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Questions tagged [projective-schemes]

This tag is for questions relating to "projective scheme".

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Does the spectrum of the stalk of a point of a noetherian scheme contain an open subscheme about that point?

I am just proving the local consistency of Hilbert polynomials on flat families. All the proofs I have found till now, they use a map $ i_{y}: \operatorname{Spec}(\mathscr O_{Y,y})\rightarrow Y$ and ...
Anubhab Pahari's user avatar
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1 answer
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Counterexamples to theorems about projective space when Noetherian hypotheses are removed

It is well known that over a Noetherian ring $A$, $H^i(\mathbb{P}^n_A, \mathscr{F})$ is a finitely generated $A$-module for every coherent sheaf $\mathscr{F}$ of modules on $\mathbb{P}_A^n$ and $i \...
Emory Sun's user avatar
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Clarify on Propositions 3.36 and 3.38 in Qing Liu's algebraic geometry book

I have two questions, that I put together as they are related to the same topic, the projective scheme of a graded $A$-algebra $B$. Question 1. Do you confirm that points (a), (b) and (c) of ...
Ezio Greggio's user avatar
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1 answer
80 views

Uniqueness of the $A$-scheme structure on $\operatorname{Proj} B$

Let $B$ be a graded $A$-algebra, and consider the topological space $X:=\operatorname{Proj} B$. Call $(X,O_X)$ the usual structure of scheme on $X$. It is a structure of $A$-scheme as well, because $...
Ezio Greggio's user avatar
2 votes
0 answers
76 views

Serre duality for arbitrary sheaves?

Let $X$ be a connected proper Cohen-Macaulay scheme over a field $k$ of dimension $n$. Is there an example of an $\mathcal{O}_X$-module $\mathcal{F}$ on $X$ such that Serre duality $$ \operatorname{...
fish_monster's user avatar
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1 answer
63 views

Proj and affine base change: find canonical morphisms from Proj of a graded ring

This is Vakil's FOAG (ed. july 2023), exercise 17.2.A. Suppose $A \to B$ is a map of rings and $S$ is a $\mathbb{Z}_{\geq 0}$-graded ring over $A$. I am trying to find a canonical isomorphism $$\...
Marc-André Brochu's user avatar
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1 answer
64 views

Showing that $\mathbb P^n_A\rightarrow \operatorname{Spec}A$ is proper.

$\newcommand{Proj}{\operatorname{Proj}}\newcommand{Spec}{\operatorname{Spec}}\newcommand{\p}{\mathfrak p}\newcommand{\P}{\mathbb P}\newcommand{\Z}{\mathbb Z}$ Ok, so I understand what we are doing in ...
Chris's user avatar
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92 views

When is thickening of scheme Cohen-Macaulay?

I have the following question. All schemes below are assumed to be Notherian, and we can also assume that these schemes are varieties over some field. Suppose there is Cohen-Macaulay closed subscheme $...
abcd1234's user avatar
3 votes
1 answer
69 views

Example and intuition for Hartshorne exercise II.2.14(c)

Hartshorne's exercise II.2.14(c) has the reader prove that if $\varphi : S \to T$ is a graded homomorphism of graded rings, and $\varphi_d : S_d \to T_d$ is an isomorphism for all $d \geq d_0$, then ...
stillconfused's user avatar
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Projective morphisms are quasi compact and quasi separated or not

I am very new to algebraic geometry. I was reading about different kinds of morphisms between schemes. I am wondering about the following question: Is every projective morphism quasi-compact and ...
KAK's user avatar
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Isomorphism in codimension $1$ and Cartier divisor

We work over $\mathbb{C}$. Let $X,Y$ be normal projective varieties, and let $f: X\dashrightarrow Y$ be a birational map among them. Assume that $f$ is an isomorphism in codimension $1$, that is the ...
ark's user avatar
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0 answers
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$V_+(I) \subset V_+(J)$ implies that $J \cap B_+ \subset \sqrt{I}$ for $\operatorname{Proj} B$

I am trying to understand the proof in Liu's book that $V_+(I) \subset V_+(J)$ implies that $J \cap B_+ \subset \sqrt{I}$, where $B$ is a graded ring, $V_+(I) = \\{ \text{homogeneous prime ideals ...
stillconfused's user avatar
2 votes
1 answer
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Different Notions of Degree Agree

Vakil's (late February 2024 edition) notes define the degree of a line bundle $\mathscr{L}$ on projective curve $C$ over field $k$ to be $\chi(C, \mathscr{L}) - \chi(C, \mathscr{O}_C)$. On the other ...
Rough L's user avatar
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Closed Embedding into Projective Bundle Implies Projective Morphism

This is problem 17.3.A of Vakil's (late February 2024) Rising Sea. A morphism $\pi \colon X \to Y$ is defined as projective when there is an isomorphism of $Y$-schemes $X \cong \underline{\...
Rough L's user avatar
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What is $\operatorname{Proj}A[x]$ when $A[x]$ has the trivial grading?

