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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1answer
104 views

Vakil's FOAG, Exercise 9.2.K: Transcendental Complex Numbers

How does one realize a transcendental complex number as a maximal ideal of $\mathbb{Q}(t) \otimes_{\mathbb{Q}} \mathbb{C}$? This is the essence of Exercise 9.2.K in Vakil's FOAG. Here is what I've ...
4
votes
2answers
204 views

Hartshorne generically finite morphisms (II, 3.7)

I have a question concerning one of the exercises of Hartshorne, Ch. II. Namely: Exercise 3.7 about gerneically finite morphisms. A morphism $f: X \rightarrow Y$ with Y irreducible and $\eta$ ...
2
votes
1answer
34 views

Functions on Finitely-Generated, Nilpotent Free, k-Algebras Determined by Values on Closed Points

I am working (slowly and with much labor) through Vakil's Algebraic Geometry and came upon this problem. Suppose $k$ is an algebraically closed field, and $A = k[x_1,... ,x_n]/I$ is a finitely ...
1
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0answers
61 views

Spec$(R)$ a scheme of finite type over $\mathbb{C} \implies R$ is a finitely generated algebra over $\mathbb{C}$.

Suppose $(\text{Spec}(R), \tilde{R})$ is a scheme locally of finite type over $\mathbb{C}$. We want to show that $R$ is a finitely generated $\mathbb{C}$-algebra. Since $(\text{Spec}(R), \tilde{R})$ ...
0
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1answer
33 views

Multiplicity of Cartier divisor on locally noetherian scheme is only non-zero at generic point

I'm following chapter 7 in Qing Liu's book 'Algebraic Geometry and Arithmetic Curves' about 'Divisors and applications to curves'. My question concerns Definition 1.27: Let $A$ be a Noetherian ...
0
votes
0answers
29 views

Comparing covers

Considering the Zariski topology, let $$V = \bigcup_{i \in I} U_i$$ be a maximal open cover of $V$ by basic open sets. Similarly, let $$V' = \bigcup_{j \in J} W_j$$ by the maximal open cover of $V'$ ...
2
votes
1answer
53 views

Families as fibres of a morphism

In both Algebraic Geometry by Hartshorne and Geometry of Schemes by Eisenbud and Harris, the authors describe the notion of a family of schemes as being the fibres of a morphism $f:X\to Y$. Or as ...
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0answers
50 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 ...
1
vote
0answers
20 views

Existence of scheme quotient

I have a morphism of schemes $X\to S$ which is very nice: flat, proper, finitely presented. I also have a finite group $G$ acting (faithfully, but not necessarily freely) on $X/S$. 1) I am quite ...
2
votes
0answers
27 views

Tensor product between an invertible sheaf and a constant sheaf.

This question is a natural extension this one. Consider an irreducible scheme $X$ with function field $K(X)$ and let $\mathscr L$ be an invertible sheaf over $X$. Then define the presheaf $$U\...
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0answers
39 views

Is there a “strong” Chow lemma where “dense” means “scheme theoretically dense”?

Recall Chow's lemma: Chow's Lemma: If $X$ is a scheme that is proper over a noetherian base $S$ then there exists a projective $S$-scheme $X'$ and a surjective $S$-morphism $f : X'\to X$ that ...
1
vote
1answer
60 views

Tensor product of the structure 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 ...
0
votes
0answers
52 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?
0
votes
0answers
53 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
votes
0answers
41 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 $G$...
7
votes
0answers
190 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} ...
2
votes
1answer
68 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^{...
1
vote
1answer
65 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 ...
1
vote
1answer
57 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.
0
votes
1answer
107 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 ...
3
votes
4answers
194 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 ...
1
vote
2answers
52 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 $\...
0
votes
2answers
59 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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votes
0answers
20 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 ...
0
votes
1answer
59 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: ...
0
votes
0answers
35 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 ...
0
votes
0answers
35 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 ...
0
votes
1answer
41 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 ...
0
votes
1answer
50 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 ...
0
votes
1answer
71 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 ...
0
votes
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 ...
4
votes
1answer
78 views

Geometrical meaning of $\operatorname{Spec}\widehat{\mathcal O}_{X,x}$

Let $X$ be a complete smooth irreducible variety over a field $K$ and consider a closed point $x \in X$. Moreover let $\widehat{\mathcal O}_{X,x}$ be the $\mathfrak m_x$-adic completion of the local ...
0
votes
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-...
0
votes
0answers
38 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 ...
1
vote
0answers
43 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 ...
1
vote
1answer
71 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\...
6
votes
1answer
86 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-...
0
votes
0answers
42 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
vote
1answer
58 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 $f:Y\longrightarrow X$ ...
6
votes
1answer
259 views

“This property is local on” : properties of morphisms of $S$-schemes

I am learning schemes theory at school and I have for now only lectures notes that I am taking during the course. The professor is quite often using following expressions, without having defined them :...
2
votes
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
61 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 ...
1
vote
1answer
34 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
votes
0answers
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
94 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 \...
2
votes
1answer
70 views

Synthetic differential geometry and formally étale morphisms?

Upon looking throug Kostcki's synthetic differential geometry notes, I stumbled upon the following definition. (Here $R$ is the geometric line, $W$ is a Weil algebra, and $\operatorname{Spec}_RW$ is ...
1
vote
1answer
39 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 ...
2
votes
0answers
80 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 $X(\mathbb C)$...
1
vote
3answers
85 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}(...
4
votes
1answer
187 views

Usefulness of the notion of Hilbert scheme in algebraic geometry.

Could someone tell me why and how Hilbert schemes and relative Hilbert schemes are important and useful in algebraic geometry? Could anyone give me some applications of this notion in concrete terms? ...