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2
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2answers
84 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
31 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
39 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 ...
2
votes
1answer
52 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
66 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 ...
3
votes
0answers
43 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
52 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
53 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: ...
0
votes
0answers
75 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
28 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 ...
6
votes
2answers
109 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 ...
3
votes
1answer
81 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
57 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 ...
1
vote
1answer
88 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
82 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
152 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 ...
0
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0answers
31 views

proof of Hartshorne on basic open sets of projective spectrum Proj S

In the proof of proposition 2.5 of Hartshorne's Algebraic Geometry, Chapter II, Section 2 it is written (somewhere in the middle): "The properties of localization show that $\phi$ is bijective as a ...
3
votes
1answer
58 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
95 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 ...
2
votes
1answer
72 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
68 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. ...
3
votes
1answer
107 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
37 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
58 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 ...
2
votes
1answer
53 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 ...
0
votes
1answer
50 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
51 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
80 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
49 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
54 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 ...
2
votes
2answers
252 views

Projective varieties and irreducibility

The "modern"(schematic) definition of a projective variety is the following: Let $k$ be an algebraically closed field. A projective variety over $k$ is a closed subscheme of $\mathbb ...
3
votes
1answer
251 views

Torsion sheaves on a curve

This is probably a silly question, but I'm a bit confused. Regarding exercises 6.11 and 6.12 of Chapter II of Hartshorne: Let $X$ be a nonsingular projective curve over an algebraically closed field ...
0
votes
1answer
50 views

The structure morphism of a projective variety induces a morphism of $k$-algeras

Suppose that $k$ is an algebraically closed field and that $X=\textrm{Proj}{\frac{k[T_1,\ldots,T_n]}{(f_1,\ldots,f_n)}}$ is a projective variety with a structural morphism $p:X\rightarrow\textrm{Spec} ...
2
votes
0answers
107 views

Local complete intersection scheme, conormal sheaves and differentials

Let $X$ be a smooth projective variety over $\mathbb{C}$ and $Z \subset X$ be a local complete intersection subscheme in $X$. Denote by $I_Z$ the ideal sheaf of $Z$ in $X$ and $\Omega^1_X$ the sheaf ...
3
votes
1answer
80 views

Degree of ample bundle over projective curve is positive

(From Vakil's notes, Exercise 18.4.K) If $C$ is an integral projective curve over a field $k$, and $\mathscr{L}$ is an ample line bundle on $C$, why is the degree of $\mathscr{L}>0$? If $C$ is ...
1
vote
1answer
67 views

Twisting relative proj (exercise from Vakil)

I'm stuck on another problem (17.2.G) from Vakil's notes, and I'm wondering if somebody could get me started. Specifically, we are given a scheme $X$, an invertible sheaf $\mathscr{L}$ on $X$ and a ...
2
votes
1answer
39 views

complex automorphisms acting on a projective variety

Consider a complex projective variety $X=\operatorname{Proj}\frac{\mathbb C[T_1,\ldots,T_n]}{(f_1,\ldots,f_n)}$ with $f_1,\ldots,f_n$ homogeneous polynomials. If $\sigma\in\operatorname{Aut}(\mathbb ...
5
votes
1answer
109 views

Closed subschemes of projective space

We work over the complex numbers. If $X$ is an integral closed subscheme of $\mathbb P^n$, then there exists a homogeneous ideal $I$ of $A=\mathbb C[x_0,\ldots,x_n]$ such that $X = V_+(I) =$ ...
1
vote
1answer
46 views

Structure sheaf of a divisor

I have heard that from a given divisor $D \hookrightarrow X$ (where $X$ is a projective scheme and $D = \sum n_i Y_i$ such that each $Y_i$ is smooth and $Y_i \hookrightarrow X$ is a closed immersion), ...
1
vote
1answer
65 views

Projective varieties are the common zeros of some homogeneous polynomials

definition: A projective variety over a field $k$ is a closed subscheme (over $k$) of $\mathbb P^n_k=\operatorname{Proj}(k[T_0,\ldots,T_n])$. Now it can be proved (I have done it) that if ...
3
votes
0answers
46 views

Associated projective bundle on a projective scheme

Suppose that $X$ is a quasi-projective $k$-scheme with a right $B$-action (where $B$ is a linear algebraic group or Lie group) and that the quotient $X/B$ exists. Let the canonical projection $X \to ...
3
votes
1answer
88 views

$d$-branched coverings and finite morphisms of degree $d$

Consider a smooth projective curve $X$ over $\mathbb C$ (so $X$ is a projective $\mathbb C$-scheme, integral, of finite type....), and let $t: X\longrightarrow\mathbb P^1_{\mathbb ...
2
votes
1answer
498 views

global sections of structure sheaf of a projective scheme X over a field k?

Some may have already asked this question. What are the global sections of structure sheaf of a projective scheme $X$ over a field $k$? By Hartshorne page 18, Chapter 1, Theorem 3.4, global sections ...
2
votes
1answer
82 views

questions about Global Proj

Can someone explain the construction of the global $\mathbf{Proj}$ to me? Although this question has been asked here, I still have several questions. For each open affine subset $U = \mathrm{Spec} ...
5
votes
1answer
80 views

Existence of a morphism of schemes from $X$ to $\mathbb P_A^n$?

In exercise ┬žII.III.VIII. of Algebraic geometry and arithmetic curves by Qing.Liu. one is asked the following: Let $X$ be a scheme over a ring $A.$ Let $f_0, \cdots,f_n\in\mathcal O_X(X)$ be ...
2
votes
1answer
61 views

If $B$ is a graded $A$-algebra, then $\operatorname{Proj}B$ is an $A$-scheme

If $B$ is a graded ring, then for me is clear that $\operatorname{Proj}B$ with affine covering given by $D_+{(f)}\cong\operatorname{Spec} B_{(f)}$ is a scheme. The problem arises when $B$ is a graded ...
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0answers
58 views

Construction of projective space of a graded ring

Let $S=k[x_0,x_1,x_2]$, $k$ an algebraically closed field and $ S_d=k[x_0^d,x_0^{d-1}x_1,\ldots,x_2^{d-1}x_1, x_2^d]$. $\mathbb{P}S_d $ is identified with the projective space $\mathbb{P}^{N_d} $, ...
3
votes
1answer
72 views

Reducibility of a Hilbert scheme in projective space

My question concerns the computation of the Hilbert scheme $\mathsf{Hilb}_{3}^{2x+1}$, which parametrizes all curves of degree $2$ and genus $0$ in $\mathbb{P}^{3}_{k}$, with $k$ algebraically closed. ...
0
votes
1answer
88 views

In the Proj construction, are the homogeneuos prime ideals determined up to an element in the base ring?

Consider the point $[a_0,...,a_n]\sim [a_0\lambda,...,a_n\lambda] \in \Bbb P^k_n$. How do you write the corresponding homogeneous prime ideal in the graded ring $S:=k[x_0,...,x_n]$? Well, the ...
1
vote
1answer
98 views

Conditions under which a variety to remains smooth after base change (if p > 0)

Let $k$ be an arbitrary field of positive characteristic and let $V$ be a smooth projective (irreducible) variety over $k$. Suppose that $K/k$ is a field extension such that $V_K:=V\times_{\text{Spec ...