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1
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1answer
49 views

Question on the definition of sheaves.

When defining a sheaf of $O_X $-modules, or sheaves in general, I have nearly always seen it given as a functor from the category of all the open sets to another category with the usual properties. ...
3
votes
1answer
51 views

Base-points and invertible sheaves

Once again I am confused after thinking too much about something I thought I already understood... Let $\mathcal{L}$ be an invertible sheaf on a smooth projective curve $X$ such that $\deg ...
3
votes
1answer
75 views

Torsion sheaves on a curve

This is probably a silly question, but I'm a bit confused. Regarding exercises 6.11 and 6.12 of Chapter II of Hartshorne: Let $X$ be a nonsingular projective curve over an algebraically closed field ...
0
votes
0answers
29 views

Book Suggestion - Complex algebraic surfaces

I am studying for an exam of algebraic geometry, in particular, I am dealing with ruled surfaces and numerical invariants, rational surfaces, Castelnuovo's Theorem and its application. I am reading ...
2
votes
1answer
41 views

Very ampleness of pullbacks of twisting sheaves

Suppose we are given $X =\mathbb{P}^1_k\times_{k}\mathbb{P}^1_k$ with its canonical projections $\pi_i$ to $\mathbb{P}^1_k$. I'd like to show that ...
1
vote
1answer
34 views

Vanishing of local cohomology groups

Let $k$ be a field and let $X$ be a smooth separated $k$-variety. Let $T$ be a closed integral subscheme of $X$ of generic point $\eta$. The object of interest here is the local cohomology group $$ ...
3
votes
1answer
60 views

Pullback of an ample line bundle through a projection

Assume $X_1$ and $X_2$ are two smooth projective curves and let $M_i$ be an ample line bundle on $X_i$, for $i=1,2$. Further, denote the natural projection map on the $i$-th factor by ...
2
votes
1answer
71 views

When is the morphism between global sections surjective

Let $C$ be a smooth projective irreducible curve. Let $Z$ be a closed subscheme of $C$ consisting of a finite set of points. Denote by $i:Z \to C$ the closed immersion. Note that this is a proper ...
4
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0answers
42 views

Extension of morphism of Coherent sheaves over the projective space

Let $\mathcal{F}_1, \mathcal{F}_2$ be coherent sheaves over $\mathbb{P}^n_{\mathbb{C}}$ for $n \ge 3$. Denote by $U_i$ the fundamental affine schemes defined by the non-vanishing of the coordinates ...
2
votes
2answers
113 views

Source: Coherent locally free sheaves and projective modules

What is a good and very quick and concise article for the proof of the equivalence of the categories of locally free sheaves on $\mathrm{Spec}(A)$ and finitely generated projective $A$-modules?
1
vote
1answer
54 views

Associated sheaf to a $\mathbb{C}$-module

In section II.5 Hartshorne describes the notion of a sheaf $\tilde{M}$ associated to a module M over a ring A. Now in the case $A:= \mathbb{C}$, we have $\operatorname{Spec}(\mathbb{C}) = {0}$. ...
1
vote
0answers
62 views

$\operatorname{Hom}(f^*F, G) = \operatorname{Hom}(F, f_*G)$

For topological spaces $X, Y$ and the continuous morphism $f \colon Y \to X$, consider the sheaf $F$ on $X$ and $G$ on $Y$. A very famous formula says that ${\mathrm{Hom}_{\mathrm{sh}}}(f^*F,G) = ...
1
vote
1answer
127 views

Reflexive sheaves and determinants

Let $X$ be a complex manifold. I just recall some notions, for my question below to be clear. Let $F\in\textrm{Coh}(X)$ be coherent of rank $r$. If it is torsionfree, it has a well-defined ...
2
votes
1answer
143 views

Pushforward commutes with external tensor product?

Let $f:X\rightarrow X'$ and $g:Y\rightarrow Y'$ be morphisms of varieties. Let $\mathcal F$ be a coherent sheaf on $X$ and $\mathcal G$ be a coherent sheaf on $Y$. Is it true that $$(f\times g)_* ...
3
votes
0answers
44 views

When does one have $f_\ast(Im(a))=Im(f_\ast a)$?

Let $f:X\to Y$ be a morphism of schemes, and $a:F\to G$ a morphism of coherent sheaves on $X$, with image $Im(a)=E\subseteq G$. I am trying to understand under which conditions one has ...
3
votes
0answers
53 views

Coherent sheaves on $\mathbb P^r$ of very high and very low dimension

Let $\mathcal F$ be a coherent sheaf on $\mathbb P^r$, the $r$-dimensional projective space over an algebraically closed field $k$. The support of $\mathcal F$, namely $$\textrm{Supp }\mathcal ...
2
votes
0answers
43 views

Questions about the details in the construction of virtual fundamental class

Let $\pi :D \subset \mathcal{X} \to S$ be a flat family of stable curves of genus $g$ with marked points $D$. Let $\mathcal{X} \to X$ be a flat family of stable morphism in the sense of Kontsevich ...
7
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0answers
169 views

Reality check: $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(1)$ and $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(-1)$ are both Möbius strips?

