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8
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0answers
108 views
+200

How to complete Vakil's proof that the composition of projective morphisms are projective when the target is quasicompact?

For this question, a morphism $\pi : X \rightarrow Y$ is projective iff there exists a finite type quasicoherent sheaf $\mathcal{E}$ on $Y$ such that $X$ is isomorphic (as a $Y-$scheme) to a closed ...
1
vote
0answers
35 views

The $\mathrm{Proj}$-construction and inverse limits

I have a couple of questions about existence of certain inverse limits in the category of schemes (I am also happy about links to relevant literature... in the stacksproject I only found the affine ...
1
vote
0answers
20 views

Existence of scheme quotient

I have a morphism of schemes $X\to S$ which is very nice: flat, proper, finitely presented. I also have a finite group $G$ acting (faithfully, but not necessarily freely) on $X/S$. 1) I am quite ...
0
votes
0answers
27 views

Sheafifying direct sum of twists

Let be $X\subseteq \mathbf{P}^r$ a smooth projective variety and let be $\mathscr E$ an invertible sheaf over $X$. Let $$M=\bigoplus_{n\geq 0} H^0(\mathscr E(n))$$ as a module over the polynomial ...
0
votes
1answer
41 views

Well-definedness of homogeneous coordinate ring of projective scheme [closed]

Let $I,J \subset S=k[x_0,...,x_n]$ be homogeneous ideals. How can I show that $\operatorname{Proj}S/\bar{I}=\operatorname{Proj}S/\bar{J}$ iff $\bar{I}=\bar{J}$? (Here $\operatorname{Proj}R$ is the ...
5
votes
2answers
129 views

Projective bundle of a sum of ample line bundles

Let $X$ be a quasi-projective variety, and let $L_1,...,L_n$ be ample line bundles on $X$. Is that true that if $E= \oplus L_i$, then $$\mathbb{P}(E) \cong Proj(\bigoplus_{i_j \in \mathbb{N}^n} H^0(...
0
votes
0answers
33 views

How are non-homogenous elliptic curves projective varieties?

So if I am given an elliptic curve such as $Y^2Z=X^3$ then I see how it can be realized as the projective variety $Proj(k[X,Y,Z]/(Y^2Z-X^3))$. But, given an elliptic curve like $Y^2 = X^3 + X$, then $...
0
votes
1answer
73 views

Is the Grassmannian bundle projective?

Let $X$ be a smooth projective complex variety and $E$ a complex vector bundle of rank $n$ on $X$. Write $\mathbb G(k,E)$ for the fiber bundle whose fibers are the Grassmannians $\mathbb G(k,E_x)$ ...
0
votes
1answer
53 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 $k(f)\...
3
votes
1answer
107 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
187 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? ...
0
votes
0answers
18 views

Give $1^{st}$ order reduced model obtained by Galerkin projection, given the state space system

We have $\delta(t) = 1$ if $t = 0$ and $\delta(t) = 0$ if $t \neq 0$ and $x(t+1)= \begin{bmatrix} 1 &&{-0.5}\\0.5&&0 \end{bmatrix}x(t)+\begin{bmatrix} 1\\1\end{bmatrix}\delta(t)$ $...
2
votes
1answer
109 views

Relation between stalks of twisted sheaf and structure sheaf

Let $A$ be a ring, $B = A[T_0,\dots, T_d]$, and $X = \textrm{Proj } B$. Then at every point $x \in X$, $$\mathcal{O}_X (n)_x \cong \mathcal{O}_{X,x}$$ Let $x$ correspond to a homogeneous prime ...
3
votes
0answers
104 views

Global sections of Proj

In constructing the relative Proj for a graded ring $S$, one inevitably has that the distinguished opens, $D_+ (f)$, where $f \in S_+$ gives rise to a scheme that is isomorphic to $\textrm{Spec } S_{(...
3
votes
1answer
53 views

What is the “module of twisted global sections”?

