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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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
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1answer
70 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
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1answer
65 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
50 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
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 ...
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0answers
40 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 ...
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2answers
188 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
49 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
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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 ...
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1answer
194 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
85 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
36 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
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 $mK_S$ ...
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0answers
61 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 ? ...
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1answer
75 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 ...
0
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0answers
45 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
78 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
68 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
52 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 ...
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1answer
43 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
63 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
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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
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1answer
59 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 ...
7
votes
1answer
126 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 ...
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0answers
38 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 $ ...
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0answers
31 views

An open subset of a scheme, schematically dense.

Let $U$ be an open subset of a scheme $X$, and let $Y$ be its complement. $U$ is called schematically dense, if for any other open set $V$ of $X$ the restriction map : $ \rho : \Gamma ( V , ...
3
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0answers
77 views

Sheaves with surjective map from the ring of global sections to stalks

Let $M$ be a smooth manifold, $\mathscr O_M$ the sheaf of vector fields. Then, by an argument involving partition of unity, we can show that the map $$\mathscr O_M(M)\to\mathscr O_{M,x}$$ is ...
6
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1answer
86 views

Functoriality of Morphisms to Affine Scheme

It is known that $\text{Mor}(X,Y)$ is in bijective correspondence with $\text{Hom}(\mathcal{O}_Y(Y), \mathcal{O}_X(X))$, provided $Y$ is an affine scheme. I do not understand a small but crucial part ...
1
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1answer
93 views

What does the Grothendieck's period conjecture mean?

I would like to know how is the Grothendieck's period conjecture about algebraic cycles, defined explicitly ? and, what link has it with the Hodge conjecture for smooth complexe algebraic projective ...
0
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0answers
35 views

Projective general linear group over a ring

For $n$ a positive integer, I would like to define $PGL_n$ as a group scheme. One candidate is $X:=\mathrm{Proj}\big(\mathbb{Z}[x_{ij}:1\leq i,j\leq n]\big)\,\backslash \,\mathbb{V}(det)$, but I am ...
2
votes
1answer
33 views

subschemes and subobjects

In scheme theory, there are terms "open subscheme" and "closed subscheme", and in category theory, there is a term "subobject". I want to know relation between them. Are open subschemes and closed ...
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0answers
69 views

Dimension of integral schemes of locally finite type over a field

In Exercise 3.20 of Algebraic Geometry, Hartshorne makes several claims about the dimension of an integral scheme of finite type over a field. For instance, he claims that the dimension is equal to ...
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0answers
36 views

Isomorphism (?) of polynomial rings with different gradings and their Proj

It's a (relatively) well-known fact that, if $a_0,\ldots,a_n\in\mathbb{N}$ share a common factor $d\in\mathbb{N}$ then $$\operatorname{Proj}k_a[x_0,\ldots,x_n] \cong ...
0
votes
1answer
88 views

map between projective schemes induced by rational points

Given a map between $\mathbb P^n_{\mathbb C}$ and itself, given by $$ [x_0:\dots:x_n] \mapsto [p_0(x):\dots:p_n(x)]$$ where $p_i$ are homogeneous polynomials of the same degree, how do I find the ...
2
votes
0answers
24 views

Is this always quasi-projective?

Let $\phi:(X,x)\to (T,t)$ be a morphism of germs of algebraic schemes over $\mathbb{C}$. How can we always (without "quasi-projective") have an open embedding of the germ $(X,x)$ into a closed ...
1
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0answers
42 views

Is it true that $\mathrm{codim}(Z \cap Y, Y) \leq\mathrm{codim}(Z,X)$ for closed subsets $Z,Y$ of a scheme $X$?

This might be a standard thing but I'm not so sure. Say $X$ is an irreducible affine scheme, $Y$ is an irreducible closed subset of $X$, and $Z$ a closed subset of $X$. If $Z \cap Y \neq ...
1
vote
1answer
106 views

normalization of a curve with a node is not flat

Given the ring $$A = \frac{K[x,y]}{y^2-x^2(x+1)}$$ I know that its normalization is $K[t]$, where $$x\mapsto t^2-1\qquad y\mapsto t^3-t$$ I have to show that the normalization map is not flat. I know ...
1
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0answers
116 views

When does a principal divisor have degree 0?

We know that on a smooth curve over an algebraically closed field principal divisors have degree 0. This holds true also for the projective space over any field (in this case equality holds too). My ...
0
votes
1answer
21 views

cardinality of fiber in a finite morphism of schemes

Given $f:X\to Y$ a finite morphism of schemes, with $Y$ locally noetherian, let's take a point $q\in Y$, and an affine noetherian open set $$q\in U=Spec(B)\subseteq Y$$ Then ...
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0answers
61 views

Tangent space of an algebraic variety $ X $ in a point $ x \in X $.

Let $X$ be an algebraic variety, and $ \mathcal{O}_{X,x} $ its local ring at a point $x \in X$, and $ \mathfrak{m}_x $ its maximal ideal. Let set $ k_x = \mathcal{O}_{X,x} / \mathfrak{m}_x $ the ...
3
votes
0answers
94 views

When is “number of points in the fiber” semicontinuous?

Let $f:X \rightarrow Y$ be a finite morphism of schemes. For $y \in Y$ let $n(y)$ be the cardinality of the set-theoretic fiber $f^-1(y)$. This is finite since $f$ is finite. I think that the ...
0
votes
0answers
41 views

Global section on product of sheafifications

Say I have two presheaves $\mathcal F$ and $\mathcal G$ of $\mathcal O_X$-modules on a scheme $X$. Let $\mathcal F^+\!, \mathcal G^+\!$ be their respective sheafifications. We then have a commitative ...
0
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0answers
48 views

Simple question about closed immersions

I would like to know if I understand closed immersions correctly. Let $f:X \rightarrow Y$ be a morphism of schemes (or locally ringed spaces in general). The morphism $f^\sharp:\mathscr{O}_Y ...
2
votes
1answer
92 views

Every commutative ring is a quotient of a normal ring?

In the book Étale cohomology by Milne I found on p. 37 (in the context of constructing the henselization of a local ring) the following claim: "Every ring is a quotient of a normal ring". The same is ...
2
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0answers
83 views

About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$

I've been trying to find some useful categorical facts about the category of schemes, locally ringed spaces and ringed spaces (that I shall denote by $\text{Sch}$, $\text{LRS}$ and $\text{RS}$ ...
1
vote
1answer
102 views

Spectrum of the Ring of Continuous Functions on a Space

I was wondering when exactly we can recover the topological space, $X$, from its ring of continuous functions into $\mathbb C$ (or some sort of sufficient topological group). For any topological ...
0
votes
1answer
123 views

Are Monomorphisms injective?

In the categories of topological spaces, rings, groups and sets I know that a morphism is a monomorphism iff it's injective. Things are different for schemes. In fact I know that a scheme injective ...
1
vote
1answer
48 views

Lifting of partial section

Does someone know if the following is true, and if so could you provide a reference. Let $X$ be a smooth $S$-scheme, let $Z$ be a closed subscheme of $S$, and let $t : Z \to X$ be an $S$-morphism. ...
5
votes
1answer
99 views

Uniqueness of the structure sheaf

Given a ring $A$ and his Spectrum $X=Spec(A)$ seen as a topological space with the Zariski topology, it's possible to build a sheaf on $X$ satisfying the conditions $O_X(X)=A$ $O_X(X_f)=A_f$ where ...