$\newcommand{\Proj}{\operatorname{Proj}} \newcommand{\p}{\mathfrak{p}} \newcommand{\Spec}{\operatorname{Spec}}$ Let $A$ be a commutative ring, and $A[x]$ has the trivial grading where every element ...
Chris's user avatar
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2 votes
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44 views

For prime divisors $V,W\subseteq X$ in a smooth threefold $X$ and and integral curve $C\subseteq V\cap W$ with $i(V,W;C)>1$, do we have $V.C=W.C$?

Let $X$ be a smooth projective variety of dimension $3$ over an algebraically closed field. Let $V,W\subseteq X$ be prime divisors, i.e. two integral closed subvarieties of dimension $2$. Let $C\...
imtrying46's user avatar
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Quotient of Graded Rings $S \to S/f$ inducing Homeomorphism on Proj

Let $A$ be a Noetherian ring, and let $X$ be a closed subscheme of of $\mathbb{P}^r_A$. We define the homogeneous coordinate ring $S:=S(X)$ of $X$ for the given embedding to be $A[x_0, ..., x_r]/I$ (...
user267839's user avatar
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Coordinate Ring of the Cone of an Integral Projective Scheme a Domain (Generalization of Hartshorne's Ex II.5.14)

I have several questions around a part of Exercise 2.5.17(a) from Hartshorne's Algebraic Geometry (p 126). The setting: Let $k$ be a field, and let $X$ be a closed subscheme of of $\mathbb{P}^r_k$. We ...
user267839's user avatar
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A point inside a projective variety gives a (quasi) projective morphism

Let $X\to \operatorname{Spec } K$ be a projective integral variety. Let $L|K$ be a field extension and consider an $L$ point $x\colon \operatorname{Spec} L\to X$ in $X$ (morphism over $K$). Under ...
manifold's user avatar
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2 votes
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Structure of a scheme on $\operatorname{Proj} A$

Let $A$ be $\mathbb Z_{\geq 0}$ graded commutative ring. Now, in almost every resource I read, we define the structure a scheme on $\operatorname{Proj} A$, by taking the distinguished basis $D_+(f)$ ...
Chris's user avatar
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38 views

What went wrong computing the normal bundle of a section in projective bundle?

Suppose $X$ is a scheme (possibly smooth / $\mathbb C$), $E$ is a vector bundle of rank $e$ on $X$ with corresponding projective bundle $p:\mathbb PE \to X$, and $s: X \to \mathbb PE$ is a section. I ...
red_trumpet's user avatar
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On the homeomorphism $U_f\subset \operatorname{Proj}A\leftrightarrow \operatorname{Spec}(A_f)_0$

Let $A$ be a $\mathbb Z_{\geq 0}$ graded ring. Then we have that $\operatorname{Proj}A$ is the set of homogenous prime ideals which do not contain the irrelevant ideal $A_+$. We put a topology on this ...
Chris's user avatar
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2 votes
1 answer
123 views

On the bijection between homogenous prime ideals of $A_f$ and prime ideals of $(A_f)_0$

Let $A$ be a $\mathbb{Z}^{\geq0}$ graded ring, and $f$ an element of positive degree. It is well known that the homogenous prime ideals of $A_f$ are in bijection with prime ideals of $(A_f)_0$, that ...
Chris's user avatar
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0 answers
82 views

Inverse image of a $0$-dimensional subscheme under a finite morphism

Let $f\colon X\rightarrow Y$ be a finite surjective morphism of complex projective varieties of the same dimension. We may suppose $Y$ is smooth. Let $Z\subset Y$ be a $0$-dimensional closed subscheme ...
MaryMoon's user avatar
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Reduced, irreducible and integral schemes

I am studying properties of schemes and I have the following problem. Let $R$ be a ring, $S=Spec(R)$, $n\in\mathbb{N}$ an integer. Show that the following are equivalent: $S$ is reduced (resp ...
Mario's user avatar
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2 votes
2 answers
104 views

[Cor 5.3.24 in Qing Liu]If $X$ is a projective scheme over a DVR $A$, then any coherent sheaf of ideal of $\mathcal{O}_X$ is flat over $A$?

