Questions tagged [projective-schemes]

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

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14 views

injective sheaves on a projective scheme cannot be coherent [duplicate]

Let $X$ be a projective scheme. If it helps (e.g. gives way to a short/elegant answer) fix a base field $k$ and assume smoothness. It is often said that the category of coherent sheaves over $X$ does ...
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23 views

If $R_0 \not\cong S_0$ but $\exists n \in \mathbb N$ with $R_d \cong S_d$ for all $d \ge n$, then $\operatorname{Proj}R \cong \operatorname{Proj}S$? [duplicate]

Let $R=R_0 \oplus R_1 \oplus \cdots$ and $S=S_0 \oplus S_1 \oplus \cdots$ be finitely generated graded rings with $R_0 \not\cong S_0$. If there exists a positive integer $n$ such that $R_d \cong S_d$ ...
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1answer
48 views

Calculating normalization of the projective cusp $V(x^2z-y^3)$

I've understood the normalization of the affine cusp $V(x^2-y^3)\subset\mathbb{A}^2$ is just $\phi:\mathbb{A}^1\to \operatorname{Spec}(\mathbb{C}[x,y]/(x^2-y^3))$ coming from the algebra map $$ \phi^*:...
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1answer
58 views

Global sections of $\operatorname{Proj} S$ when $S$ is an integrally closed domain

Let $S=\bigoplus_{n \geqslant 0}S_n$ be a finitely generated $\mathbb{Z}_{\geqslant 0}$-graded algebra over $\mathbb{C}$ with $S_+ \neq 0$, here $S_+:=\bigoplus_{n>0}S_n$. Let us also assume that $...
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1answer
73 views

Confusion about a standard rational map $\operatorname{Proj} A[x_0,\dots,x_n] \dashrightarrow \operatorname{Proj} A[x_0,\dots,x_{n-1}]$.

From Vakil's FOAG: Definition 6.5.1: A rational map $\pi$ from $X$ to $Y$, denoted $\pi: X \dashrightarrow Y$, is a morphism on a dense open set, with the equivalence relation $(\alpha: U \to Y) \sim ...
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2answers
37 views

Are there any homogeneous prime ideals of $k[x,y,z]/(xz,yz,z^2)$ not containing $z$?

Let $R=k[x,y,z]/(xz,yz,z^2)$. I would like to find the homogeneous prime ideals of $R$ not containing $z$. In other words, I would like to find the points of $D_+(z) \subset \operatorname{Proj} R$. ...
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44 views

Is the quasi-projective scheme morphism separated and finite type?

Is the quasi-projective scheme morphism separated and finite type? In Hartshorne's book, it is shown when it is noetherian, but I was wondering if this assumption is necessary.
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13 views

Sections of Serre's twisting sheaves

My understanding of Serre's twisting sheaves is quite shaky. If $B = \bigoplus_{d\geq 0}B_d$ is a graded module, $X = \operatorname{Proj}B$, and $f\in B_1$, then I know that $\mathcal{O}_X(n)|_{D_+(f)}...
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164 views

Asymmetric Numeral Systems Arithmetic

Recently, I've read about the ANS, where each symbol get described on a basis with size that is relative to its frequency. I was wondering if it is possible to obtain meaningful arithmetic operations ...
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54 views

Proof of Fundamental Theorem of Elimination Theory, from Vakil's FOAG

I've been working the last section of Chapter 7 of Vakil's FOAG, namely the one on the Fundamental Theorem of Elimination Theory. I'm not entirely sure if I've understood the proof correctly, ...
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30 views

What is a Trivial Family

Let $S\subset \mathbb{P}^{d}$ be a smooth projective surface and let $\mathbb{P}^{d*}$ be the dual projective space of $\mathbb{P}^{d}$ (remember, $\mathbb{P}^{d*}$ parametrizes hyperplanes in $\...
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19 views

Hypersurfaces of Proj and quasi-coherent sheaf

Let $F$ be a homogeneous polynomial of degree $d$ and consider the closed subscheme $ X = Proj(k[x_0, \dots, x_n] / (F) ) \subset \mathbb{P}^n_k$. Now we have an exact sequence: $ 0 \longrightarrow \...
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2answers
74 views

