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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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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1answer
81 views

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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29 views

Tilting object and mutation in Coh $(X)$

I'm studying the article "The cluster category of a canonical algebra" of Barot, Kussin, and Lenzing. I would like to understand an argument about mutation. I wrote the definition below: Let $T = ...
0
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1answer
54 views

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 ...
2
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63 views

if the Hilbert polynomial of coherent sheaf is constant number,then locally free?

I'm reading "The Geometry of Moduli Spaces of Sheaves" (Huybrechts Lehn). I want to prove [$Quot_{X/S}(H,l)=Grass_S(H,l)$]where X=S and l is a number. Let T be a S-scheme and [ρ:$H_T \rightarrow ...
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32 views

Checking a global property over the closed point of a local ring

Let $k$ be a field and let $T=\textrm{Spec }R$ be the spectrum of a local $k$-algebra $(R,\mathfrak m)$. Let $X\to T$ be a proper flat map from a $k$-scheme $X$, and suppose given a morphism of ...
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2answers
74 views

Commutation of pushforward and pullback along cartesian squares

I am sure this is well-known, but I cannot find a reference. Consider a cartesian square $$\require{AMScd} \begin{CD} P @>{v}>> X\\ @V{g}VV @VV{f}V \\ Z @>{u}>> Y \end{CD}$$ where ...
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1answer
49 views

Conditions for the map of sheaves $F\to F\otimes_{\mathcal O_Y}f_\ast \mathcal O_X$ to be an isomorphism

Let $f:X\to Y$ be an immersion of locally noetherian separated schemes, for instance (not necessarily reduced) varieties over a field $k$. We can us assume $Y$ is affine. Let $F$ be a quasi-coherent ...
2
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1answer
104 views

Support and stalks at generic points

Let $X$ be an noetherian scheme, $Y$ an irreducible closed subscheme of $X$ with generic point $y$ and $\mathscr G$ a coherent sheaf of $\mathscr O_X$-modules. Consider the following statement: If ...
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42 views

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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33 views

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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1answer
30 views

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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1answer
67 views

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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1answer
77 views

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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1answer
41 views

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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80 views

Higher direct images along the blowup

Let $S$ be a smooth projective surface and $p:X\to S\times S$ be a blowup along the diagonal with the exceptional divisor $E$. How to compute $Rp_*\mathcal{O}_X(-2E)$? Is it true that ...
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38 views

Is the dual of a flat module flat

Let $k$ be an algebraically closed field, $T$ be a integral, regular, projective $k$-scheme and $X$ another projective, integral $k$-scheme. Let $\mathcal{F}$ be a coherent (pure) sheaf on $X \times_k ...
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99 views

Coherent sheaves on $\mathbb{P}^1$

Let $F$ be a coherent sheaf on $\mathbb{P}^1$. How to show that there exists a unique exact sequence of the form $$0\to\mathcal{O}_{\mathbb{P}^1}(-1)^{\oplus a}\to\mathcal{O}_{\mathbb{P}^1}^{\oplus ...
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371 views

Intersection of twisted cubics in $\mathbb{P}^3$

Suppose we have two twisted cubics $C_1$, $C_2$ in $\mathbb{P}^3$ such that both of them lie in some cubic surface, which means that $h^0(\mathbb{P}^3, I_{C_1\cup C_2}(3))>0$. I want to show that ...
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30 views

Cohomologies of certain vector bundle on $\mathbb{P}^3$

Consider the collection of $m$ pairwise disjoint lines $L_1,\ldots,L_m$ in $\mathbb{P}^3$ and pose $Z=L_1\sqcup\cdots\sqcup L_m$. Consider the rank-$2$ vector bundle on $\mathbb{P}^3$ which is given ...
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138 views

Null-correlation and Tango bundles on $\mathbb{P}^3$

Let $V$ be a four-dimensional complex vector space and $\mathbb{P}^3=\mathbb{P}(V)$. There are two interesting bundles $N$ and $T$ on $\mathbb{P}^3$, both of rank 2, called respectively a ...
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49 views

