The concept of scheme is from algebraic geometry: A locally ringed space that is glued together (much like a manifold) from affine schemes, that is spectra of rings with Zariski topology and structure sheaf.

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Weak Kodaira Vanishing - Hartshorne III.7.1

In the Serre Duality section of Algebraic Geometry by Robin Hartshorne, the following exercise is posed: If $X$ is an integral projective scheme over a field $k$, prove that an ample invertible sheaf ...
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Quotient of group schemes and its rational points.

At the moment I have some difficulties in understanding the quotient of group schemes and so exact sequences. I am aware that precise answers would be difficult to be given without speaking of sheaves ...
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Is the fiber product of the connected component of a group scheme connected?

Let $G$ be a group scheme over a field $k$. Let $G^0$ be the connected component containing the identity. Is it true that $G^0\times_k G^0$ is connected? I know that this is true if $G^0$ is ...
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Check a non-projective morphism

The following is from the wiki: http://en.wikipedia.org/wiki/Proper_morphism Projective morphisms are proper, but not all proper morphisms are projective. For example, it can be shown that the ...
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The Chow variety of conics in $\mathbb{P}^{3}_{k}$

Consider the family of all conics in $\mathbb{P}^{3}_{k}$, with $k$ an algebraically closed field. Such curves all have degree $2$ and genus $0$, and they can be uniquely defined to be all curves in ...
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Counting the dimension of a component of $\mathsf{hilb}^{2t+1}_{3}$

Consider the Hilbert scheme $\mathsf{hilb}^{2t+1}_{3}$, parametrizing varieties of degree $2$ and genus $0$ in $\mathbb{P}^{3}_{k}$, with $k$ an algebraically closed field. Consider the component $ ...
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Hilbert Scheme and Chow variety in the case of Conics in $\mathbb{P}^{3}$

My question concerns the relationship between chow varieties and hilbert schemes in the case of conics in $\mathbb{P}^{3}_{k}$. More precisely, consider the Hilbert scheme $\mathsf{hilb}^{2t+1}_{3}$, ...
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Morphism between two $K$-schemes restricted to an affine subscheme

Suppose that $f:X\longrightarrow Y$ is a morphism between two $K$-schemes. If $U\subseteq X$ is an affine open set, then can we conclude that $f(U)$ is contained is some affine open subset of $Y$? ...
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Complexifications of degree 3 subschemes in $\mathbb A^2_{\mathbb R}$

I am trying unsuccessfully to solve exercise II-20 (page 65) from the book "The geometry of schemes" by Eisenbud and Harris. In this exercise it is stated that there are two non-isomorphic subschemes ...
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Motivating examples of Spec(R) where R is not a finitely generated k-algebra

A scheme is a ringed space locally isomorphic to an affine scheme. An affine scheme is the spectrum of a commutative unital ring together with a structure sheaf. What are examples of "interesting" ...
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Geometric reducedness (integral) versus reducedness (integral)

All the schemes here are over $\mathbb{C}$. Suppose $X \to Y$ is a morphism of varieties, then the geometric reducedness (integral) of the generic fibre implies the geometric reducedness (integral) ...
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Connectedness in Unipotent Groups

Are all subgroups of an unipotent group over a finite field (as a scheme) connected?
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A morphism from $\mathbb P^1_\mathbb C$ to $\mathbb P^1_\mathbb C$

Consider the projective scheme $\mathbb P^1_\mathbb C$ that is different from the projective line $\mathbb P^1(\mathbb C)$. Now look at the following lemma: I don't understand what is a ...
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A scheme with not-numerable affine covering.

This question maybe finds the answer in the definition. Let $X$ be a scheme with not hypothesis on it (not noetherian, not affine,... ). Can I find a scheme that has a not numerable affine covering? ...
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Does $\operatorname{Proj}(\sigma)$ fix some points?

Consider a subfield $K\subseteq\mathbb C$, then by some properties of the fibered product of schemes we have that: $$\mathbb P^1_\mathbb C\cong\mathbb ...
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Confused with $\mathbb Q$-rational points

Look at the following enlightened part of a proof extracted from a paper (here $C$ is an algebraically closed field of characteristic $0$): Clearly $\mathbb Q\subset C$ and $\mathbb P^1_C$ is ...
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global sections of structure sheaf of a projective scheme X over a field k?

