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57 views

Segre embedding and Hilbert polynomial of coherent sheaves

Let $X \subset \mathbb{P}^n$ and $Y \subset \mathbb{P}^m$ be smooth, projective subvarieties, $F$ and $G$ coherent, torsion-free, sheaves on $X$ and $Y$ with Hilbert polynomials $P_{F}$ and $P_G$, ...
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71 views

Semicontinuity of cohomology of torsion-free sheaves restricted to divisors

Let $X$ be a smooth projective variety, $\mathcal{E}$ a torsion-free coherent sheaf on $X$ and $\mathfrak{d}$ a linear system of divisors in $X$. I would like to show (at least when $X$ is a surface) ...
3
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1answer
147 views

A non-rational variety with a full exceptional collection?

Does there exist a smooth non-rational projective variety whose bounded derived category of coherent sheaves admits a full exceptional collection? I could not find any examples in the literature (for ...
4
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0answers
112 views

Generators of unbounded derived categories of (quasi-)coherent sheaves

An object $T$ in a triangulated category $\mathcal{D}$ is called a generator if $T^\perp=0$, which means that for any nonzero $X$ in $\mathcal{D}$, there are $i\in\mathbb{Z}$ and a nonzero morphism $T[...
4
votes
1answer
117 views

Gluing finitely presented quasi coherent sheaves

Let $X$ be a quasi-compact, separated scheme, and $\{\text{Spec}(A_i)\subset X\}_{i=1,\ldots, n}$ a finite affine open cover. Suppose a quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ is such ...
4
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1answer
497 views

Proper mapping theorem

Let $Z\to X$ be a closed immersion of schemes. Assume $\mathcal{O}_Z$ and $\mathcal{O}_X$ both are coherent sheaves of $\mathcal{O}_Z$, resp. $\mathcal{O}_X$-modules. In particular, the coherent ...
4
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0answers
189 views

Question about Corollary II.5.18 in Hartshorne

Corollary II.5.18 in Hartshorne says that if X is a projective scheme over a Noetherian ring, then any coherent sheaf on X is a quotient of a finite direct sum of twisted structure sheaves. The ...
2
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0answers
153 views

Singularities of reflexive sheaves

I am studying reflexive sheaves (on $\mathbb{P}^3$) by the Hartshorne's paper ''Stable reflexive sheaves''. As far I understood, reflexive sheaves fail to be locally free at a finite number of points (...
6
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1answer
144 views

Question on condition for a sheaf to be locally free in Orlov 2004

In "Triangulated Categories of Singularities and D-Branes in Landau-Ginzburg Models", Orlov twice mentions the following criterion for a sheaf $P_1$ to be locally free: If for all closed points $t:x ...
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0answers
108 views

Loci in the algebraic stack of coherent sheaves

Let $S$ be a scheme, $X$ and $V$ a algebraic spaces over $S$ with $X\to V$ separated, finitely presented and flat. Then $\mathcal{Coh}_{X/V}$ is an algebraic stack. Let us now assume $S = V$ is a ...
3
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0answers
124 views

Is the relative moduli space of semi-stable sheaves on families of curves fine

Let $\pi:X \to B$ be a family of smooth, projective curves. Fix coprime integers $r,d$. Denote by $\mathcal{M}(r,d)$ the relative moduli functor corresponding to rank $r$, degree $d$, semi-stable ...
5
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1answer
347 views

Pushforward maps for cohomology of coherent sheaves

Let $X$ be a smooth projective algebraic variety over a field $k$, of dimension $n$, and let $Z$ be a smooth closed subvariety of dimension $m$, with $i: Z \hookrightarrow X$ the inclusion map. For ...
3
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0answers
143 views

Obstruction to lifting coherent sheaves on discrete valuation ring

Let $R$ be a discrete valuation ring with algebraically closed residue field $k$. Let $K:=\mathrm{Frac}(R)$ the fraction field of $R$. Suppose $K$ is of characteristic zero. Denote by $\overline{K}$ ...
1
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1answer
177 views

Pushforward of coherent sheaves and field extensions

This is a continuation of the discussion in the mathoverflow, Pushforward of semi-stable sheaves. Let $X$ be a smooth projective variety over a field $k$ and $L$ be a finite field extension of $k$. ...
1
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1answer
162 views

Isomorphism of sheaves in families of projective varieties

Let $\pi:\mathcal{X} \to S$ be a flat, family of projective varieties (here $\mathcal{X}$ and $S$ are noetherian). Let $E$ and $F$ be two locally free sheaves on $\mathcal{X}$ such that for all $s \in ...
3
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0answers
158 views

Examples of varieties with every stable sheaf simple

Are there examples of projective varieties over a non-algebraically closed field such that every geometrically stable sheaf on the variety is simple? I see, for example in Huybrechts-Lehn and in some ...
2
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0answers
129 views

Base change, descent theory and coherent sheaves

Let $k$ be a field of characteristic zero and $X$ a smooth, projective $k$-variety. Let $E_{\overline{k}}$ be a coherent sheaf on $X_{\overline{k}}$ ($\overline{k}$ denotes the algebraic closure of $k$...
1
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1answer
162 views

