Questions tagged [projective-schemes]

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

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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):=...
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Coherent subsheaf of $\mathcal{O}(-1)$ on $\mathbb{P}^1$

I'd like to ask that if there is a classification of coherent subsheaf of $\mathcal{O}(-1)$ on $\mathbb{P}^1$. Thanks.
Jean's user avatar
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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
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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 $C\notin\{C_1,\ldots,C_n\}$. ...
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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 ...
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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
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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
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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\...
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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,...
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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 \...
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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
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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
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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
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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 ...
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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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What is a basis for the fibred product of two projective spaces viewed as schemes?

The context is that I am interested in trying to understand Beilinsons resolution of the diagonal, as proved in https://johncalab.github.io/stuff/beilinson.pdf p65-67. I have most of the details ...
Philip O'Donnell's user avatar
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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^...
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Homogeneous prime ideals in $(S_\star)_f$ map bijectively to homogeneous prime ideals in $(S_\star)$ not containing $(1,f,f^2,...)$?

As the title suggests, why do homogeneous prime ideals in $(S_{\star})_{f}$ map bijectively to homogeneous prime ideals in $S_{\star}$ not containing $(1,f,f^2,...)$? What I tried: I know that prime ...
kid111's user avatar
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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 ...
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How do I interpret the intersection of a variety with a "non-closed hyperplane?"

I am trying to understand Vakil's statement and proof of Bertini's theorem, which has been updated since many of the questions related to it were posted on this website (for what it's worth, I'm not ...
Emory Sun's user avatar
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How do I define a map from points on a curve to lines through them?

I am struggling with part (a) of Exercise 17.5.B in Vakil's Foundations of Algebraic Geometry. The statement of the problem goes as follows: Suppose $C = V(f(x, y, z)) \subseteq \mathbb{P}_k^2$ is a ...
Emory Sun's user avatar
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Understanding morphism of schemes between projective spaces defined by homogeneous coordinates

I'm trying to solve the following exercise: Let $k$ be a field and $f : \mathbb{P}_k^1 \longrightarrow \mathbb{P}_k^1$ a morphism of schemes defined by (in homogeneous coordinates) $$ [x:y] \mapsto [x^...
Bolito2's user avatar
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Does Ample imply direct image algebra finitely generated?

Let $S$ be a Noetherian scheme, $f: X \to S$ be a projective morphism and $\mathcal{L}$ an $f$-ample invertible $\mathcal{O}_X$-module. Is the $\mathcal{O}_S$-algebra $\bigoplus_{m ∈ \mathbb{Z}_{\geq ...
SeparatedScheme's user avatar
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Weil Conjectures for schemes

I have two questions. What class of schemes are the Weil conjectures true for? Projective schemes? Do I need additional hypotheses? I know from my question (The Weil conjecture involving Betti ...
Joseph Harrison's user avatar
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Proj of the direct sum of two graded rings

Let us fix the notation. Let $V$ be a 4-dimensional complex vector space and let $$\mathbb P(V)=\operatorname{Proj}(\operatorname{Sym}V)=\operatorname{Proj}(\mathbb C[x,y,z,w]).$$ Consider the $\...
Bobech's user avatar
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Quasi-coherent sheaves on projective space.

I would like to better understand the characterization of quasi-coherent sheaves on $X=\mathbb{P}_A^d = \operatorname{Proj}A[x_0,...x_d]$, where I call $B = A[x_0,...x_d]$ We define $\mathcal{F}[k] = \...
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1 answer
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Quasi-coherent $\mathcal{F} \cong \tilde{M}$ on $\operatorname{Proj}B$ does not determine $M$

I would like to understand in detail the fact that given $\tilde{M}\cong \tilde{N}$ on $\operatorname{Proj}B$, then it is not always true in general that $M \cong N$. I know this has to do with ...
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the meaning of $[x_0, x_1, \ldots, x_n]$ formed by projective coordinates on projective space $\Bbb{P}^n_A$ in Vakil's FOAG 7.3.F

In Vakil's FOAG, the projective space $\Bbb{P}^n_A$ is defined to be $\operatorname{Proj} A[x_0, x_1, \ldots, x_n]$. (there is another definition in the book too, just glueing $n+1$ affine space). ...
onriv's user avatar
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does natural structure morphism from Proj $S$ to Spec $A$ require $S$ a finitely generated graded ring?