Some may have already asked this question. What are the global sections of structure sheaf of a projective scheme $X$ over a field $k$? By Hartshorne page 18, Chapter 1, Theorem 3.4, global sections ...
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$K$-schemes as varieties: the importance of the structural morphism

Consider a variety $p:X\longrightarrow\operatorname{Spec K}$ where $X$ is an integral scheme and $p$ is a separated morphism of finite type. Now chose an element ...
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Morphisms between schemes such that every point in the codomain has at most $n$ preimages.

Consider a finite morphism $f:X\longrightarrow Y$ between two integral and Noetherian schemes. If $\operatorname {deg}(f)=[K(X):K(Y)]=n$, is it true that for every $y\in Y$ then $|f^{-1}(y)|\le n$? ...
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What is a field of definition of a morphism?

An algebraic variety $X$ over a field $K$ is a $K$-scheme integral separated and of finite type over $K$. Definition: If $L$ is a subfield of $K$ we say that $X$ is defined over $L$ if there ...
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Geometric interpretation and computation of the Normal bundle

My question concern the definition, geometric meaning, and usage of the normal bundle in algebraic geometry. Let $X$ be a nonsingular variety over an algebraically closed field $k$, and $Y\subseteq ...
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Group Scheme - equivalent ways of defining it

I know that a group scheme is an $S$-scheme $G$ equipped with one of the equivalent sets of data a triple of morphisms $μ$: $G$ ×S $G$ → $G$, $e$: $S$ → $G$, and $ι$: $G$ → $G$, satisfying the ...
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Can Hom(X,Y) be given a natural scheme structure?

Let $X$ and $Y$ be $S$-schemes. Is there a "natural" scheme structure on $\operatorname{Hom}_S(X,Y)$, maybe subject to some conditions on the schemes in question? Interpret "natural" however you'd ...
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Closed immersion definition

Hartshorne defines a closed immersion as a morphism $f:Y\longrightarrow X$ of schemes such that a) $f$ induces a homeomorphism of $sp(Y)$ onto a closed subset of $sp(X)$, and furthermore b) the ...
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Example of Gluing Schemes

Let $k$ be a field. $U_0 = \mathbb{A}^1_k = \operatorname{Spec}(k[T])$ and $U_1=\mathbb{A}^1_k = \operatorname{Spec}(k[S])$. $U_{01} = D(T) = \mathbb{A}^1_k\backslash \{0\} = ...
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Unipotent Group Scheme

Can someone give me a reference of why this is true? Let A an associative k-algebra in wich every element is nilpotent, (non unital) and free of finite rank k-module, then for every commutative ...
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Closed points of a scheme (locally) of finite type over an algebraically closed field

If $X$ is an arbitrary scheme, I can prove that the set $X(k)$ of $k$-valued points is in bijective correspondence with the set $$\{(x,\iota) \, | \, x \in X, \, \iota:\kappa(x) \hookrightarrow k\},$$ ...
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Noetherian schemes and varieties

What types of varieties (e.g. projective, affine,...) over a field $k$ (char = $0$) are Noetherian schemes?
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Schemes to the rescue?

I am reading chapter 1 of Gille and Szamuely's book Central Simple Algebras and Galois Cohomology, on quaternion algebras. In it they prove (remark 1.3.1 on p. 18) that the conic associated to a ...
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Existence of a morphism of schemes from $X$ to $\mathbb P_A^n$?

In exercise §II.III.VIII. of Algebraic geometry and arithmetic curves by Qing.Liu. one is asked the following: Let $X$ be a scheme over a ring $A.$ Let $f_0, \cdots,f_n\in\mathcal O_X(X)$ be ...
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Infinitesimal thickening of a smooth closed subscheme

Let $A$ be a noetherian ring (if it is useful I can assume that $A$ is an algebra of essentially finite type over a field) and $I \subset A$ is an ideal s.t. $A/I$ is smooth. Is it true that extension ...
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If $B$ is a graded $A$-algebra, then $\operatorname{Proj}B$ is an $A$-scheme

If $B$ is a graded ring, then for me is clear that $\operatorname{Proj}B$ with affine covering given by $D_+{(f)}\cong\operatorname{Spec} B_{(f)}$ is a scheme. The problem arises when $B$ is a graded ...
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When is the global section functor exact?

Given sheaves $\mathcal{F}_1,\mathcal{F}_2, \mathcal{F}_3$ on some scheme $X$, and an exact sequence $$ 0\rightarrow \mathcal{F}_1\rightarrow \mathcal{F}_2\rightarrow \mathcal{F}_3\rightarrow 0 ...
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Why is the preimage of a Spec $A$ the Spec of its integral closure in a morphism of curves?