Pushforward of semi-stable sheaves under finite field extension

Let $k$ be a field of characteristic zero and $X$ be a non-singular rationally connected variety over $k$. Let $L$ be a finite field extension of $k$. This induces a proper morphism $p:X_L \to X_k$. ...
5
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0answers
203 views

Torsion-free sheaves over nodal curves

Let $X$ be an irreducible nodal curve (over $\mathbb{C}$) with exactly one node, say at $x$. Let $F$ be a torsion-free, rank $n$ sheaf on $X$. We know that $F_x \cong \mathcal{O}_{X,x}^{\oplus n-a} \...
2
votes
1answer
346 views

Galois descent for absolute Galois group

Let $K$ be a field of characteristic zero, $\bar{K}$ its algebraic closure and $X$ a smooth, projective $K$-scheme. We know the Galois descent theory for quasi-coherent sheaves defined on $X_L$ for a ...
1
vote
2answers
158 views

Is relative torsion freeness an open condition?

Let $S$ be an integral scheme and $X \to S$ be a smooth scheme of finite type over $S$. Let $\mathcal{E}$ be a coherent sheaf on $X$, and $\eta$ be the generic point of $S$. Assume that restriction $\...
3
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0answers
124 views

de Rham isomorphism with holomorphic forms

For a non-compact Riemann surface $X$ there is an isomorphism: $$\Omega(X)/\mathrm d \mathcal O(X)\simeq H^1(X,\mathbb C)$$ where $\Omega$ is the sheaf of holomorphic forms on $X$. The group on the ...
4
votes
2answers
497 views

Different definition of sheaf cohomology

It could be related to my previous question here. Let $\mathcal F$ be a sheaf on a topological space $X$. Hartshorne in his book on Algebraic geometry defines the sheaf cohomology by $$ H^i(X, \...
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0answers
84 views

Stable restrictions of sheaves

Let $X$ be a projective variety and $Y$ a subvariety. If $E$ is a stable sheaf on $X$, then under certain circumstances (e.g. the theorems of Flenner, Mehta-Ramanathan, Bogomolov) the restriction $E|...
4
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0answers
198 views

When is a coherent subsheaf determined by its global sections

I am reading an article in which a proof is based on defining a subsheaf by only giving its global sections. The exact setting is that, one has a surjective finite morphism $f:Y\to X$ between ...
3
votes
1answer
370 views

Vanishing of some Ext groups of coherent sheaves

We call a coherent sheaf 'of pure support' if it has no subsheaves with support of smaller dimension. Now, let $X$ be a smooth projective variety, $F$ and $G$ coherent sheaves of pure support on $X$....
13
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0answers
320 views

Why do people study unbounded derived category of quasi-coherent sheaves rather than focus on bounded derived category of coherent sheaves?

Let $X$ be a scheme and let $D_{qoch}(X)$ and $D^b_{coh}(X)$ be the unbounded derived category of quasi-coherent sheaves and bounded derived category of coherent sheaves on $X$, respectively. $D^b_{...
2
votes
0answers
61 views

Minimum number of generators for a gloablly generated sheaf over a curve

Let $X$ be a smooth, projective curve over an algebraically closed field and $E$ be a globally generated locally free sheaf of rank $r$. Is it always possible to write $E$ as the quotient of $r+1$ ...
1
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0answers
101 views

Is sheaf stability an open condition?

Let $X$ be a smooth projective variety. If $E$ is a coherent sheaf on $X$, we write its Hilbert polynomial: $$P_E(m) = \alpha^E_dm^d + O(m^{d-1}).$$ We say $E$ is Gieseker stable if $E$ is pure and $...
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0answers
145 views

Is quotient by maximal destabilizing sheaf, torsion-free?

Let $k$ be an infinite field (not necessarily algebraically closed), $X$ a smooth, projective curve over $k$ and $F$ a pure, coherent sheaf on $X$. Let $F'$ be the maximal destabilizing sheaf of $F$. ...
1
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0answers
146 views

Classifying length $4$ modules over $\mathbb C[x,y]$

I am trying to classify all modules of length $4$ over the ring $A=\mathbb C[x,y]$, supported at the origin $0\in \mathbb C^2$, up to ($A$-linear) isomorphism. Let $\mathfrak m=(x,y)$ be the ideal of ...
2
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0answers
77 views

Singularities of $Spec(Sym^* E^{\vee})$ for $E$ a coherent sheaf

Let $X$ be a smooth complex algebraic variety, and $\mathscr{E}$ a torsion-free coherent sheaf on $X$. Which type of singularities can the total space $\mathrm{Tot}(\mathscr{E}):=\underline{\mathrm{...
0
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0answers
104 views

Fiberwise injective resolution of coherent sheaf

Let $k$ be an algebraically closed field (of characteristic zero) and $X, Y$ be projective $k$-varieties. Let $F$ be a coherent sheaf on $X \times_k Y$, flat over $Y$. Does there exists a coherent $\...
2
votes
1answer
167 views

coherent ring whose nilradical is not finitely generated

Let $A$ be a commutative ring with $1$.We say that $A$ is coherent if and only if every finitely generated ideal of $A$ is finitely presented. Does there exist a coherent ring such that nil-radical ...
5
votes
1answer
170 views

Is the dual of a compact generator also a compact generator of the derived category of a variety?

