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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5
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1answer
72 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 ...
0
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0answers
36 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
38 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
50 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
54 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
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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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
90 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 ...
0
votes
1answer
67 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 ...
1
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1answer
37 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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0answers
38 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
votes
0answers
77 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 ...
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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
votes
2answers
57 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
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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
25 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 ...
1
vote
1answer
40 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 ...
1
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1answer
25 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 ...
1
vote
2answers
50 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
56 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 ...
0
votes
0answers
16 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
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0answers
30 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 ...
1
vote
1answer
53 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 ...
0
votes
0answers
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
34 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 ...
1
vote
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 ...
3
votes
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 ...
0
votes
1answer
56 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 ...
3
votes
1answer
58 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. ...
2
votes
1answer
62 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 ...
1
vote
0answers
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 ...
0
votes
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 ...
1
vote
2answers
72 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 ...
0
votes
1answer
39 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
votes
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
votes
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 ...
0
votes
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 ...
2
votes
0answers
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
votes
1answer
42 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 ...
2
votes
1answer
37 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 ...
3
votes
1answer
105 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
votes
1answer
47 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
vote
1answer
34 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
votes
0answers
45 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
votes
0answers
44 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
votes
1answer
39 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 ...
1
vote
0answers
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
votes
1answer
104 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 ...
1
vote
1answer
29 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 ...