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 $$M=\bigoplus_{n\geq 0} H^0(\mathscr L(n))$$ as a module over the polynomial ...
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24 views

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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68 views

Coherent sheaves with no cohomology over a hypersurface

Let $X_d \subset \mathbb{P}^{n+1}$ be a smooth hypersurface of degree $d$. How one can describe all coherent sheaves on $X_d$ with no cohomology i.e. $$ H^i(X_d, F) \cong 0, $$ for all $i \in \mathbb{...
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85 views

Cohomologies with line bundle vs. coherent coefficients

I recently learned in a lecture that the derived category of a smooth variety is generated/spanned by (complexes of) locally free sheaves. (Unfortunately I haven't been able to find a more precise ...
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71 views

Pushforward of sheaves on the blowup of $\mathbb A^2$ to $\mathbb P^1$

In http://arxiv.org/abs/1210.2564 Example 4.12 it is written that for $Y$ the blowup of $\mathbb A^2$ at the origin (i.e. $Y \cong \mathrm{Tot} \, \mathcal O_{\mathbb P^1} (-1)$), and $π \colon Y \to \...
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59 views

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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85 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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56 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 ...
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68 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:$H_T \rightarrow F$]$...
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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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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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50 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 ...
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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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51 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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32 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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70 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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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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42 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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86 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 $p_*\mathcal{O}...
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40 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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102 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 b}...
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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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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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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 null-...
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50 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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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 ...
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126 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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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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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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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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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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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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23 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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62 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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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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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. $H^i(\...
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318 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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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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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 $F_1,....
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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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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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53 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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141 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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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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136 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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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 $U_{\...
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102 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 ...