Let $X$ be a projective variety. Suppose we have computed the graded modules corresponding to $\Omega_{\mathbb P^n}$ (the cotangent sheaf) and $\mathcal O_X$. One way to get a representation for $\...
1
vote
0answers
32 views

Construction of relative projective space via glueing

I would like to gain further practice on glueing schemes by constructing projective space over a ring. I am considering the following: I wonder how we get that $\mathcal{O}_{X_i} (X_{ij}) = \...
1
vote
0answers
35 views

Are $ \mathbb{A}^n (k ) = k^n $ and $ \mathbb{P}^{n} (k) = \mathrm{Proj} \ k[X_0 , \dots , X_n ]$ irreducibles?

Are $ \mathbb{A} (k ) = k^n $ and $ \mathbb{P}^{n} (k) = \mathrm{Proj} \ k[X_0 , \dots , X_n ]$ irreducibles when $ k $ is a domain ? Thanks in advance for your help.
1
vote
0answers
23 views

twist and product of projective spaces

On $\mathbb P^n_k\times_k \mathbb P^m_k$, is it true that $T_{\mathbb P^n_k\times_k \mathbb P^m_k}\otimes \mathcal O_{\mathbb P^n_k\times_k \mathbb P^m_k}(d,e)\simeq p_1^*T_{\mathbb P^n_k}(d)\oplus ...
1
vote
1answer
78 views

Very basic question on projective bundles: Why is the fiber a vector space?

Consider an integral scheme $(X,\mathcal O_X)$ of dimension $n$ and a "good" locally free sheaf of $\mathcal O_X$-modules $\mathcal E$ of rank $n+1$. Then, thanks to the "global proj construction" we ...
1
vote
0answers
62 views

Why is $ \mathrm{Spec} \ A [T_1 , \dots , T_n ] \to \mathrm{Proj} \ A [T_0 , \dots , T_n ] $ an embedding?

Let $ A $ be a commutative ring with identity. I would like to know how to establish that : $ \mathrm{Spec} \ A [T_1 , \dots , T_n ] \to \mathrm{Proj} \ A [T_0 , \dots , T_n ] $ is an embedding ? ...
0
votes
1answer
89 views

map between projective schemes induced by rational points

Given a map between $\mathbb P^n_{\mathbb C}$ and itself, given by $$ [x_0:\dots:x_n] \mapsto [p_0(x):\dots:p_n(x)]$$ where $p_i$ are homogeneous polynomials of the same degree, how do I find the ...
2
votes
2answers
124 views

Exercise 6.5.F in Ravi Vakil's notes: Showing conic $x^2 + y^2=z^2$ in $\mathbb{P}_k^2$ is isomorphic to $\mathbb{P}_k^1$

I have been stuck on Exercise 6.5.F in Ravi Vakil's notes for a little while now, and I would greatly appreciate any hints/comments/solutions! Let $k$ be a field that is not of characteristic $2$. I ...
0
votes
0answers
46 views

Projective variety defined by a non-radical ideal.

In the context of the Exercise 5.3.D in Vakil's notes, I want to show that there are examples of a reduced graded ring $A$ and a non-radical homogeneous ideal $I$ such that $\text{Proj}(A/I)$ is a ...
2
votes
1answer
44 views

On projection $\mathbb{P}_A^n \rightarrow \mathbb{P}_A^{n-1}$

I am learning about rational maps at the moment, and the notes I am reading gives an example, the projection $\mathbb{P}_A^n \rightarrow \mathbb{P}_A^{n-1}$ given by $[x_0, ..., x_n] \rightarrow [x_0,...
2
votes
1answer
64 views

I need help in this proof of this exercise from Fulton's book

I'm reading Fulton's algebraic curves book. I'm trying to understand this solution which I found online of the question 4.17 on page 97. What I didn't understand is why $V(J_z)$ are exactly those ...
1
vote
1answer
106 views