Relative proj base ring, closed immersions via Vakil

This is an annoying question and I apologize in advance. Vakil defines a relative proj sheaf of algebras on $X$ to be such that there is a cover of $\operatorname{Spec}(A)$ s.t above each we have $S^A$...
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1answer
71 views

$k$-rational points of $\mathbb{P}^n_k$

Let $k$ be a field. Show that the $k$-rational points of $\mathbb{P}^n_k=\operatorname{Proj}[x_0,x_1,...,x_n]$ are in bijective correspondence with the points $[a_0,a_1,...,a_n], a_i\in k$ of the ...
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75 views

Proj of Almost Same Graded Rings are Isomorphic (Exercise from Vakil's FOAG)

I'm trying to solve Exercise 6.4.F from Vakil's FOAG: 6.4.F. Exᴇʀᴄɪsᴇ.$\quad$Show that if $R_\bullet$ and $S_\bullet$ are the same finitely generated graded rings except in a finite number of nonzero ...
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1answer
112 views

Show that different notions of projectivity are distinct

Let $X\longrightarrow S$ be a morphism of (in my case: Noetherian) schemes. Then $X$ is called projective over $S$ if $X$ is isomorphic to a closed subscheme of $\mathbb{P}^{n}_{S}$. By a slightly ...
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1answer
47 views

When does a fibration has the geometrically integral generic fiber?

Backgrounds: Let $k$ be an algebraically closed field. A fibration $\pi:X \to B$ is a surjective morphism from a surface(= an integral smooth projective scheme of dimension $2$ over $k$) to a integral ...
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1answer
39 views

Are all embeddings of varieties essentially the same?

Let $X$ be a nice projective variety, suppose we have two different embeddings to projective space $e_1, e_2$ of $X$ in $P^{n_1}, P^{n_2}$. Are there always embeddings $E_1: P^{n_1} \to P^{N}, E_2: P^{...
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40 views

Hyperplane $\Phi_X$ in Chow construction (Algebraic Geometry by Joe Harris)

I'm trying to figure out what Harris wanted to say in following construction, called Chow construction (Algebraic Geometry by Joe Harris, p. 269): The first construction of the parameter space for ...
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459 views

Exercise 1.24 from Joe Harris' Algebraic Geometry: First Course

I have a question about solvability of Exercise 1.24 (p. 14) from Joe Harris' Algebraic Geometry: A First Course and correctness of following 'synthetic' construction which according to the book (or ...
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51 views

Ideal sheaf of a point in $\mathbb{P}^1$

Let $k$ be a algebraically closed field, $X = \mathbb{P}^1_k = \mathrm{Proj} k[x,y]$ the projective space and $p = (ax+by) \in X$ a closed point. Let $\mathcal{I}_p = \ker( {\mathcal{O}_X \to p_* \...
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1answer
81 views

Injection into double dual is isomorphism away from codimension $\geq 2$ subscheme

Let $X$ be a quasi-projective integral scheme and let $F$ be a torsion-free coherent sheaf over $X$. Then there is an injective map $$\phi\colon F\hookrightarrow F^{\vee\vee}.$$ Moreover, in my ...
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36 views

Projection as limit of flat family (Hartshorne III.9.8.3)

In example III.9.8.3, Hartshorne considers the projection $\varphi:\mathbb{P}^{n+1}\setminus P\to\mathbb{P}^n$ (projective space over an algebraic closed field $k$) where $P=(0,\ldots,0,1)$ ie $\...
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1answer
117 views

"Stacks project" - construction of Segre embedding - a clarification

I am currently trying to understand the construction in the Stacks project of the Segre embedding: https://stacks.math.columbia.edu/tag/01WD. They use the correspondence between invertible sheaves and ...
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1answer
37 views

Projectivity of a morphism between schemes implies the existence of an exact sequence

If $X\longrightarrow S$ is projective over a Noetherian scheme $S$ and $\mathsf{E}$ is a coherent sheaf on $X$, then for any affine $U\subseteq S$ there exists an exact sequence \begin{equation*} E_{0}...
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1answer
54 views

A more concrete description of the functor of points of projective space.

