The concept of scheme mimics the concept of manifold obtained by glueing pieces isomorphic to open balls, but with different "basic" glueing pieces. Properly, a scheme is a locally ringed space locally isomorphic (in the category of locally ringed spaces) to an affine scheme, spectrum of a ...

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76 views

Is this solution to Hartshorne Exercise III.10.2 correct?

Exercise III.10.2 in Hartshone's Algebraic Geometry is discussed in this question. The accepted answer proceeds by showing that the smooth locus is open on $X$, then transferring this to $Y$ using the ...
1
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0answers
28 views

Are $ \mathbb{A}^n (k ) = k^n $ and $ \mathbb{P}^{n} (k) = \mathrm{Proj} \ k[X_0 , \dots , X_n ]$ irreducibles?

Are $ \mathbb{A} (k ) = k^n $ and $ \mathbb{P}^{n} (k) = \mathrm{Proj} \ k[X_0 , \dots , X_n ]$ irreducibles when $ k $ is a domain ? Thanks in advance for your help.
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2answers
57 views

Sheafyness and relative chinese remainder theorem

The relative chinese remainder theorem says that for any ring $R$ with two ideals $I,J$ we have an iso $R/(I\cap J)\cong R/I\times_{R/(I+J)}R/J$. Let's take $R=\Bbbk [x_1,\dots ,x_n]$ for $\Bbbk $ ...
1
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1answer
21 views

Fiber product with diagonal morphism [duplicate]

Stacks tag 01KR states that the diagram of schemes is "by general category theory" "a fibre product diagram". I tried to show this using the universal property, but didn't obtain anything useful. ...
1
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1answer
37 views

Very ample divisors over non-algebraically complete field

For a projective scheme $V$ over an algebraically-closed field $k$ it is a well-known fact fact that a base-point-free linear system of divisors is very ample iff it separates $k$-points and tangent ...
5
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2answers
106 views

If $A=\Gamma(X,O_X)$ and $Spec(A)=X$ then $Spec(A)=X$ as schemes.

Let $X$ be a scheme and $A$ a commutative Ring. Assume that we have an isomorphism $A \rightarrow \Gamma (X, O_X)$ and the induced map $X \rightarrow Spec(A)$ is a homeomorphism. Question: Is it ...
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2answers
37 views

Why does $\operatorname{Proj}(B)$ not contain ideals containing the irrelevant ideal?

Let $A$ be a ring and let $B = \bigoplus_{d\geq 0} B_d$ be a graded $A$-algebra.In Liu's book Algebraic Geometry and Arithmetic Curves the set $\operatorname{Proj}B$ is defined as the set of all ...
6
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81 views

Vanishing of Tor sheaf on a union of subschemes with vanishing Tor.

Suppose $X$ is a scheme and $Y$ and $Z$ are closed subschemes such that $Y$ is a finite union of closed reduced irreducible subschemes $C$ that satisfy ...
4
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2answers
99 views

What does the case of $\operatorname{Spec}C^{\infty}(M)$ tell us about the relavance of scheme theory to general rings?

I guess a lot of people with previous exposure to differential geometry have had this naive question pop out in their mind when studying schemes for the first time. The category of compact smooth ...
2
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1answer
55 views

Coproduct decomposition of the quot scheme functor

I am reading FGA explained and stuck on this argument: The functor $\mathfrak{Q}uot_{E/X/S}$ naturally decomposes as a co-product $$\mathfrak{Q}uot_{E/X/S}=\coprod_{\Phi\in \mathbb{Q}[\lambda]} ...
2
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1answer
52 views

Hartshorne Exercise III.8.4(c)

Let $Y$ be a noetherian scheme, and let $\mathcal E$ be a locally free $\mathcal O_Y$-module of rank $n+1$, with $n\ge 1$. Let $X=\mathbb P(\mathcal E)$ [the projective bundle over $\mathcal E$], with ...
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0answers
21 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, ...
0
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1answer
68 views

Morphisms of schemes are $\mathcal{O}-$modules

The following might not be true and part of the purpose of this question is to verify my understanding so far. Let me start with a few statements: Since the sheafication of a presheaf only takes ...
2
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0answers
47 views

How should I think of schemes of the form $\operatorname{spec} k[f_1, \ldots, f_n]$, $f_i \in k[x_1, \ldots, x_m]$?

