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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47 views

Geometric and arithmetic Frobenius

I read in Serre's "Lectures on $N_X(p)$" that when $X$ is a scheme defined over $\mathbb{F}_q$ (a finite field), the geometric Frobenius $F: X \mapsto X$ is defined by fixing every element of the ...
2
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0answers
38 views

Do finite groups act admissibly on separated schemes of finite type over k

Background: Recall from SGAI that a group $G$ acts admissibly on a scheme $X$ if the quotient $X \to X/G$ exists and is an affine morphism of schemes. This is the case if and only if every orbit of $G$...
2
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1answer
65 views

Covering by open subfunctors and epimorphisms of sheaves.

I am trying to learn about the functor of points approach to algebraic geometry. Given the category of locally ringed spaces $GSp$ (geometric spaces) we have a functor $$\mathcal{G}: GSp \to Set^{...
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45 views

Fibres of a morphism

Let $f:X\rightarrow S$ a proper morphism, and $s\in S$ a point. If $S$ is locally Noetherian, then what are the properties of the fibre scheme $X_s$ over the Spec of the residue field at $s$? Is this ...
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1answer
50 views

What does it mean for a scheme to be proper?

What exactly does it mean for a scheme to be proper? I can't seem to find an actual definition of this anyway despite the term being frequently used.
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1answer
61 views

The non-existense of the fine moduli scheme of vector bundles. Why?

The reference I am using is this enter link description here. The question is about the moduli space of vector bundles. I am trying to understand why the fine moduli scheme does not exist. Let $C$ a ...
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1answer
38 views

Tensor product of the stricture sheaf with the function field

Consider an irreducible scheme $X$ with function field $K(X)$. Then define the presheaf $$U\mapsto \mathscr O_X(U)\otimes_{\mathscr O_X(U)} K(X)$$ for every open set $U\subset X$. Is this presheaf ...
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1answer
98 views

Strict transform of blow up

Let $X$ be a smooth projective variety over $\mathbb{C}$. Consider the blow up of $X$ about a closed subvariety $Z$. Let $X'=Bl_Z(X)$. Let $Y$ be a smooth irreducible divisor of $X$ properly ...
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2answers
51 views

restrictions of rational sections of an invertible sheaf.

Let $(X,\mathscr O_X)$ be an integral scheme with function field $K(X)$ and let $\mathscr L$ be an invertible sheaf on $X$. Moreover suppose that $\{U_i\}$ is an open covering of $X$ such that $\...
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182 views

Is there a universal statement for the construction of global Proj?

Let $X$ be a scheme and $\mathcal{A}$ be a sheaf of $\mathbb{Z}_{\ge 0}$-graded $\mathcal{O}_X$-algebras. From the data above there is a construction which gives the "global Proj" $Proj \mathcal{ A} ...
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4answers
186 views

Diagonal morphism of regular variety is a regular embedding

Let $X$ be a regular $k-$variety (i.e. all of its local rings are regular) of pure dimension $d$. Then I would like to show that the diagonal morphism $X\rightarrow X\times_k X$ is a regular embedding ...
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45 views

Covering maps of schemes.

A curve $X$ is modular if there is a finite covering $X_0(N)\rightarrow X$. What does covering mean in this context, and for more general morphisms of schemes? Just covering as topological spaces?
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2answers
54 views

Scheme theoretic 'class inclusions'

For lack of a better name*, I call the following two things class inclusions: $$1) \quad\textbf{Magma}\supset \textbf{Semigroup}\supset\textbf{Monoid}\supset \textbf{Group}\supset \textbf{Abelian ...
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19 views

What is the linear preserving criterion?

I have a question regarding the linearity preserving criterion: From an article I read, is said that the linearity preserving criterion required that the discretization scheme is exact whenever the ...
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1answer
55 views

Morphism of finite type schemes not surjective. Is there a closed point in the complement of the image?

I have the following proplem: Let $k$ be an algebraically closed field and let $X,Y$ be schemes of finte type over $k$. Now let $f:X\to Y$ be a morphism of schemes that is not surjective. Question: ...
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0answers
34 views

Vanishing set of a pullback section

Let $f\colon X\to Y$ be a morphism of schemes and let $\mathcal{F}$ be an $\mathcal{O}_Y$-module. Let $s\in H^0(Y,\mathcal{F})$. If I'm not mistaken, then $(f^{-1}s)_x=s_{f(x)}$ for all $x\in X$ and ...
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0answers
31 views

Closed point in the generic fiber of an arithmetic surface

Let $S$ an irreducible Dedekind scheme of dimension $1$, and let $\pi:X\to S$ be a regular, integral fibered surface. We assume that $\pi$ is a flat morphism and that $X_\xi$ is the generic fiber over ...
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1answer
65 views

Doubt in Hartshorne Example 7.17.3, Chapter 2

Let $X$ be a smooth projective variety over $\mathbb{C}$ of dimension $n$. Let $L$ be a line bundle on $X$, and let $V\subset H^0(X,L)$ be a subspace of sections. Suppose that $s_1,..,s_{l+1}$ is a ...
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1answer
48 views

Push-forward of quasi-coherent sheave on affine scheme is quasi-coherent

Let $X=$ Spec$R$, $Y=$ Spec$S$, $f:X \to Y$ be a morphism of schemes. Let $M$ be a $R$-module, and let $\mathcal{F}=\tilde{M}$ be the sheaf on $X$ induced by $M$. How can I show that the pushforward ...
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0answers
12 views

Gröbner degeneration, flat families and morphism of varieties.