The question is already asked here natural structure morphism from Proj $S$ to Spec $A$ . And It’s exercise 7.3.I in Vakil’s FOAG: 7.3.I. EASY EXERCISE. If $S$ is a finitely generated graded A-...
onriv's user avatar
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Why is the projective space an integral scheme

I would like to prove that the $Proj(k[x_{0},\dots, x_{n}])$ is an integral scheme ($k$ is a field). I know a preposition that says a scheme is integral if and only if the image of the structure sheaf ...
T. Wildwolf's user avatar
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1 answer
131 views

Closed points of $\text{Proj}(\mathbb R[x,y,z]/(x^2+y^2+z^2))$

Consider $X=\text{Proj}(\mathbb R[x,y,z]/(x^2+y^2+z^2))$, I want to show that $X(\mathbb R)=\emptyset$. I have two problems with this. The first one is that I'm not sure of the definition of $X(\...
raisinsec's user avatar
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Morphisms from laurent series to Projective space

Let $k$ be a field and $k((x))$ be its field of formal laurent series. I'm trying to understand morphisms $f : Spec \ k((x)) \rightarrow \mathbb{P} = Proj \ k[T_0, T_1]$. For example, does specifying ...
Steven Mai's user avatar
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Maps of graded rings inducing isomorphism between Proj.

Let $\Bbbk$ be a field. Let $S,T$ be graded $\Bbbk$-algebras, and let $\phi:S\to T$ be a graded ring map. We can assume finite generation if things become simpler. We have that if for sufficently ...
Display Name's user avatar
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Is $\mathbb{P}^1$ a group scheme?

It's known that the set of meromorphic functions (functions to $\mathbb{C}\cup\{\infty\}$) on a complex variety $X$ forms a field, called the function field of $X$. Edit: Thanks to the comment by @...
Z Wu's user avatar
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For divisor $D$, how to describe isomorphisms $H^0(X,\mathcal{O}_D(nD)) \xrightarrow{\sim} H^0(X,\mathcal{O}_D\left((n+1)D\right)$?

Let $X$ be a scheme, $D$ an effective divisor on $X$ with structure sheaf $\mathcal{O}_D$, and $U = X\setminus D$. I think I need $D$ either ample or affine. If necessary we can assume $\mathcal{O}_X$ ...
Somatic Custard's user avatar
3 votes
0 answers
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Algebraic Projective Noether normalization

Reading through the literature I see that there are two proofs for the Noether normalization for projective varieties (over infinite fields). The first, "geometric", by iterating a ...
user4231's user avatar
1 vote
0 answers
114 views

Global sections of a torsion sheaf

Let $X$ be a smooth projective curve, $\mathcal{F}$ a torsion sheaf on $X$, $\Gamma(X,\mathcal{F})$ finitely generated by $ \{s_1,\ldots,s_n\}$ (or $n:= \dim \Gamma(X, \mathcal{F}) $). Is $\mathcal{F}$...
fish_monster's user avatar
1 vote
1 answer
115 views

Sections of a torsion sheaf

I am trying to get a better understanding of torsion sheaves over projective schemes. Is it true that if $\mathcal{F}$ is torsion over a projective scheme $X$, then $$\Gamma(X,\mathcal{F}) \cong \...
fish_monster's user avatar
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Hartshorne, Exc II 7.9: Do I need regularity to show $\operatorname{Pic} \mathbb P(E) = \operatorname{Pic} X \times \mathbb Z$?