The functor $\mathbb{P}^n_{\mathbb{Z}}: \text{CRing}\to \text{Sets}$ corresponding to projective $n$-space over $\mathbb{Z}$ has the following description: For each commutative ring $A$, $\mathbb{P}^...
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145 views

Question on Smooth Completion of curves

Let $X$ be a smooth irreducible curve of finite type $C$ over a separably closed field $k$. For such curves is known that they always have a smooth compactification, that is there exist a smooth ...
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1answer
60 views

Smoothness is a geometric property

I read that "smoothness is a geometric property" meaning that if $k\subset K$ is a field extension and $X_k$ is a scheme over $k$ then $X_k$ is smooth over $k$ if and only if $X_K = X_k \...
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2answers
106 views

$\mathbb{P}^1$ isomorphic to conic in $\operatorname{Proj}(K[x,y,z])$ context

I have recently started to study schemes and I found my self on the follow situation: I want to prove that $\mathbb{P}^1$ is isomorphic to a conic. I have used the morphisim $\mathbb{P}^1\...
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1answer
41 views

Irreducible components of the closure of a locally closed subset

I would like to ask for a reference of the following fact: the irreducible components of the closure of a locally closed subset are the closures of the irreducible components of the subset. Thank you ...
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53 views

Projection $\pi: \mathbb{P}^n \dashrightarrow \mathbb{P}^{n-1}$ away from center $p$ maps

Let $X \subset \mathbb{P}^n$ be a projective variety and $p \in \mathbb{P}^n$ an arbitrary point not contaned in $X$. The rational projection with center $p$ map $\pi: \mathbb{P}^n \dashrightarrow \...
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1answer
78 views

Quotient of $ \operatorname{Proj} A$ by the action of a finite group

Let $X$ be $ \operatorname{Proj}(A)$ for some graded ring A, and let $G$ be a finite group acting on $A$ with morphisms of graded rings; consequently $G$ acts on $X$. I know I can write $X = \bigcup_{...
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8 views

Direct proof that global functions on a projective scheme over k form a finitely generated k-algebra

Suppose that $X$ is a finite type separated projective scheme over a field $k$. Then the space of global sections $\Gamma(X,\mathcal{O}_{X})$ has finite $k$-dimension. I am only aiming to show ...
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52 views

Failure of rigidity for non-proper morphism.

Debarre's version of rigidity from Higher-dimensional algebraic geometry is Lemma 1.15 Let $X, Y$ and $Y'$ be varieties and let $\pi: X \to Y$ and $\pi': X \to Y'$ be proper morphisms. Assume $\pi_* \...
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1answer
117 views

What is the projective bundle over $(\Bbb P^1\times\Bbb P^1)/(\Bbb Z/2)$ which isn't $\Bbb P(\mathcal{E})$ for some vector bundle $\mathcal{E}$?

I'm thinking about when a (Zariski-locally trivial) $\Bbb P^n$-bundle comes from the projectivization of a vector bundle, and I've found Sasha's answer here on MO stating that $(\Bbb P^1\times\Bbb P^1)...
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65 views

Short Exact Sequence of a Hyperplane in the Projective Space

Let $H$ be a hyperplane in $\mathbb P^n$ where $f:H\rightarrow \mathbb P^n$ is the closed immersion. Let $\mathcal F$ be a coherent subsheaf of $\mathcal E = \oplus\mathcal O_{\mathbb P^n}(l)$. There ...
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48 views

isomorphic $\operatorname{Proj}(S)$ with nonisomorphic $S$

Let $S$ be a graded ring. To understand $\operatorname{Proj}(S)$ better, I want to know under what changes of $S$ will $\operatorname{Proj}(S)$ remains unchanged. All answers from a proposition or ...
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77 views

If $\mathcal{L}$ is an invertible sheaf on $\mathbb{P}(\mathcal{E})$, then is the restriction to the fibers constant?

Let $X$ be a Noetherian regular scheme, and $\mathcal{E}$ a locally free sheaf of rank $n+1\geq 2$ on $X$. Let $\pi:\mathbb{P}(\mathcal{E})\to X$ be the natural morphism. Then for every $x\in X$, the ...
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80 views

If a global morphism of sheaves induces isomorphisms on fibers, then is it an isomorphism?

