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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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 $1$. Let $\mathfrak ...
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34 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
73 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 ...
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33 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 ...
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59 views
+100

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 ...
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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: ...
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47 views

The sheaf (of stalks) of meromorphic functions, why don't we use a more natural definition?

If $A$ is a commutative ring with $1$, let's denote with $R(A)$ the set of regular elements of $A$. Let $(X,\mathcal O_X)$ be a locally Noetherian scheme, then the sheaf (of stalks) of meromorphic ...
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2answers
55 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))$. ...
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1answer
35 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 ...
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24 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
35 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
23 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
48 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 ...
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1answer
54 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 ...
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15 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 ...
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25 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
52 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 ...
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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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33 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 ...
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1answer
41 views

Being a regular embedding is an open condition for locally Noetherian schemes

Exercise $8.4.G$ of Vakil's algebraic geometry notes asks us to prove: If a locally closed embedding $\pi:X\rightarrow Y$ of locally Noetherian schemes is a regular embedding at $p$, then it is a ...
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2answers
105 views

Intersecting two pencils of plane curves

In $\Bbb{P}^2$, let $D_1,D_2$ be two curves of degree $d_1,d_2$ respectively. Choose two pencils $|D_1(t)|\subset|D_1|$ and $|D_2(t)|\subset|D_2|$ (free of fixed components) parametrized by the same ...
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1answer
55 views

Scheme $\{x\in \mathbb{R}^{d}:|x^{2}|\leq \varepsilon\}$ glued by which affine schemes?

I am trying to see why the $\{x\in \mathbb{R}^{d}:|x^{2}|\leq \varepsilon\}$ (equipped with a ring, say sheaf of continuous functions on that set) is a scheme i.e.to which affine schemes is it locally ...
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1answer
56 views

Differential forms on a projective curve: two constructions

Let $X$ be a projective curve on a perfect field $k$ ($k$-scheme integral, separated, of finite type) and let $K$ be the function field of $X$. Let's compare the following two constructions: 1. ...
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1answer
57 views

if the Hilbert polynomial of coherent sheaf is constant number,then locally free?

I'm reading "The Geometry of Moduli Spaces of Sheaves" (Huybrechts Lehn). I want to prove [$Quot_{X/S}(H,l)=Grass_S(H,l)$]where X=S and l is a number. Let T be a S-scheme and [ρ:$H_T \rightarrow ...
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32 views

Checking a global property over the closed point of a local ring

Let $k$ be a field and let $T=\textrm{Spec }R$ be the spectrum of a local $k$-algebra $(R,\mathfrak m)$. Let $X\to T$ be a proper flat map from a $k$-scheme $X$, and suppose given a morphism of ...
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1answer
25 views

Details about the definition of: “deformation of a family”

Let $f:X\to Y$ be a flat, surjective morphism of $k$-schemes with connected fibres i.e. $f$ is a family. Definition: Let $T$ be a $k$-scheme. A deformation of $f$ (over $T$) is a family $g:\mathfrak ...
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2answers
70 views

Commutation of pushforward and pullback along cartesian squares

I am sure this is well-known, but I cannot find a reference. Consider a cartesian square $$\require{AMScd} \begin{CD} P @>{v}>> X\\ @V{g}VV @VV{f}V \\ Z @>{u}>> Y \end{CD}$$ where ...
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1answer
37 views

Intersection of locally closed subschemes

Let $S$ be a Noetherian scheme and $Y,Z$ two locally closed subschemes. What is the scheme theoretic intersection of $Y$ and $Z$. I am asking because in Mumford's "Lectures on curves on an algebraic ...
3
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1answer
41 views

sheaf defined on the support of a sheaf

Let $X$ be a topological space and $\mathcal{F}$ a sheaf on $X$. Let $Y$ be the support of $\mathcal{F}$ (Hartshorne, exercise 1.14), i.e., $Y = \left\{ P \in X: \, F_P \neq 0\right\}$. Is it true ...
3
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1answer
40 views

on the necessity of gluing conditions

Suppose we are given a family of schemes $\left\{X_i\right\}_i$, with $U_{ij}$ open in $X_i$ such that there exists isomoprhism $\phi_{ij}: U_{ij} \rightarrow U_{ji}$. Why do we need the condition ...
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1answer
49 views

Maps to $\mathbb{P}^1$ induced by rational functions.

I am reading from Hartshorne, Corollary II.6.10 page 138. Given a nonsingular (is this necessary?) curve $X$ over a field $k$, let $f\in K(X)^*\setminus k$. Then the inclusion of fields ...
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67 views

Visualize the affine to projective map $\mathbb{A}^{n+1}_k-\{O\}$ to $\mathbb{P}^n_k$

Classically, the natural map $\mathbb{A}_\mathbb{C}^{n+1}-\{O\} \rightarrow \mathbb{P}^n_\mathbb{C}$ maps every point in $\mathbb{A}_\mathbb{C}^{n+1}-\{O\}$ to the affine line that contains it. For ...
3
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1answer
41 views

Total space of a finite rank locally free sheaf, Vakil's 17.1.4 & 17.1.G

If $X$ is a scheme and $\mathcal{F}$ is a locally free sheaf of rank $n$, then in Vakil's book the total space of $\mathcal{F}$ is defined to be $Spec(\text{Sym}^\bullet \mathcal{F}^\vee)$, the ...
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1answer
36 views

Is the tangent bundle of a smooth $\Bbbk$-scheme an infinitesimal extension of it?