I am trying to understand why a map is actually a morphism of varieties. The map, from the affine line to the Hilbert scheme of points, is given by a Gröbner degeneration from an ideal I to its ...
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1answer
39 views

Well-definedness of homogeneous coordinate ring of projective scheme [closed]

Let $I,J \subset S=k[x_0,...,x_n]$ be homogeneous ideals. How can I show that $\operatorname{Proj}S/\bar{I}=\operatorname{Proj}S/\bar{J}$ iff $\bar{I}=\bar{J}$? (Here $\operatorname{Proj}R$ is the ...
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0answers
24 views

Image of base change of immersion

I'm revising for my exams now and struggling with the following exercise: Let $f: X \to S$ be an open or closed immersion and $g: S' \to S$ another morphism where $X,S,S'$ are schemes. Then the base-...
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0answers
36 views

Grassmannian as schemes

I would like to understand the Grassmannian as a scheme. If $V$ is a vector space over the complex numbers, then $\mathbb{C}$-valued points of the Grassmannian $\mathbf{Grass}(r,V)$ consists of all ...
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34 views

Formula relating dimension of fiber of morphism between varieties

Let $f: X \to Y$ be a morphism of (irreducible) varieties, where the dimension of every fiber dim$f^{-1}(y)=n$ is the same. Must it follow that dim$X=$ dim$Y+n$? The reason I am asking this is that ...
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1answer
83 views

Cohomology Class of a Subvariety

I'm working on question 7.4 of Chapter III.7 in Hartshorne's Algebraic Geometry. The question is about the cohomology class of a subvariety. The setup is as follows: $X$ is an $n$-dimensional non-...
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0answers
38 views

Cartier divisor and map to associated line bundle

Given a Cartier divisor $D$ on an integral, separated scheme $X$ of finite type over an algebrically closed field $k$. Does such a divisor always induce a map $\mathcal{O}_X \to \mathcal O_X(D)$? I ...
1
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1answer
50 views

Push forward of an exact sequence of sheaves under blow up

Let $X$ be a smooth projective variety of dimension $n\geq 3$. Let $Z$ be a smooth subvariety of $X$ of codimension at least 3. Let $Y$ be the blow up of $X$ along $Z$ and let $f:Y\longrightarrow X$ ...
2
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0answers
54 views

On the definition of principal Cartier divisors

In Liu's Algebraic Geometry and Arithmetic Curves, Definition 7.1.17, a few lines after the definition of principal Cartier divisor (as one in the image of $\Gamma (X,K_X^*) \to \Gamma (X,K_X^*/ \...
2
votes
1answer
58 views

Intuition behind fibers of a morphism of schemes

Let $X,Y$ be schemes over a field $k$ and $f:X\rightarrow Y$ a morphism. Let us suppose that the fiber $f^{-1}(y)$ of $f$ at a point $y\in Y$ has two connected components $Z_{1},Z_{2}$. I have read ...
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1answer
33 views

Does every open covering of a scheme also contain an affine covering?

Let $U_i$ denote a family of open subsets of a scheme $X$, which is not affine, so that $X = \bigcup U_i$. Can this covering be somehow transformed into an affine covering of $X$?
2
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48 views

What properties single out $ \operatorname{Spec}(\mathbb{k}) $-schemes that are quasi-projective varieties over $ \mathbb{k} $?

I have a question in algebraic geometry that I would like to ask: Let $ \mathbb{k} $ be an algebraically closed field. Is there a property $ P $, phrased in the language of schemes, such that ...
4
votes
2answers
93 views

Fiber of morphism homeomorphic to $f^{-1}(y)$

I want to solve exercise 3.10 (a) of Hartshorne's book, chapter II, which asks to prove the following: Let $f\colon X\to Y$ be a morphism of schemes and let $y\in Y$, then $X_y=X\times_Y \...
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1answer
69 views

Intuition about formal brances of a curve at a point

Consider an algebraic surface $X$ and a curve $Y\subset X$. Here $X$ is a $K$-scheme integral of finite type of dimension $2$ and $Y$ is a closed subscheme of dimension $1$. Fix a closed point $x\...
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1answer
38 views

