6
votes
1answer
60 views

exact sequence induced by restriction to closed subscheme

I'm somewhat embarrassed to be confused about this issue after a while studying algebraic geometry, but here goes. Let $\iota: Y \hookrightarrow X$ be the inclusion of a closed subscheme, $U$ is the ...
1
vote
0answers
38 views

Describing a locally free sheaf sitting between two locally free sheaves which are given as extensions

Assume $X$ is a two dimensional scheme with $Pic(X)=\mathbb{Z}$ such that every rank two locally free sheaf $\mathcal{E}$ is given by an exact sequence $0\rightarrow \mathcal{O}_X(n)\rightarrow ...
2
votes
1answer
37 views

Why is $H^0(C,\mathcal O(D))$ a vector space?

Given a divisor $D$ on a smooth curve, one can define the sheaf $\mathcal O(D)$ by the prescription $\Gamma(U,\mathcal O(D) :=$ $\{$meromorphic functions on $U$ that satisfy $(f) + D \ge 0\}$. Then, ...
4
votes
1answer
206 views

Global sections of a tensor product of vector bundles on a smooth manifold

This question is similar to Conditions such that taking global sections of line bundles commutes with tensor product? and Tensor product of invertible sheaves except that I am concerned with ...
4
votes
0answers
61 views

Cohomology of $\mathcal O(k)$

I am reading a paper in which it is claimed that $H^1(\mathcal O(-k),\mathcal O)=0$, where $k\geqslant 1$. Moreover, the argument also requires that $H^2(\mathcal O(-k),\mathcal O)=0$. Here ...
4
votes
1answer
97 views

Identifying a line bundle on $\mathbb{P}^1$

I have a geometric line bundle $L$ on $\mathbb{P}^1 = \{[x_0:x_1]\}$. With respect to the standard affine cover $U_0 = \{x_0 \neq 0\}$ and $U_1 = \{x_1 \neq 0\}$, I have the transition function ...
3
votes
0answers
109 views

A question of extension of vector bundles.

Fix $p \in \mathbb{P}^1$. Let $X=\mathbb{P}^1\times \mathbb{P}^1$, $C_1=\mathbb{P}^1\times \{p\}$ and $C_2=\{p\}\times \mathbb{P}^1$. Since $\mathrm{Ext}^1(\mathcal{O}_{C_2},\mathcal{O}_{C_1})\cong ...
8
votes
1answer
201 views

When is the pushforward / direct image of a reflexive sheaf locally free?

I have seen a number of theorems that guarantee the direct image of a reflexive sheaf to be reflexive again, or for the direct image of a locally-free sheaf to be locally free again. This makes me ...
4
votes
2answers
404 views

Exterior power of a tensor product

Given 2 vector bundles $E$ and $F$ of ranks $r_1, r_2$, we can define $k$'th exterior power $\wedge^k (E \otimes F)$. Is there some simple way to decompose this into tensor products of various ...