Let $\phi:X\to Y$ be a morphism of nonsigular complete curves. Let Spec $A:=U\subset Y$ be an open set. Why is $\phi^{-1} (U) =$ Spec $B$, the spectrum of the integral closure of $A$ in $K(X)$? Can we ...
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About two isomorphic schemes

My question is related to an answer I read on MO: http://mathoverflow.net/questions/157973/classical-algebraic-varieties-vs-k-schemes-vs-schemes In the accepted answer, the user Julian Rosen claims ...
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What is an algebraic automorphism over $k$?

I'm reading some notes about the action of finite groups on algebraic varieties, and I've found this sentence. Let $Y$ be a scheme of finite type over a field k, and let $G$ be a finite group, ...
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From the residue field at a point to a scheme

Consider a scheme $X$; for every $x\in X$ with residue field $k(x)$, we have the canonical surjection $\mathcal O_{X,x}\longrightarrow k(x)$ that induces the morphism of affine schemes ...
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Example of nonempty scheme with no closed points

I know that when a scheme $X$ is quasicompact, every point has a closed point in it's closure. This of course means that every nonempty quasicompact scheme has a closed point. If we drop the ...
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What is the zero subscheme of a section

Let $X$ be a scheme, $\mathcal F$ a locally free sheaf of rank $r$ and $s \in \Gamma(X, \mathcal F)$ a global section of $\mathcal F$. Question: What is the zero subscheme of $s$? I can't believe ...
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Groups acting on schemes: the quotient scheme doesn't always exist.

Preliminary notion: Consider the action of a group $G$ on an object $X$ of some category $\mathcal C$. We have a group homomorphism $\rho:G\longrightarrow\operatorname{Aut}(X)$ which sends $g$ in ...
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$K$-rational points and algebraic sets

Notation: If $X$ is a $K$-scheme, then a point $x\in X$ is said $K$-rational if its residue field $k(x)=\frac{\mathcal O_{X,x}}{\mathfrak m_{X,x}}$ is isomorphic to $K$. The set of all $K$-rational ...
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Closed subsets and closed subschemes

Consider a scheme $(X,\mathcal O_X)$; a closed subscheme of $(X,\mathcal O_X)$ is a scheme $(Z,\mathcal O_Z)$ such that: $Z$ is a closed subset of $X$ There is a morphism of schemes ...
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$\operatorname{Spec} (\cdot)$ is functorial [duplicate]

If $A$ is a ring (with unity) I'm trying to prove that the assignement $A\mapsto\operatorname{Spec}A$ defines a contravariant functor from the category of rings to the category of affine schemes. If ...
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Exercise 2.3 from Hartshorne's algebraic Geometry.

2.3) A scheme $(X,\mathcal{O}_X)$ is reduced if for every open set $U\subset X$, the ring $\mathcal{O}_(U)$ has no nilpotent element. b) Let $(X,\mathcal{O}_X)$ be a scheme. Let ...
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The two projection maps are different?

I'm reading Vistoli's notes, and I came across something on the bottom of page 31, starting with "The reader might find our definition of sheaf pedantic..." Essentially my problem is the following ...
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How to think of the Zariski tangent space

The Zariski tangent space at a point $\mathfrak m$ is defined as the dual of $\mathfrak m/\mathfrak m ^2$. While I do appreciate this definition, I find it hard to work with, because we are not given ...
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The projective line is defined over $\mathbb Q$

Notations: A variety $X$ over a field $C$ is an integral $C$-scheme such that the structure morphism $p: X\longrightarrow \textrm{Spec } C$ is separated and of finite type. We say that a variety $X$ ...
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Global generation of $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ and $\mathcal{E}$

I'm trying to prove that $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ is generated by global sections on $\mathbb{P}(\mathcal{E})$ if and only if $\mathcal{E}$ is generated by global sections ($\pi: ...
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Dimension of the closure of points on a scheme

I would like to prove the following fact. Let $X$ be a scheme, and $x\in X$. Show that $\text{dim}(\mathcal{O}_{X,x})=\text{codim}(\bar{\{x\}},X) $, with $\bar{\{x \}}$ the closure of the subset ...
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Reducibility of a Hilbert scheme in projective space

My question concerns the computation of the Hilbert scheme $\mathsf{Hilb}_{3}^{2x+1}$, which parametrizes all curves of degree $2$ and genus $0$ in $\mathbb{P}^{3}_{k}$, with $k$ algebraically closed. ...