# Tagged Questions

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space. (Def: http://en.m.wikipedia.org/wiki/Coherent_sheaf)

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### Sheafifying direct sum of twists

Let be $X\subseteq \mathbf{P}^r$ a smooth projective variety and let be $\mathscr E$ an invertible sheaf over $X$. Let $$M=\bigoplus_{n\geq 0} H^0(\mathscr E(n))$$ as a module over the polynomial ...
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### Do there exist torsion sheaf over moduli spaces?

Usually people bother with studying moduli spaces of (coherent) torsion free sheaves that live on a topological space $X$. These spaces, actually stacks, are badly behaved topological spaces. Still, ...
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### If a divisor $D$ satisfies that $D^{2}=1$, is it true that the morphism induced by $|D|$ is birational?

Let $X\subset \mathbb{P}^{5}$ be a non-degenerate algebraic surface. Let us suppose that $D\subset X$ is a curve such that $D^{2}=1$. I would like to know if the rational map induced by the complete ...
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### Section of a coherent sheaf vanishing outside a point

I need some help in understanding an argument, probably basic, about coherent sheaves, which I've read in a paper, and as far as I understand can be described as follows: Let $\mathcal F$ be a ...
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### Is Coherent sheaf acyclic?

I am no expert in sheaf theory so the following question may be trivial. Let $X$ a complex manifold, and let $\mathcal{F}$ a coherent sheaf on $X$. Is $\mathcal{F}$ acyclic? If not: can you give a ...
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### Global sections of a proper ideal sheaf are 0

Let $X$ be a projective variety and $\mathcal{I} \subset \mathcal{O}_X$ is an ideal sheaf on $X$ not equal to $\mathcal{O}_X$. I'm supposed to show $\Gamma(X,\mathcal{I}) = 0$. My first thought was ...
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### Section of a coherent algebraic sheaf being zero over a principal open

I have a question on Proposition 6 of §43 of Serre's Coherent Algebraic Sheaves (page 51-52 in the link). The proposition is stating that, given $X$ any irreducible algebraic variety, $Q$ a regular ...
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### Coherent algebraic sheaf on a closed subvariety

I am reading Serre's Algebraic Coherent Sheaves. I can't see why it holds the remark at the end of chapter 39 (page 48 in the link): "Let $\mathcal{G}$ be a coherent algebraic sheaf on V which is ...
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### Definition of the degree of a sheaf

Let $C$ and $D$ be smooth closed curves on a projective smooth surface $X$ of finite type over an algebraically closed field. I'm looking for a definition of the term $\deg_C(\mathcal{O}(D)_{|C})$. ...
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### Is surjectivity preserved in open neighborhoods?

Let $X,S$ be schemes of finite type over a field and let $f:X\times S\to S$ be the projection. Suppose we have a morphism of coherent sheaves $\phi:\mathscr E\to \mathscr F$ on $X\times S$. Is it ...
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### Torsion coherent sheaf on a curve has finite support

I would like to show that a torsion, coherent sheaf $\mathcal{F}$ on a regular integral curve $C$ is supported at a finite number of closed points. This is from Ravi Vakil's notes, namely part 13.7.G. ...
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