Let $X$ be a Noetherian scheme (regular if needed), and let $\mathcal{E}$ be a locally free sheaf of rank $2$ on $X$. Let $\pi:\mathbb{P}(\mathcal{E})\to X$ be the natural morphism, and let $f:\...
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24 views

Applying Bertini's Thm: the intersection of $X\subset\mathbb{P}_k^m$ with $n$ general hyperplanes consists of a finite number of reduced points

This question is Vakil 12.4.C (http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf). Take a projective variety $X\subset\mathbb{P}_k^m$ of dimension $n$ (and with $k$ algebraically closed). ...
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1answer
106 views

How to prove $ ((S\times_A T)(1))_{(f\otimes g)}\cong S(1)_{(f)}\otimes_{S_{(f)}}(S\times_A T)_{(f\otimes g)}\otimes_{T_{(g)}} T(1)_{(g)}$?

I'm currently trying to solve Exercise 5.11 in chapter 2 of Hartshorne: Let $S,T$ be $\mathbb{Z}_{\geq 0}$-graded rings with $S_0=T_0=A$, and define their Cartesian product $S\times_A T$ to be $$ S\...
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1answer
94 views

Definition of projectively normal in Harshorne Ex II.5.14

In Hartshorne, Ex II.5.14, he defines that a closed subscheme $X \in \mathbb{P}^r_A$ is called projectively normal if the homogeneous coordinate ring $S(X)$ is integrally closed. This is also given by ...
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1answer
158 views

Projective scheme over a ring

I’m reading Qing Liu’s Algebraic Geometry and Arithmetic Curves. I’m confusing my double definitions of projective morphism over a ring $A$. In definition 2.3.42, projective scheme over $A$ is an $A$-...
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1answer
76 views

Weighted projective space and projective space are isomorphic

From Vakil's book: Exercise 8.2.N Show that the weighted projective space $\mathbb{P}(m, n) = Proj(k[x, y])$ (where $x$ and $y$ have degrees $m$ and $n$ respectively) is isomorphic to $\mathbb{P}^1$. ...
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1answer
93 views

Explicit proof that $\mathbb{P}_B^n\cong\mathbb{P}_A^n\times_{\operatorname{Spec}A}\operatorname{Spec} B$

On page 103 of Hartshore, just before the definition of projective morphisms, he states that if $\varphi:A\to B$ is a ring morphism and $(f,f^{\#}):\operatorname{Spec}B\to \operatorname{Spec}A$ is the ...
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1answer
66 views

Are these rings the same? A detail in the Proj construction

The following detail is needed in the affine-by-affine gluing construction of the Proj of a graded rings. Every single reference that I know of takes the claim for granted. Let $A$ be a nonnegatively ...
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0answers
74 views

Map from $B$-scheme to projective space

I am trying to understand exercise 6.3.M(a) in Vakil's algebraic geometry notes. It goes as follows: Suppose $B$ is a ring. If $X$ is a $B$-scheme, and $f_0,...,f_n$ are $n+1$ functions on $X$ with no ...
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0answers
33 views

Uniqueness of Projective Spectra for Ideals with Fixed Degree Generators

I am a beginner studying algebraic geometry on the level of Hartshorne. By Chapter II Exercise 2.14(c), the projective spectra $\mathrm{Proj}\,S$ can be isomorphic (as locally ringed spaces) to $\...
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1answer
100 views

The structure sheaf on $\operatorname{Proj}S$

Suppose $S$ is a $\mathbb{Z}_{\geq 0}$ graded ring that that $f\in S_+$ where $S_+$ is the irrelevant ideal. I know that we may associate $D_+(f)$ with the set of prime ideal of the zero degree ...
2
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1answer
190 views

Divisor of a rational section in Ravi Vakil's notes

The following is from Exercise 14.2.A of Ravi Vakil's algebraic geometry notes (page 401 here). The exercise asks us to consider the rational section $\frac{x^2}{x+y}$ of the sheaf $\mathcal{O}(1)$ on ...

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