$\operatorname{Proj}k[x,y,z]/(xz,yz,z^2)$ isomorphic to $\mathbb{P}^{1}_{k}$

While dealing with the Proj construction, I encountered with this seemingly-simple question, but somehow I can't get the point at this moment. Is the scheme $$\operatorname{Proj}k[x,y,z]/(xz,yz,z^...
3
votes
0answers
45 views

Normal projective varieties and its coordinate ring

Let $k[X_0,...,X_n]$ be a polynomial ring over an algebraically closed field of characteristic zero and $I$ an ideal of $k[X_0,...,X_n]$ generated by homogenous polynomials. Denote by $X$ the ...
0
votes
1answer
79 views

Linear Systems And invertible Sheaf

Hello Fellow Mathematicians/Algebraic Geometer , This is question has two parts one which is more conceptual and the other more straight forward: i) Let $X$ be a non-singular protective variety ...
1
vote
1answer
99 views

Exercise in R.Vakil 18.4.L: Ample line bundles and finite morphism

Here's the question: R.Vakil, Exercise 18.4.L: Suppose $\mathcal{L}$ is a base-point free invertible sheaf on a proper variety $X$, and hence induces some morphism $\phi: X\rightarrow\mathbb{P}^n$....
0
votes
0answers
123 views

Dimension of the set of forms of degree $d$ in the homogeneous coordinate ring of $V$ (Fulton, Exercise 4.10)

I need help to solve the following problem that appears on page 55 of the book of William Fulton entitled Algebraic Curves. Exercise : Let $ R = k[X,Y,Z] $, $F \in R$ an irreducible form of degree $...
1
vote
0answers
34 views

Connection between Chladni Plates and Projective Geometry?

Does anybody know of a connection between Chladni Plates and Projective Geometry? Any published research on the subject? I am interested to know if Chladni Plates (or surfaces) can be explained/...
6
votes
2answers
168 views

Perfect complexes and the derived category of a smooth projective variety

I know that on a smooth projective variety any coherent sheaf has a finite locally free resolution. I read somewhere that this implies that any object in $D^b(X)$ for $X$ smooth projective is then ...
5
votes
2answers
174 views

Isomorphism of Proj schemes of graded rings, Hartshorne 2.14

This question is based on exercise $2.14$ of chapter $2$ of Hartshorne. Suppose $\varphi:S\rightarrow T$ is a graded homomorphism of graded (commutative, unital) rings such that $\varphi_d := \...
0
votes
1answer
85 views

How to see $\Gamma(\mathscr{O}_S,\operatorname{Proj} S)$ as a ring?

Let $S$ be a finitely generated graded $A$-algebra. For each homogeneous $f\in S_+$, we have a scheme structure $D(f)\cong \operatorname{Spec} S_{(f)}$ where $S_{(f)}$ denotes the zeroth piece of the ...
2
votes
1answer
122 views

Rational maps from noetherian $k$-scheme to projective $k$-scheme extend over regular codimension 1 sets

I am trying to solve and am currently stuck on Vakil's problem 16.5.B which is asked as follows: Suppose $X$ is a Noetherian $k$-scheme and $Z$ is a irreducible codimesion $1$ subvariety whose ...
2
votes
2answers
148 views

Understanding Hartshorne's proof that every projective morphism is proper.

In chapter $II$ of Hartshorne, theorem $4.9$ shows that every projective morphism is proper, using the valuative criterion for properness. I understand how the required morphism is constructed, but I ...
5
votes
1answer
214 views

Blow-up and base change

Consider a complex smooth (projective) surface $X$ and a blow-up $\epsilon:S\longrightarrow X$ at a point $x\in X$. Let $\sigma\in\text{Aut}(\mathbb C)$ be a field automorphism and moreover let $$\...
3
votes
1answer
67 views

“Twist” of $\mathbb P^n_K$ through a field automorphism.

This question is closely related to this recent one. Suppose that $s:X\longrightarrow\text{Spec}\, K$ is a variety over $K$ (i.e. a $K$ scheme, separated, proper and geometrically integral) and ...
2
votes
1answer
156 views

The importance of the structural morphism of a projective variety.