The problem comes from the proof of Corollary 5.3.24 in Qing Liu's book Algebraic Geometry and Arithmetic curve. Let $\DeclareMathOperator{\Spec}{Spec}S=\Spec A$ be the spectrum a discrete valuation ...
Z Wu's user avatar
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1 answer
59 views

Is projective morphism over a discrete valuation ring always flat?

Let $A$ be a discrete valuation ring with a uniformizing parameter $t$. Let $f:X\to \DeclareMathOperator{\Spec}{Spec}\Spec A$ be a projective morphism (i.e. $X$ is a closed subscheme of $\mathbb{P}^...
Z Wu's user avatar
  • 1,785
1 vote
0 answers
130 views

Effective divisors of $X$ with $\operatorname{Pic}X\cong \mathbb Z$

Consider $X$ a projective variety with $\operatorname{Pic}X\cong \mathbb Z$. So $\mathcal L\cong \mathcal O_X(m)$ for any invertible sheaf $\mathcal L$ and for some $m\in \mathbb Z$. I want to show ...
raisinsec's user avatar
  • 463
0 votes
0 answers
74 views

image of Segre embedding is cut out (scheme-theoretically) by vanishing minors

I am doing exercise 10.6.B of Vakil's FOAG (July 31, 2023 version). Basically, I need to show that the image of the Segre embedding is cut out by equations so that the matrix \begin{bmatrix} a_{00}&...
Mizutsuki's user avatar
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0 answers
44 views

Morphism of schemes and closed points [duplicate]

I apologize for the vagueness of my question. In the case of morphisms between schemes (that locally look like an $A$-algebra maybe), the textbook (Vakil's FOAG, July 31, 2023 version) often uses the &...
Mizutsuki's user avatar
  • 494
0 votes
0 answers
102 views

What is $k((t))$?

Let $k$ be a field, what is the notation $k((t))$? I know that $k[[t]]$ is the ring of formal power series, but I've never seen $k((t))$. Sorry for such a stupid question. For context, I'm supposed to ...
Chris's user avatar
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2 votes
0 answers
85 views

Problem showing the saturation map of a graded ring f.g. in degree 1 induces an isomorphism of projective schemes and O(1) (The Rising Sea 15.6.G)

I've had problem working on this exercise from the July 31, 2023 version of Vakil's The Rising Sea: 15.6.G. Exercise. Show that the map of graded rings $S_\bullet \to \Gamma_\bullet\widetilde{S_\...
AprilGrimoire's user avatar
2 votes
0 answers
46 views

global section of a B-scheme defines a morphism to projective $B$-scheme

This is problem 7.3.O in Vakil's AG notes (July 31, 2023 version). I am not sure how to define the map locally. Problem 7.3.O. Let $B$ be a ring and $X$ a $B$-scheme. Suppose $f_0,\cdots,f_n$ are n+1 ...
Mizutsuki's user avatar
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0 votes
0 answers
24 views

Example of a family of ample divisors $\{A_m\}_{m\geq 1}$ on a smooth projective variety $X$ such that $mA_m$ has a basepoint?

I know this famous example due to Kollár: Take $E$ an elliptic curve, and on $E\times E$ consider a horizontal fiber $F_1$, a vertical fiber $F_2$ and the diagonal $\Delta$. Let $X$ be a triple cover ...
imtrying46's user avatar
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0 answers
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Vanishing ideal of a subset of $\operatorname{Proj} S_\bullet$

I am reading FOAG by Vakil. In 4.5.K, I am asked to define $I(Z)$ where $Z\subseteq\operatorname{Proj}S_\bullet$ and $S_\bullet$ is a graded ring. The immediate idea of mine is to define it as $$I(Z):=...
Mizutsuki's user avatar
  • 494
0 votes
1 answer
56 views

Intuition of projective line via glueing

I am reading the construction of projective line (4.4.6) following Prof. Vakil's Algebraic geometry notes FOAG. However, I fail to understand few lines during the discussion. Let $X=\operatorname{Spec}...
Mizutsuki's user avatar
  • 494
0 votes
0 answers
41 views

Does Kleiman's Theorem work on quasi-projective varieties?

Let $X$ be a projective variety and let $C_1,\ldots,C_n\subseteq X$ be some integral curves contained in $X$. Let $D$ be a divisor such that $D.C\geq 0$ for all curves $C\notin\{C_1,\ldots,C_n\}$. ...
imtrying46's user avatar
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1 vote
0 answers
26 views

For $D$ very ample on a threefold $X$, is it true that $D^2.S\geq(\operatorname{mult}_x D)^2\cdot\operatorname{mult}_x S$?