Problem in understanding the proof of lemma $7.2.5$ in Liu's book

Let's analyze the proof of the following lemma: Lemma: Let $X$ be an integral, Noetherian scheme and let $f\in K(X)^\ast$, then for all but finitely many points $x\in X$ of codimension $1$ we have ...
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3answers
82 views

Scheme over a not algebraically closed field

I'm trying to make a scheme out of an affine variety $V\subset k^n$, when the base field $k$ is not algebraically closed. I wonder if $V$ is the spectrum of its ring of regular functions $\mathcal{O}(...
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0answers
42 views

Base change and relative dimension

Let $B$ a ring of Krull dimension $1$ and suppose that $\text{Spec }B$ is irreducible. And let $f:X\to\text{Spec}(B)$ a $B$ scheme with the following properties: $X$ is projective, regular and ...
2
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0answers
78 views

Rational sections of invertible sheaves and hermitian inner products

Notations: Let $X$ be a $\mathbb C$-scheme of finite type, projective, integral and of dimension $1$ (i.e. an algebraic curve) and with function field $K(X)$. The set of closed points is $X(\mathbb C)$...
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0answers
34 views

Finite type assumption necessary for this property of very ample sheaves?

The question $\mathcal{L}$ is very ample, $\mathcal{U}$ is generated by global sections $\Rightarrow$ $\mathcal{L} \otimes \mathcal{U}$ is very ample asks for a proof of the following statement: ...
2
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2answers
59 views

Why are there no $\mathbb{R}$-valued points on a complex curve?

We say a $K$-valued point on a scheme is a map Spec$(K) \to S$, so in particular, a real valued point on the parabola $y = x^2$ should be a map Spec$(\mathbb{R}) \to $Spec$(\mathbb{C}[x,y]/(y-x^2))$. ...
1
vote
1answer
37 views

Schemes not determined by morphisms from one object

I was reading a bit about the functor of points of a scheme, and it was mentioned that there does not exist a scheme $Y$ so that Hom$(Y,X)$ determines the points $X$ for all schemes $X$. This is in ...
0
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1answer
27 views

Representation of an effective Cartier divisor: is there an imprecision in Liu's book?

I will use the notation of Liu's book which is also the standard one. If you don't feel comfortable with it please ask more information in the comments Let $X$ be a scheme, and consider a Cartier ...
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1answer
43 views

Exercise I-5 of Eisenbud-Harris

I have just started learning about schemes, so please do not be too harsh with me. I am trying to do Exercise I-5 of Eisenbud-Harris, "The geometry of schemes". Suppose $X$ is the topological space $...
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1answer
28 views

Morphism smooth over the function field. What does it mean?

Look at the following lines I found in the book "Moriwaki - Arakelov geometry" (beginning of chap. 4): Let $S$ be a connected Dedekind scheme with function field $K$. Let $\pi:X\to S$ be a ...
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2answers
55 views

Sheaf associated to a Cartier divisor

This question is motivated by a construction, unclear for me, related to Cartier divisors. But, in the end it can be reduced to a question involving only sheaves on topological spaces. Let $X$ be a ...
2
votes
1answer
58 views

Image of divisors under étale cover

Let $f\colon X\to Y$ be an étale cover of degree $d$ between two smooth projective varieties. If $V\subset X$ is an effective reduced and irreducible divisor, does $f$ restrict to an isomorphism $V\...
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0answers
17 views

Affine morphism question

For a scheme $X$ and ring $A$, we have $\text{Hom}_{\text{Sch}}(X, \text{Spec}A) \cong \text{Hom}_{\text{Ring}}(A, \Gamma(X,\mathcal{O}_X))$. For $X$ a scheme over $Y$, how can this be generalized ...
2
votes
0answers
31 views

Quotient of a scheme under infinite group action

Let $X=\mathbb{A}^2\setminus\{(x,y)\}$ be the affine scheme minus the origin (say over a field $k$). Consider the action of the group $G=k^*$ given by $(x,y)\mapsto (\alpha x,\alpha^{-1}y)$ for any $\...
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1answer
57 views

Geometric xample of a one object cover which does not satisfy the sheaf condition?

For topological spaces, a one object cover automatically satisfies the sheaf axiom because open inclusions are injective, which is equivalent to the projections of the kernel pair (= self intersection)...
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41 views

Categories of defintion for sites on spaces and sites on schemes

Starting with example 2.26 in Vistoli's Notes he defines topological sites on $\mathsf{Top}$, while sites on schemes are always defined on a slice category $\mathsf{Sch}/X$. Why is this?
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1answer
35 views

Are representables on the étale site on topological space sheaves?

The gluing lemma says representables on the classical site on $\mathsf{Top}$ are sheaves. Basic scheme theory says the same is true for the small Zariski site of a scheme. Are representables on the ...