to B\to C\to 0$$ of vector bundles on $X$. I have seen in different papers the following type resolution of wedge product of $C$ ...
28
votes
2answers
1k views

Symmetric powers and duals of vector bundles in char p

Suppose that $X$ is a smooth projective variety (eg $P^n$) and $E$ is a vector bundle (eg the tangent bundle). If the characteristic is zero, then taking symmetric powers "commutes" with taking duals: ...
21
votes
4answers
2k views

Using linear algebra to classify vector bundles over P^1

There is a theorem of Grothendieck stating that a vector bundle of rank $r$ over the projective line $\mathbb{P}^1$ can be decomposed into $r$ line bundles uniquely up to isomorphism. If we let ...
4
votes
3answers
1k views

Endomorphisms of vector bundles

I'm a bit stuck, and I'm hoping someone can help me out. I have a vector bundle $E$ on an algebraic curve (the ones I am interested in are holomorphic, but I'm sure that doesn't matter so much for ...