Let $X$ be a variety (or more generally a quasi-compact, separated scheme) and $D(X)$ be the derived category of complexes of $\mathcal{O}_X$-modules with quasi-coherent cohomologies. Let $\mathcal{E}$...
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0answers
264 views

A criterion for purity

I have started reading the book "The Geometry of Moduli Spaces of Sheaves" by Huybrechts and Lehn. This is a statement in this book at page no.3 the last line. "$E$ is pure if and only if all ...
4
votes
1answer
184 views

Zero locus of a family of morphisms of vector bundles

Let $X$ be a variety and $\varphi : F_1 \to F_2$ be a morphism of vector bundles over $X$. Then it is easy to check that the locus on $X$ for which $\varphi$ vanishes is a closed subscheme of $X$. ...
14
votes
1answer
326 views

Which category of sheaves on a manifold remembers the manifold?

Given a not too nasty topological space $X$, the category of sheaves of sets on $X$ remembers $X$. Given a scheme $S$, the category of quasicoherent sheaves on $S$ remembers $S$. Given a smooth ...
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0answers
169 views

What is the support of a coherent sheaf on $X\times Y$ if it is invariant by tensoring a very ample line bundle on $X$?

Let $X$ be a smooth projective variety over a field $k$ with char$k=0$ and $\mathcal{L}$ be a very ample line bundle on $X$. Let $\mathcal{F}$ be a coherent sheaf on $X$. It is well-know that if $\...
18
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1answer
943 views

When do real analytic functions form a coherent sheaf?

It is known that, in general, the sheaf of real analytic functions on a real analytic manifold is not coherent. However, there are some examples, where we have coherence: for example, if $X$ is a ...
7
votes
1answer
525 views

Reference request: tangent space to moduli space of coherent sheaves is $\operatorname{Ext}^1(E, E)$

Is there a standard reference for the fact that, in an appropriate algebraic-geometrical context, the tangent space at the point $[E]$ to the moduli space $\mathcal M$ is something like $\operatorname{...
1
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1answer
146 views

Extending locally free sheaves and compatibility with fibers

Let $X$ be a smooth, projective variety over an algebraically closed field $k$ (of characteristic zero), $B$ a connected, noetherian scheme (possibly non-reduced) and $U$ an open subscheme of $X \...
5
votes
1answer
223 views

When is a sheaf coherent if its image under a Fourier-Mukai transform is coherent?

Let X and Y be to varieties and $F\colon D\mathrm{QCoh}(X) \to D\mathrm{QCoh}(Y)$ a continuous functor between the corresponding unbounded derived categories of quasi-coherent sheaves (given by a ...
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0answers
131 views

Destabilizing subsheaf: length of $0$ dimensional subscheme

I have a very specific question (quite elementary, sorry!) Let $G$ be a rank $2$ torsion free sheaf on an algebraic surface $X$ (normal maybe?) Let $L\otimes \mathcal{I}_Z$ be a Gieseker ...
2
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0answers
210 views

G-equivariant coherent sheaves and their quotients

Let $X$ be a smooth complex quasiprojective variety and let $G$ be a finite group acting on $X$. The quotient $X/G$ is then a variety. Let $\mathcal{F}$ be a $G$-equivariant coherent sheaf on $X$. ...
2
votes
1answer
220 views

Do finite flat sheaves define families of $0$-cycles?

Let $X$ be a smooth projective $\mathbb C$-variety and let $X^{(n)}$ denote the symmetric product $X^n/S_n$, parametrizing effective $0$-cycles of degree $n$ on $X$. Question. Let $S$ be a ...
2
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0answers
160 views

Double dual of ample sheaf

Let $X$ be a projective manifold. Then we can define ample sheaves on $X$, and many results of ample vector bundles still hold in this more general case (See K. Kubota, Ample sheaves). Now I was ...
5
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1answer
142 views

Is locally freeness of a sheaf (of fixed rank) around a divisor detectable from a first order neighbourhood?

Assume you have a smooth projective variety $X$ over the complex numbers, a smooth prime divisor $D$ on it, and a torsion free coherent sheaf $E$ on $X$ of rank $r>0$. Let $E|_{2D}:=E\otimes_{\...
3
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1answer
209 views

Torsion free sheaves in flat families

Let $R$ be a dvr, $X$ a flat, projective, integral, normal $R$-scheme such every closed fiber is again integral, normal. Let $F$ be a torsion-free coherent sheaf on $X$, flat over $R$. Is it true that ...
2
votes
1answer
106 views

On a morphism between reflexive sheaves

Let $X$ be a normal, projective variety and $U$ be the regular locus of $X$. Let $\mathcal{F},\mathcal{G}$ be reflexive sheaves on $X$ and $f:\mathcal{F} \to \mathcal{G}$ be a morphism. Suppose that ...