How should I think of schemes of the form $\operatorname{spec} k[f_1, \ldots, f_n]$, $f_i \in k[x_1, \ldots, x_m]$? What of the map on spectra induced by the inclusion $i : k[f_1, \ldots, f_n] \to ...
2
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1answer
32 views

Non-noetherian scheme with infinitely many irreducible components passing through a point? Point still a domain?

Is there a non-noetherian scheme with infinitely many irreducible components passing through a point? (I expect the answer to be yes, but I do not know of an example.) For extra internet points, I ...
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0answers
32 views

Is the set of regular points in a scheme open in general?

In the situation when smooth/k coincides with regularity (for a finite-type k scheme edit: k also perfect, thanks Remy), I think this should be true (?). But I am not sure about the situation for a ...
2
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0answers
37 views

Is there an example of a scheme $X$, irreducible, with a dense open subset $U$ so that $\dim U < \dim X$?

Is there an example of a scheme $X$, irreducible, with a dense open subset $U$ so that $\dim U < \dim X$? What about just for topological spaces? I know for varieties this doesn't happen (though ...
2
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1answer
51 views

Intuition for the valuative criterion for properness of morphisms?

I've always been told that the intuition for the valuative criterion for properness is something like this: a morphism $X\rightarrow Y$ is proper if, given a map of a small disk $D$ into $Y$ and a ...
4
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0answers
44 views

Hartshorne problem III.5.2(a)

Consider problem III.5.2(a) in Hartshorne's Algebraic Geometry: Let $X$ be a projective scheme over a field $k$, let $\mathcal O_X(1)$ be a very ample invertible sheaf on $X$ over $k$, and let ...
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1answer
54 views

Why is the rank of a locally free sheaf same everywhere if $X$ is connected?

Let $(X,\mathcal{O}_X)$ be a connected scheme.Let $\mathcal F$ be a locally free sheaf on $X$. This means that $X$ can be covered by open sets $U$ for which $\mathcal F|_U$ is a free $\mathcal O_X|_U$ ...
3
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1answer
52 views

Spectrum of infinite product of rings

$\def\Z{{\mathbb{Z}}\,} \def\Spec{{\rm Spec}\,}$ Suppose $R$ a ring and consider $\Spec(\prod_{i \in \mathbb{Z}} R)$. Now for the finite case, I know that holds $\Spec(R \times R) = \Spec(R) \coprod ...
0
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1answer
38 views

Image of not dominant morphism in Spec Z is finite

I have a very simple question that I seem not to be able to answer by myself. I want to understand the following: "If the structure morphism $f: X \to \operatorname{Spec}\mathbb{Z}$ is not dominant, ...
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1answer
68 views

Very basic question on projective bundles: Why is the fiber a vector space?

Consider an integral scheme $(X,\mathcal O_X)$ of dimension $n$ and a "good" locally free sheaf of $\mathcal O_X$-modules $\mathcal E$ of rank $n+1$. Then, thanks to the "global proj construction" we ...
0
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1answer
29 views

Maps induced by divisors and degree

I feel very stupid to ask this question but I've serached everywhere and I've not found a clear (to me) answer. Is a well known result in Curve theory (over $k=\bar{k}$) that a Divisor $D$ on a curve ...
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2answers
48 views

The Hirzebruch surface $\mathbb F_n$ contains a unique irreducible curve with negative self-intersection.