The normal bundle of conic

Let $C\subset\mathbb{P}^2\subset\mathbb{P}^n$ be a smooth conic (everything is over the field $\mathbb{C}$). I want to compute $T_{\mathbb{P}^n|C}$ and $N_{C/\mathbb{P}^n}$. Let $z_0,z_1,...,z_n$ be ...
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1answer
75 views

$\Gamma(X,-)$ for Quasi-Coherent/Coherent Sheaves Maps to R-modules/Finitely Generated R modules

I'm trying to show that the functor $\Gamma(X,-)$ from the category of quasi-coherent sheaves maps a quasi-coherent sheaf to an R-Module, and also that for coherent sheaves the same functor takes the ...
4
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1answer
118 views

Vector bundle as locally free coherent sheaves

I am studying coherent sheaves and was looking for a geometric motivation. Hence, in wikipedia and although here is stated that it can be seen as a generalization of vector bundles, which is quite ...
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158 views

Normal bundle of twisted cubic.

Let $C$ be a twisted cubic in $\mathbb P^3$. I'd like to compute the splitting type of normal bundle $N_{C/\mathbb P^3}$? I understood that $T_{\mathbb P^3}|_C=\mathcal O(4)^{\oplus 3}.$ So we have an ...
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52 views

Some propreties about $ \mathfrak{Coh}_X $ and $ \mathfrak{QCoh}_X $.

I would like to know : why is the category $ \mathfrak{Coh}_X $ of coherent scheaves the smallest abelian category containing line bundles ? Why is the category $ \mathfrak{QCoh}_X $ of quasi ...
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42 views

Which curves have reflexive structure sheaf?

Let $C$ be a curve, i.e. a purely one-dimensional scheme, embedded in a smooth projective threefold $X$. For a coherent sheaf $E$ of codimension $c$ on $X$, let $E^D=\mathscr Ext_X^c(E,\omega_X)$ be ...
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28 views

Is the tensor product of a flasque sheaf with a locally free sheaf necessarily flasque?

Is the tensor product of a flasque sheaf with a locally free sheaf necessarily flasque? I found myself asking this while working an exercise in Hartshorne. I suspect the answer is 'No' in general, ...
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35 views

How can I understand instantons as sheaves?

In specific, instantons are considered torsion free coherent sheaves. Why is that the case? Is there a nice way to understand this relation and of course also understand how the two moduli spaces ...
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53 views

Reference Request: Quasi-Coherent Sheaves, Picard Group, etc.

I'm a rookie in algebraic geometry, trying to learn. Recently I noticed that I really don't understand the following topics very well. I am asking you if there is comprehensive references (one maybe ...
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1answer
21 views

Annihilator ideal of image sheaf

Let $(f,\tilde{f}): (X, \mathscr{O}_X) \rightarrow (Y, \mathscr{O}_Y)$ be a holomorphic map between complex spaces, such that $f_*(\mathscr{O}_X)$ is $\mathscr{O}_Y$-coherent. Define $\mathscr{I} = ...
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Geometric intuition for coherent rings, modules, and sheaves

Throughout, all rings are commutative. Definition 1. A ring $R$ is coherent if the solutions $\mathbf x=(x_1,\dots,x_n)$ to a linear equation $\mathbf{rx}=0$ are a finitely generated $R$-submodule of ...
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1answer
61 views

Extensions of sheaves with isomorphic middle terms

Let $\mathcal{F}$ and $\mathcal{G}$ be two coherent sheaves on a variety $X/k$. If I know that $\dim_k \operatorname{Ext}^1(\mathcal{F}, \mathcal{G})=1$, and I have two different nontrivial (not ...
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48 views

Direct limit of rings and coherent sheaves

Let $R$ be a discrete valuation ring, $\{B_i\}_{i \in I}$ is an inductive system of finitely generated $R$-algebras and $B$ the direct limit of the inductive system. Let $X$ be a projective scheme, ...
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72 views

Cohomological criterion for being a vector bundle

The Theorem of Horrocks sais that a locally free sheaf $\mathcal{F}$ on $\mathbb{P}^n$ splits into a direct sum of line bundles if and only if all the intermediate cohomologies vanish, i.e. ...
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296 views

Why did Serre choose coherent sheaves?