Let $X$ be a smooth projective variety over an algebraically closed field $k$, and suppose $\dim X=3$. Let $D$ be an effective, very ample divisor and let $x\in X$ be a point. Then is it true that for ...
imtrying46's user avatar
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0 votes
0 answers
34 views

Semistability and Proj quotients

Before to make my question, I'm going to introduce all necessary notions. Let $G$ be a linearly reductive group (i.e. each rational representation is completely reducible) acting regularly on an ...
wood's user avatar
  • 301
3 votes
1 answer
96 views

Help understanding what is the projection in problem II.6.3 (a) Hartshorne

In algebraic geometry by Robin Hartshorne, exercise II.6.3.a is written as follows Cones. In this exercise, we compare the class group of a projective variety $V$ to the class group of its cone (I, ...
Elad's user avatar
  • 3,247
4 votes
0 answers
57 views

For a Fano threefold $X$, is there a lower bound for $S^3$ where $S\subseteq X$ is an irreducible surface?

Let $X$ be a smooth projective variety over an algebraically closed field $k$, and assume that $-K_X$ is ample, i.e. $X$ is Fano. If $\dim X=2$, then we have that $C^2\geq -1$ for all curves $C\...
imtrying46's user avatar
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2 votes
1 answer
158 views

If a resolution $f:Y\to X$ satisfies $R^if_*\omega_Y=0$ and $R^if_*\mathcal{O}_Y=0$ for all $i>0$, then do we have $f_*\omega_Y=\omega_X$?

Let $X$ be a normal projective variety over an algebraically closed field of arbitrary characteristic (but I'm mainly interested in positive characteristic). Assume that $X$ has rational singularities,...
imtrying46's user avatar
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1 vote
0 answers
52 views

Sheaf cohomology free module

Let $X \rightarrow$ Spec$A$ be a scheme over a ring $A$. I know that if $X$ is a projective scheme over $A$, and $A$ is Noetherian. $H^p(X, \mathcal{O}_X)$ is afinitely generated A-module for all $p \...
Andarrkor's user avatar
  • 611
2 votes
1 answer
82 views

Automorphism of $k$ varieties is induced by automorphism of $\mathbb{P}_k^n$ (exercise 5.1.24 in Q Liu)

The exercise is given below. $5.1.24.$ Let $X$ be an algebraic variety over a field $k$. Let $f : X \to Y := \mathbb{P}^{n}_{k}$ be a morphism of algebraic $k-$varieties. Let $\mathcal{L} = f^{*}\...
Analyse300's user avatar
1 vote
1 answer
207 views

Two definitions of scheme theoretic dual projective space

Vakil’s FOAG gives the definition of the dual peojective space via introducing new indeterminates: (sorry that I have to quote it as a screenshot) And the answer in Scheme theoretic dual of $\mathbb ...
onRiv's user avatar
  • 1,268
4 votes
1 answer
204 views

Pushforward of the Segre embedding in K-theory

Fix $n$, $m\ge 1$, and let $d=\binom{m+n}{m}$ and $N=mn+m+n$. Consider the Segre embedding $\sigma:\mathbb{P}^m\times \mathbb{P}^n \hookrightarrow \mathbb{P}^{N}$, which has degree $d$. I'm trying to ...
Alvaro Martinez's user avatar
3 votes
1 answer
127 views

Why are these two projective schemes isomorphic?

From Ravi Vakil's Foundations of Algebraic Geometry, exercise 7.4.C: Let $R = k[x, y, z]/(xz, yz, z^2)$. Show that $\operatorname{Proj} R \cong \mathbb{P}_k^1$ (as schemes). My thought was to induce a ...
TasmanFell's user avatar
0 votes
0 answers
48 views

defining ideal of an ellipse and a curve in $\mathbb{P}^2$

I would like to find an ideal $I=(f,g)\subset k[x_0,x_1,x_2]$ with $V(I)\subset \mathbb{P}^2$ having 6 points where $\deg f=2$ (which is the homogeneous defining polynomial of an ellipse) and $\deg g=...
Mary's user avatar
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1 vote
1 answer
207 views

Dimensions of cohomology of ideal sheaf

This question follows a previous question Sheaf morphism from closed subscheme is a closed immersion, it's just another part so I'll recall everything. For $K=\bar{K}$ a field consider $X=\mathbb P^...
raisinsec's user avatar
  • 463
2 votes
1 answer
100 views

Sheaf morphism from closed subscheme is a closed immersion

For $K=\bar{K}$ a field consider $X=\mathbb P^1_K$, $Z=\{P_1,\dots,P_n\}\subseteq X$ closed points. Give $Z$ the reduced induced closed subscheme structure and write $\iota:Z\to X$ the closed ...
raisinsec's user avatar
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