Consider the $n$-th Hirzebruch surface $\mathbb F_n:=\mathbb P_{\mathbb P^1}(\mathcal O_{\mathbb P^1}\otimes\mathcal O_{\mathbb P^1} (n))$, and let $C_0\subseteq \mathbb F_n$ a section of the ruling ...
2
votes
1answer
31 views

Morphism between affine scheme corresponding with ring homomorphism

Let $X = \text{Spec} R$ and $K = \text{Spec} S$ be two affine schemes with $f,g: K \rightarrow X$. I know that a morphism between affine schemes correspond with a ring homomrphism, so denote $\varphi: ...
1
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1answer
53 views

$\mathbb P^1\times \mathbb P^1$ is minimal

I'm trying to prove that the complex surface $\mathbb P^1\times \mathbb P^1$ is minimal. I'd like to prove it directly, namely by showing that there are not exceptional curves. Suppose that $E$ is a ...
4
votes
1answer
65 views

Degree of $f:\mathbb{P}^1_k\rightarrow \mathbb{P}^1_k$

Let $k$ be an algebrically closed field and consider $\mathbb{P}^1_k$ the $1$-dimensional projective space over $k$. My question is the following: Let consider $f:\mathbb{P}^1_k\rightarrow ...
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0answers
29 views

What exactly does the phrase “the quadratic cuts out a quadratic cone”?

Let $u,w,v \in \mathbb{C}$. Then I read in a review on affine schemes that The quadratic $uw-v^2$ cuts out a quadratic cone $X \in \mathbb{A}^3$ with coordinate ring $\mathbb{C}[u,v,w]/(uw-v^2)$. ...
2
votes
1answer
54 views

Geometrically ruled surface, sections and intersection numbers.

Consider a geometrically ruled surface $\pi: S\longrightarrow C$ where both $S$ and $C$ are projective, non-singular and complex ($C$ is obviously a curve). By using Tsen theorem one can show that ...
3
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0answers
49 views

Tricks to find $\dim H^0(O(D)(n))$

Let $X$ be a curve of genus $g\neq 0$ and $D$ a divisor on $X$. If $n \in \mathbb{Z}$ and $O(D)(n)=O(D)\otimes O(n)$ is the twist of $O(D)$ by $n$, there is a manageable proceedings to find the ...
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0answers
22 views

Zariski tangent space and subschemes of length 2

Let $X$ be a scheme, $x\in X$.The Zariski tangent space of $X$ at $x$ is the dual of the vector space $\mathfrak{m}_{x}/\mathfrak{m}_{x}^2$ over $k(x)$. It is a general fact that there is a ...
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0answers
39 views

Description of Blow up via Ress Algebra

Let $X=\mathrm{Spec}(R)$ be an affine scheme and $Z=\mathrm{Spec}(R/I)$ be a closed subscheme of $X$ defined by an ideal $I$ of $R$. The blow up of $X$ along $Z$ is ...
3
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2answers
180 views

Why is the “smallest non-affine scheme” not affine?

Exercise I-25 of Eisenbud and Harris' "Geometry of Schemes" defines the following scheme, which is then claimed to be the smallest non-affine scheme: Let $X=\{p,q_1,q_2\}$ with the open subsets ...
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1answer
46 views

Projective line is not isomorphic to the affine space with a doubled origin (schemes)

In the book 'Geometry of Schemes' (Eisenbud-Harris), there is a question that says the following (page 34, I-44): put $Y = \rm{Spec\,} K[s]$ and $Z = \rm{Spec\,} K[t]$. Let $U \subset Y$ be the open ...
2
votes
1answer
27 views

If $f : X \to Z$ is a morphism of schemes, which factors through an open $i : U \to Z$ on the level of topological spaces…

If $f : X \to Z$ is a morphism of schemes, which factors through an open $i : U \to Z$ on the level of topological spaces... then is there a (unique) morphism of schemes $g : X \to U$ which makes the ...
6
votes
1answer
172 views

What does Hartshorne do wrong?

