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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2
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2answers
61 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
vote
1answer
37 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
vote
1answer
40 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
votes
2answers
114 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
50 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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0answers
95 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 $$\mathscr{Tor}_i^{\mathcal{O}_X}(\mathcal{O}_C,\...
5
votes
2answers
106 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
votes
1answer
71 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
votes
1answer
68 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 ...
0
votes
0answers
30 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
79 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
votes
0answers
49 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 k[...
2
votes
1answer
47 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 ...
1
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0answers
56 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
40 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 ...
3
votes
1answer
82 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 ...
5
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0answers
64 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 $\...
2
votes
1answer
74 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
votes
1answer
82 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
votes
1answer
41 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, ...
1
vote
1answer
77 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
37 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. A well known result in Curve theory (over $k=\bar{k}$) states that a divisor $D$ on a ...
1
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2answers
59 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
35 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
55 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 ...
5
votes
1answer
71 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 \...
0
votes
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
67 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
votes
0answers
51 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 ...
0
votes
0answers
25 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 ...
1
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0answers
45 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 (say)$\mathrm{Bl}_{Z}(X)=\mathrm{...
3
votes
2answers
194 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 $\...
1
vote
1answer
55 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
29 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
199 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
votes
1answer
91 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 ...
1
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0answers
37 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 $\mathbb{A}^...
2
votes
1answer
67 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 $mK_S$ ...
1
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0answers
62 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
78 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 $\phi:S-\...
0
votes
0answers
47 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
votes
2answers
84 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
vote
2answers
70 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 ...
1
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0answers
53 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
votes
1answer
50 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 $U$ ...
0
votes
0answers
64 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
63 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 ...
1
vote
1answer
64 views

Generic point and pull back

Let's work over $\mathbb C$, Consider a finite morphism between two integral curves $f: X\rightarrow Y$, let $\eta_Y$ (resp $\eta_X$) be the generic points Questions: What is $X_{\eta_Y}=f^{-1}(\...
7
votes
1answer
128 views

Intersection of curves on surfaces and the sheaf $\mathcal O_{C\cap C'}$

Let $S$ be a (complex) projective surface and let $C,C'\subset S$ two closed irreducible curves (namely two prime Weil divisors). It is well defined the scheme $C\cap C'$ (as fibered product of $C\...
1
vote
0answers
39 views

A special presheaf of rings over the spectra $ \mathrm{Spec} A $.

Let $ A $ be a commutative ring with unity, and $ \mathrm{Spec} A $ the spectra of $ A $. We define over $ \mathrm{Spec} A $ the presheaf of rings as follows : If $ U $ is an open subset of $ \...