In scheme-theory, "The projective $n$-dimensional space over $k$" is defined as $\mathbb P^n_k:=\text{Proj}(k[T_0,\ldots,T_n])$. Moreover $\mathbb P^n_k$ is endowed with a structure of variery over $k$...
2
votes
1answer
81 views

What is the canonical morphism of $\mathbb{P}^n_A$ to $\text{Spec }A$?

On Liu's book Algebraic Geometry and Arithmetic Curves, he gives a general definition of the projective scheme $\mathbb{P}^n_A$, for $A$ a ring, as $Proj (B)$, for $B=A[x_0,\dots,x_n]$. Later, he ...
1
vote
2answers
81 views

Structure sheaf of $Proj \ S$ in terms of compatible stalks

Let $S$ be a graded ring. I was wondering if someone could please explain me how I can interpret structure sheaf of $Proj \ S$ in terms of compatible stalks? Thank you! Edit: This is Exercise 4.5.M. ...
4
votes
1answer
166 views

Compute the cohomology of projective schemes

In Hartshorne's book, Section 3.5, the cohomology of projective spaces is computed. How to compute the cohomology of projective schemes? Maybe the general case is complicated, please look at the ...
0
votes
0answers
41 views

What is the definition of $I(Z)$ when $Z \subseteq Proj S$

Let $S$ be a graded ring. In the notes of Ravi Vakil, he only says "define $I(Z) \subseteq S_+$", when $Z \subseteq Proj S$, where $S_+$ is all the elements in $S$ that are sum of positive degree ...
1
vote
0answers
62 views

Point at infinity of the scheme $\mathbb{P}^1$

Let $\mathbb{P}^1_k$ be the scheme defined as either $\text{Proj }k[t]$ or obtained by gluing two affine lines appropriately. What is the point at infinity which in topology usually is given by $[0:1]$...
2
votes
1answer
58 views

Confused about short exact sequence involving $\mathcal{O}_{\mathbb{P}^n}$ and $\mathcal{O}_Z$ for $\pi:Z\rightarrow \mathbb{P}^n$ closed embedding

If $\pi: Z\rightarrow \mathbb{P}^n$ is a closed embedding where $Z$ is the zero set of some degree homogeneous polynomial, we have: $$0\rightarrow \mathcal{O}_{\mathbb{P}^n}(-d)\rightarrow \mathcal{O}...
0
votes
1answer
55 views

$(-1)$-curves and base change along a field automorphism.

Suppose that $E\subseteq S$ is a $(-1)$-curve inside a non-singular complex projective surface. By a $(-1)$-curve $E$, I mean that $E\cong\mathbb P^1_\mathbb C$ and $E^2=-1$. Now consider a field ...
0
votes
1answer
70 views

Notation in the Semicontinuity Theorem

In Hartshorne's book Algebraic Geometry in Chapter III Section 12 we have the following situation: $f\colon X\to Y$ is a projective morphism of schemes, $Y$ is the affine spectrume of a ring $A$ , ...
1
vote
1answer
87 views

Does the canonical morphism commute with direct image functor?

I am trying to prove the representability of the Quotient functor. I have the following problem. Let $\phi \colon T \to S$ be a morphism of noetherian schemes and let $F$ be a coherent sheaf on $\...
1
vote
1answer
54 views

Scheme theoretic dual of $\mathbb P^n_k$

Consider an algebraically closed field $k$, and define $\mathbb P^n_k:=\textrm{Proj}(k[T_0,\ldots,T_n])$. In some algebraic geometry books I see the notation ${(\mathbb P^n_k)}^\vee$ that is referred ...
2
votes
1answer
58 views

$\mathbb C$-isomorphism between two $\mathbb C$-schemes.

Consider a field automorphism $\sigma\in\textrm{Aut}(\mathbb C)$, and moreover consider the $\mathbb C$-scheme $p:\mathbb P^1_{\mathbb C}\longrightarrow\textrm{Spec}\,{\mathbb C}$ where $\mathbb P^1_{\...