This question follows up this previous question, which has an accepted answer that I am having trouble believing. (Edit: the answer has now been fixed; thanks Georges!) I have been working on giving ...
2
votes
0answers
78 views

Example of a coherent sheaf on an open subset where extension isn't trivial?

Motivation: I am working on problem II.5.15 in Hartshorne's Algebraic Geometry, which is to prove that, given a noetherian scheme $X$, an open subset $U\subset X$, and a coherent sheaf $\mathscr{F}$ ...
2
votes
2answers
215 views

Serre duality for curves, the other statement.

Here's a question from someone who's just found out what Serre duality (in the case of curves) is. It occurs to me that the popular statement which can also be interpreted as the Riemann-Roch theorem ...
1
vote
1answer
154 views

Homological dimension of skyscraper sheaf

Let $X$ be a smooth variety, $x$ a closed point, and $\mathcal{O}_x$ the skyscraper sheaf of the residue field $k(x)$ at $x$. Recall that the homological dimension of a complex of coherent sheaves ...
0
votes
1answer
50 views

Why is any rank $1$ sheaf always stable?

Let $X$ be a projective scheme over $\mathbb{C}$. For a sheaf on $X$, $$ p_E(d)=\chi(X,E(d)) $$ be the Hilbert polynomial of $E$. A sheaf $E$ on $X$ is siad to be stable if for every proper subsheaf ...
1
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0answers
81 views

Characterization of Hilbert schemes of points

Let $X=\operatorname{Proj}(R)$ be a projective scheme with a $k$-algebra $R$. Let $\mathcal{M} \in \operatorname{Coh}(X)$ be a coherent sheaf on $X$ whose Hilbert polynomial $\chi(\mathcal{M})$ is a ...
5
votes
1answer
37 views

Testing local freeness on curves

Let $X$ be a smooth variety (over an algebraically closed field, if it makes a difference), and $\mathscr{F}$ a coherent sheaf on $X$. I have heard it claimed that $\mathscr{F}$ is locally free if and ...
2
votes
2answers
135 views

Evaluation map of sheares $f^{*}(f_{*}\mathcal{F})\rightarrow\mathcal{F}$.

Let $f:X\rightarrow Y$ be a projective morphism of algebraic varieties and $\mathcal{F}$ be a coherent sheaf on $X$. Then some people say that there is a canonical evaluation map $$ ...
2
votes
1answer
149 views

Coherent Sheaves on Projective Space

I am having trouble proving the following claim and would be glad if someone could help me out. Claim: Let $\mathbb P$ denote n-dimensional projective space, and let $F$ be a coherent sheaf on ...
7
votes
2answers
216 views

connections on coherent sheaves

Let $X$ be a smooth variety over $\mathbb{C}$. If $\mathscr{F}$ is a coherent sheaf on $X$ with connection, does it follow that $\mathscr{F}$ is locally free? I can't think of any counterexamples. ...
8
votes
1answer
415 views

is the pushforward of a flat sheaf flat?

Let $f:X \to Y$ be a morphism of schemes and let $F$ be an $\mathcal{O}_X$-module flat over $Y$. Is $f_*F$ flat over $Y$? What's wrong with this argument? [EDIT: as Parsa points out, the (underived) ...
2
votes
0answers
106 views

Fourier-Mukai isomorphism exploiting projection formula

Let $X$, $Y$ be varieties, ${\cal E}\in D^b_{coh}(X\times Y)$. Define the Fourier-Mukai transform with kernel $\cal E$ to be the functor $\Phi({\cal E},-)\colon D^b_{coh}(X)\to D^b_{coh}(Y)$ to be the ...
4
votes
1answer
203 views

ample sheaf which is not very ample

I'm having trouble understanding a remark in Hartshorne: Let $X$ be the nonsingular projective cubic defined by $y^2z = x^3 - xz^2$ and put $P_0 = (0,1,0)$. The claim is that $\mathscr{L}(P_0)$ is not ...
14
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0answers
370 views

Hartshorne exercise II.5.12(b)

I've been working on the Hartshorne exercise in the title for quite a while, which goes like this: let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes, $\mathscr{L}$ a very ample invertible ...
2
votes
1answer
282 views

Higher Ext groups of skyscraper sheaf

I would like to understand the calculation of higher Ext groups of a skyscraper sheaf $\mathcal{O}_p$ at $p$. The calculation I have seen does this using a Koszul resolution. It starts out like this ...
16
votes
1answer
251 views

Formal Schemes Mittag-Leffler

Here is a question that is similar to my last one. I've been trying to learn about Grothendieck's Existence Theorem, but it seems that there aren't very many places that talk about formal schemes and ...
8
votes
1answer
2k views

What local system really is

I know a local system is a locally constant constant sheaf. But why does a local system on the topological space $X$ correspond to $\tilde{X}\times_G V$, where $G$ is the fundamental group of $X$, ...
14
votes
2answers
1k views

Precise connection between Poincare Duality and Serre Duality

The statements of Poincare duality for manifolds and Serre Duality for coherent sheaves on algebraic varieties or analytic spaces look tantalizingly similar. I have heard tangential statements from ...