I'm currently trying to learn algebraic geometry from Hartshorne's Algebraic Geometry. I've often heard it said, both on MathOverflow and in my department, that Hartshorne's treatment of certain ...
0
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1answer
75 views

Question on the proof of Hartshorne IV.3.7

Proposition 3.7 in Hartshorne's Algebraic Geometry is the following. Let $X$ be a curve in $\mathbb P^3$, let $O$ be a point not on $X$, and let $\phi: X \rightarrow \mathbb P^2$ be the morphism ...
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0answers
34 views

Fibers of the Blow up of $\mathbb{A}^6$ over the center defined by ideal defined by vanishing of rank two minors.

Let us consider set of all matrices \begin{pmatrix}{} x & u & v \\ u & y & w \\ v & w & z \end{pmatrix} where $x,y,z,u,v,w\in \mathbb{R}$ which is equivalent to ...
2
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1answer
59 views

if the canonical divisor is nef, then a multiple if effective

Let $S$ be a complex projective non-singular surface. Couldd you explain the following implication: If the canonical divisor $K_S$ is nef then there exists a number $m>>0$ such that $mK_S$ ...
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0answers
23 views

Is it true that the complete linear system of a "negative'' divisor has dimension $0$?

Let $X$ be a projective, non-singular variety and consider a divisor $D\in\text{Div}(X)$ such that $D=-D'$ for a certain effective divisor $D'$. Now Consider the complete linear system $|D|:=\{C\in ...
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0answers
60 views

Why is $ \mathrm{Spec} \ A [T_1 , \dots , T_n ] \to \mathrm{Proj} \ A [T_0 , \dots , T_n ] $ an embedding?

Let $ A $ be a commutative ring with identity. I would like to know how to establish that : $ \mathrm{Spec} \ A [T_1 , \dots , T_n ] \to \mathrm{Proj} \ A [T_0 , \dots , T_n ] $ is an embedding ? ...
1
vote
1answer
64 views

Understand the corrispondence between rational maps and linear systems.

A key fact in "birational geometry" (on $\mathbb C$) is the following theorem: Let S be a surface. Then there is a bijection between the following sets: (i) {rational maps ...
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0answers
42 views

$k$-structure on $K$-schemes

I'm reading A. Borel's "Linear Algebraic Groups". At an early point in the book, the author establishes the following concepts: (Let $K$ be an algebraically closed field, and $k$ a subfield of $K$) ...
2
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2answers
61 views

Corollary of Riemann Roch theorem

I'm studying about Riemann Roch theorem for curves and its consequences (in algebaic gemoetry). In Hartshorne I've found an exercise equivalent to this assert: Let $D$ be an effective divisor of ...
1
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2answers
58 views

What exactly does the Hilbert scheme of points parametrize?

The Hilbert scheme of points is defined as $$ \text{Hilb}^n(X) = \{ I \subset \mathbb{C}[x,y] \text{ such that } \text{dim}_{\mathbb{C}}/I = n \} $$ or, in words, the Hilbert scheme of points ...
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0answers
49 views

$\Omega_X$ locally free $\implies$ $X$ smooth

Let $X$ be $n$-dimensional scheme of finite type over an algebraically closed field. In the proof of Proposition 7.4.11 in Gathmann, the first paragraph reads If $\Omega_X$ is locally free of rank ...
0
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1answer
29 views

Rational map is not defined on a subset of codimension $\geq 2$

I have the following Lemma (http://www.jmilne.org/math/xnotes/AVs.pdf, Lemma 3.2): A rational map $f:V\dashrightarrow W$ from a normal variety to a complete variety is defined on an open subset $U$ ...
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0answers
56 views

Reduced and not reduced points of a scheme.

Here is a small paragraph of a course of "Anwar Alameddin" that i found on the net accidently, and that i try to understand clearly, about intersection theory : Let $ k $ be an algebraically ...
1
vote
1answer
57 views

There exists a divisor linearly equivalent to $P$ not containing $P$

Let $X$ be a non-singular projective curve over $\mathbb C$ and let $P\in X$ be a closed point. Which is, in your opinion, the easiest way to prove that there exists a divisor $D$ of $X$ satisfying ...