First thing - I don't know any algebraic geometry. I'm trying to understand a little bit about quasi-coherent sheaves but not for the sake of AG, so please rely on as little knowledge as possible. ...
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106 views

Quasicoherent sheaves as smallest abelian category containing locally free sheaves

On page 362 of Ravi Vakil's notes, the author says "It turns out that the main obstruction to vector bundles to be an abelian category is the failure of cokernels of maps of locally free sheaves - ...
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33 views

subsheaf of free sheaves

Let $X$ be an irreducible nodal curve, $E:=\oplus_{i=1}^r \mathcal{O}_X$ be a free sheaf on $X$ and $F \subset E$ a (coherent) subsheaf. Is it possible to write $F$ as a direct sum of subsheaves ...
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54 views

Riemann-Roch for nodal curves

Let $X$ be an irreducible, nodal curve and $E$ a coherent subsheaf of a free sheaf $\oplus_{i=1}^r \mathcal{O}_X$ on $X$ of rank strictly less than $r$. Assume that $r \ge 2$. It follows that $H^0(E)$ ...
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0answers
98 views

K-theory of projective space

Is there any way to prove that the twisting sheaves $\mathcal{O}(K)$ generate the algebraic K-theory of projective space without actually using any K-theory machinery (e.g. Bott periodicity)? Like for ...
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1answer
51 views

A question on very ample line bundle and closed immersion

Let $X$ be a projective scheme, $i:X \hookrightarrow \mathbb{P}^n$ a closed immersion, $\mathcal{L}:= i^*\mathcal{O}_{\mathbb{P}^n}(1)$ a very ample line bundle. Let $j:{\mathbb{P}^n} \hookrightarrow ...
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1answer
130 views

Additivity of the first Chern class

I have a highly elementary question: is the first Chern class additive? More specifically, given a short exact sequence of coherent sheaves on a nonsingular curve $X$ $$ 0 \to \mathscr F'\to \mathscr ...
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32 views

Locally free sheaves on reducible curves and their subsheaves

Let $X$ be a reducible (but reduced), connected, projective curve with at worst nodal singularities and $\mathcal{F}$ be a locally free sheaf of rank $r$ on $X$. Suppose that $\mathcal{F}'$ is a depth ...
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1answer
120 views

Globally generated vector bundle

Could someone give me a definition of globally generated vector bundle? A rapid search gives me the definition of globally generated sheaves, but I am in the middle of a long work and don't really ...
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1answer
66 views

Computing the sheaf of 1-forms on a toric variety

Consider projective space $P^{2}$ and its corresponding fan. We have the affine opens defined by $U_{\sigma_{0}} = Spec(\mathbb{C}[x,y])$, $U_{\sigma_{1}} = Spec(\mathbb{C}[x^{-1},x^{-1}y])$ and ...
0
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1answer
92 views

Negative degree invertible sheaves on non-singular varieties have no global sections

Let $X$ be a non-singular projective complex variety and $\mathcal{L}$ be an invertible sheaf on $X$ with negative degree. Is it true that $\mathcal{L}$ has no global sections? If so, can someone ...
3
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1answer
95 views

Are trivial vector bundles on curves semistable?

Let $C$ be an irreducible projective curve with at worst nodal singularities. Let $E$ be the trivial locally free sheaf of rank $r$ i.e., $E$ is the direct sum of $r$ copies of the trivial line ...
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1answer
43 views

If $F$ a sheaf and $S\subset F$ a subfunctor, then $S$ is a subsheaf if and only if…

This is Proposition 1 from Maclane & Moerdijk's Sheaves in Geometry and Logic, part II, section 1. Proposition 1. Let $F$ be a sheaf on $X$ and $S\subset F$ a subfunctor. $S$ is a subsheaf if ...
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82 views

Duality and Serre's criterion

Let $X$ be a projective scheme, $\mathcal{F}$ a coherent sheaf on $X$ which is $S_2$. Then under what additional conditions is its dual, $\mathcal{H}om_X(\mathcal{F},\mathcal{O}_X)$ also $S_2$?
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30 views

Reflexive sheaves on stable curves-II

This is an extension of Reflexive sheaves on stable curves. Let $C$ be a stable curve and $\mathcal{F}$ a reflexive sheaf on $C$ supported on the whole of $C$. Is the projective dimension of ...