An infinitesimal extension of an affine scheme $\operatorname{Spec}R$ is a surjection $\hat R\twoheadrightarrow R$ with nilpotent ideal. The scheme case is defined by globalizing. I read somewhere on ...
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1answer
100 views

Question about proof that the Grassmannian is a parameter space

Edit: If it is easier to give a reference where this is written down in detail, I would gladly accept that as an answer. Fix a base scheme $B$, and fix $n$ and $k$ with $k<n$. In section 28.3 ...
4
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1answer
45 views

closed immersion onto an affine scheme - showing affineness

Let $A$ be a ring, $X=\operatorname{Spec}A$ and $f: Z \rightarrow X$ a morphism of schemes such that i) $f$ is a homeomorphism of topological spaces and ii) $f^{\#}:\mathcal{O}_X \rightarrow f_* ...
1
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1answer
32 views

local equation of a divisor, localization and local ring at a point

Let $X$ be a regular, proper surface (integral, separated, of fin. type) over a perfect field $k$ and let $Z\subset X$ be an integral irreducible subscheme. Moreover $\eta$ is the generic point of $X$ ...
3
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40 views

Cohomology of structure sheaf of abelian variety

Let $X$ be an abelian variety over $\mathbb{C}$ of dimension $n$. Consider the structure sheaf $O_X$. It's Euler characteristic is zero, because $\chi(O_X)= (O_X^n)/n!$. And the self intersection of ...
3
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0answers
43 views

The hypothetical fibre over an infinite prime in $\mathbb{A}^1_{\mathbb{Z}}$

This is just a speculative question I've been wondering about; I hope that others may find it interesting too, but if it's too vague please let me know! We can picture $\mathbb{A}^1_{\mathbb{Z}} = ...
2
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1answer
35 views

Isotrivial family: different definitions

Let $f:X\to B$ a flat morphism of varieties over an algebraically closed field $k$. If $f$ is flat and with connected fibres we say that $f:X\to B$ is a family over $B$. In literature you can find ...
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35 views

Factorization through diagonal morphism

The following question came up when reading Hartshorne's proof of the valuative criterion. Let $f\colon X\to Y$ be a $Y$-scheme and let $K$ be a field. Let $g\colon \operatorname{Spec}K\to X \times_Y ...
2
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1answer
101 views

Support and stalks at generic points

Let $X$ be an noetherian scheme, $Y$ an irreducible closed subscheme of $X$ with generic point $y$ and $\mathscr G$ a coherent sheaf of $\mathscr O_X$-modules. Consider the following statement: If ...
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1answer
27 views

Trace of a linear system on a smooth projective variety

In Hartshorne's Algebraic Geometry, Chapter 2, section 7, the trace of a linear system is defined as follows. Let $i:Y\hookrightarrow X$ be a closed immersion of nonsingular projective varieties over ...
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1answer
37 views

Stalks of ringed space

Let $X$ be a locall ringed space (more narrowly a scheme, if you like) and $A=\Gamma(X,\mathcal{O}_X)$ its ring of global sections. Given a point $x\in X$, is there a prime ideal $p$ of $A$ such that ...
4
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1answer
64 views

Geometrical meaning of $\text{Spec}\,\widehat{\mathcal O_{X,x}}$

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 ...
2
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1answer
63 views

Pullback of an invertible sheaf through an isomorphism

Consider an isomorphism of schemes $(f,f^{\#})(X,\mathcal{O}_X)\to(Y,\mathcal{O}_Y)$. Moreover let $\mathcal F$ be an invetible sheaf on $Y$ and let $f^{*}\mathcal{F}$ be its pullback. Is it true ...
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1answer
60 views

Preimage of diagonal subscheme is a closed subscheme

Let $\alpha: X\to S$ and $\beta:Y\to S$ be $S$-schemes and let $\Delta\subseteq Y\times_S Y$ be the diagonal subscheme defined as follows (following Eisenbud-Harris): for each affine open subscheme ...
2
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0answers
51 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 ...
4
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1answer
173 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? ...
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1answer
51 views

Conormal bundle of Cartier divisors

Given any closed immersion of schemes $i:Z\to X$ defined by a sheaf of ideals $\mathcal{I}$ on $X$, apparently the conormal bundle is $\mathcal{C}_{Z/X}:= {\mathcal{I}}/{\mathcal{I}^2}$ "seen as a ...