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0
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20 views

Relative flat vs flat direct image

Let $Y$ be a Noetherian scheme. Let $\mathscr{F}$ be a coherent sheaf on $\mathbb{P}^n \times Y$. Denote $\pi: \mathbb{P}^n \times Y \rightarrow Y $ canonical projection. We have two notions. $\pi_* ...
4
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0answers
42 views

Help understanding the proof of a theorem about Cohomology of Vector Bundles

I am trying to understand a paper called Betti tables of graded modules and cohomology of vector bundles, but i am stuck in Proposition 6.8 which states: Let $\mathcal{E}$ be a vector bundle on ...
2
votes
2answers
61 views

Cohomology groups of coherent sheaves for very small and very big twists.

Let $\mathcal{F}$ be nonzero coherent sheaf over the projective space $\mathbb{P}_k^n$. The Serre vanishing Theorem says that $h^i \mathcal{F}(d)=0$ for $i>0$ and $d\gg 0$. I am wondering if it is ...
2
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0answers
43 views

Question about Tate resolution and cohomology groups of nonzero coherent sheaves.

Let $\mathcal{F}$ be a nonzero coherent sheaf on the projective space $\mathbb{P}_{k}^m$. I am trying to show that for every integer $d$ there is $j$ for which $h^j\mathcal{F}(d-j) \neq 0$. My ...
3
votes
0answers
68 views

Criterion of flatness for projective morphism

Let $f: X \to Y$ be a projective morphism, $\mathcal O(1)$ is a relatively very ample sheaf, then f is flat iff $f_*\mathcal O(m)$ is locally free for big $m$. I can prove flatness implies that ...
1
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0answers
50 views

Coherent sheaf on reduced scheme is free on dense open set

It should be well-known fact, but I couldn't find this in Hartshorne's "Algebraic Geometry", Mumford-Oda or Ravi Vakil's Lecture notes. Let $X$ be a reduced connected scheme and $\mathcal F$ is a ...
2
votes
1answer
30 views

Construct natural transformation $u^* R^i f_* \rightarrow R^i g_* v^*$ without assumption of quasi-coherence

I am reading Hartshorne Algebraic geometry. Chapter 3 Proposition 9.3 (in particular remark 9.3.1). It states that if we have commutative diagram in category of schemes (namely morphisms $f, g, h, u$ ...
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0answers
13 views

Reference request: Order of a vector bundle

Please could you link me to an accessible reference/set of lecture notes on the definition of the order of an algebraic vector bundle? Google just shows some very complicated definition which is ...
1
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0answers
48 views

Exceptional coherent sheaves on $\mathbb{P}^n$ are vector bundles

Let $E$ be a coherent sheaf over $\mathbb{P}^n_k$. Coherent sheaf is called exceptional if $\operatorname{Hom}(E,E) \cong k$, and $\operatorname{Ext}^{> 0}(E,E) \cong 0$. How one can show that such ...
2
votes
2answers
71 views

Stalks of the sheaf $\mathscr{H}om$?

The question is basically the title. What are the stalks of the sheaf $\mathscr{H}om_{\mathscr{O}_X}(\mathscr{F},\mathscr{G})$? If $X$ is a noetherian scheme and $\mathscr{F}$ is coherent, then ...
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0answers
53 views

Comparing two definitions of determinant of coherent sheaves

Let $f:X \to S$ be a smooth, projective morphism of $k$-schemes for some field $k$. Let $\mathcal{F}$ be a coherent sheaf on $X$ flat over $S$. We know (by Proposition $2.1.10$ of Huybrechts-Lehn, ...
1
vote
1answer
37 views

Extension of coherent sheaf from open subspace.

I'm trying to solve following problem: Let $X$ be a noetherian scheme, $U$ an open subspace of $X$, $\mathcal F \in Qcoh(X), \mathcal G\in Coh(U), \mathcal G \subset \mathcal F|_{U},$ then there ...
1
vote
1answer
68 views

Does the canonical morphism commute with direct image functor?

I am trying to prove the representability of the Quotient functor. I have the following problem. Let $\phi \colon T \to S$ be a morphism of noetherian schemes and let $F$ be a coherent sheaf on ...
2
votes
0answers
58 views

Rank, degree and slope of a general coherent sheaf

Let $(X,\mathcal O_X)$ be a ringed space and $\mathcal F$ be a coherent sheaf of $\mathcal O_X$-modules on $(X,\mathcal O_X)$. Are there the definitions of rank, degree and slope of $\mathcal F$ in ...
2
votes
0answers
35 views

G-equivariant invertible sheaves on affine curves

Let $A$ be a Noetherian integral domain, and $G$ a finite group of automorphisms acting on $A$. Let $B = A^G$, the ring of invariants. The inclusion $B \hookrightarrow A$ induces a surjective morphism ...
1
vote
1answer
65 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
68 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
127 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 ...
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0answers
44 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
46 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
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1answer
39 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 $$ ...
4
votes
1answer
100 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
82 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
44 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
159 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
57 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
65 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
206 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
186 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
45 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
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0answers
61 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
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0answers
47 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 ...
9
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0answers
185 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
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0answers
93 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
250 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
175 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
vote
0answers
84 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
48 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
140 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
177 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 ...
8
votes
2answers
237 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. ...
9
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1answer
451 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
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0answers
118 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
211 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 ...
15
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1answer
520 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
326 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
258 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 ...
9
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 $X$, ...
15
votes
2answers
2k 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 ...