Exercise II 7.9 (a) in Hartshorne's Algebraic geometry is Let $X$ be a regular noetherian scheme, and $E$ a locally free coherent sheaf of rank $\geq 2$ on $X$. Show that $\DeclareMathOperator{\Pic}{...
red_trumpet's user avatar
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Deformations of finite schemes

I am reading some texts about the tangent space to the Hilbert scheme. Apparently, $T_{[Z]}(Hilb (X)) = H^0(Z, N_{Z/X})$ for any $Z$ is a consensus which I agree with, but then regarding the hilbert ...
Aitor Iribar Lopez's user avatar
4 votes
1 answer
181 views

Vakil's proof of Bertini's theorem (Theorem 12.4.2) - where do the linear conditions on the fiber come from?

As I'm reading through Vakil's notes on algebraic geometry, I'm stumbling over some of the details in his proof of Bertini's theorem. The proof in question is discussed already on this site (1, 2 and ...
daniel gratzer's user avatar
1 vote
1 answer
246 views

Quasi-projective A-schemes are locally of finite type over A, The Rising Sea, Ex.5.3.D

A projective $A$-scheme is a $\operatorname{Proj} S_{\bullet}$ where $S_{\bullet}$ is a finitely generated graded ring over $A=S_0$. A quasi-projective $A$-scheme is an open quasicompact subscheme of ...
Potitov06's user avatar
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1 answer
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Hartshorne example III.9.8.3

In this example we consider a closed subscheme $X_1\subset \mathbb{P}^{n+1}\setminus \{P\}$ where $P=(0:\ldots:0:1)$. For every $a\in \mathbb{A}^1\setminus 0$ we have an automorphism of $\mathbb{P}^{n+...
frogorian-chant's user avatar
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A $k$-dimensional reduced subscheme $Y \subset \mathbb{P}^n$ of degree $1$ is a linearly embedded $k$-plane

In following all schemes $X$ will be considered as separated, of finite type over an algebraically closed field $K$ of characteristic $0$. Recall that a Hilbert function $HF_S: \mathbb{N}_0 \to \...
user267839's user avatar
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Definition of $ \ \mathcal{O}_{\mathbb{P}_k^n}(1)$ [duplicate]

I've found references to the object $\ \mathcal{O}_{\mathbb{P}_k^n}(1), \ $ where $\mathbb{P}_k^n \ $ is the usual scheme, but I'm not able to find its proper definition. Is it just a line bundle ...
Alessandro's user avatar
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3 votes
1 answer
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Show that the dimention of the intersection of projective linear sub-spaces of dimentions $d_1$ and $d_2$ of $\mathbb{P}^n$ is bigger than $d_1+d_2-n$

Proposition: Let $L$ and $M$ linear projective subspaces of $\mathbb{P}^n$ of dimention $d_1$ and $d_2$ respectively. Prove that $\operatorname{dim}(L\cap M)\geq d_1+d_2-n$ (we consider the dimention ...
Alan Jr's user avatar
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1 answer
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Does the degree of a projective curve depend on the embedding into projective space?

I'm studying the classification of curves in $P^3$ in chapter IV of Hartshorne. I've already studied chapter I, chapter II (sections 1 to 6), and chapter III (sections 1 to 7) of Hartshorne in my ...
izzy's user avatar
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nth Veronese subring embedding induces isomorphism of schemes (Vakil FOAG 6.4.D)

Exact exercise is: show the embedding map of graded rings $S_{n\bullet} \to S_\bullet$ for $S_\bullet$ finitely generated graded ring induces an isomorphism $\operatorname{Proj} S_\bullet \to \...
categoryisschemes's user avatar
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Are complete varieties in the sense of Weil the same as proper varieties?

This is a continuation of a previous question; I will use the same notation and terminology. An abstract variety (or Variety, with a big V) in the sense of Weil consists of the following data: A ...
Zhen Lin's user avatar
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Do ample sheaves descend along limits / noetherian approximation?

Suppose $X \to S$ is a proper (projective) morphism of schemes, $S$ is quasi-compact and quasi-separated, and $\mathcal L$ is a relatively ample sheaf on $X$. By stacks, tag 01ZA, the scheme $S$ is a ...
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