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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 ...
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1answer
40 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: ...
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0answers
66 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 ...
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0answers
24 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 ...
5
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2answers
87 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
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1answer
68 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 := ...
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1answer
44 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 ...
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1answer
80 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 ...
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2answers
70 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
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1answer
130 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 ...
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0answers
28 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 ...
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1answer
56 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
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1answer
76 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
67 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 ...
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2answers
56 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
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1answer
82 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 ...
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0answers
34 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 ...
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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
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1answer
49 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 ...
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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 ...
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1answer
47 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 $A$ , ...
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1answer
73 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 ...
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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 ...
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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
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2answers
166 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
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1answer
182 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 ...
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1answer
43 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
99 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
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1answer
67 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 ...
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1answer
59 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
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1answer
38 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
83 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 $X = V_+(I) =$ ...
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1answer
41 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), ...
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1answer
63 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 ...
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0answers
42 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
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1answer
76 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
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1answer
372 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 ...
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1answer
69 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} ...
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1answer
75 views

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

In exercise §II.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
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1answer
60 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
50 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} $, ...
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1answer
60 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. ...
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1answer
77 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 ...
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1answer
76 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 ...
3
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1answer
123 views

On the Hilbert function of projective schemes

Let $X \subset \mathbb{P}^n$ be a projective subscheme (not necessarily reduced or irreducible). Denote by $I_X$ the ideal of $X$ i.e., $\Gamma_*(\mathcal{I}_X)$. There are two definitions of Hilbert ...
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votes
3answers
171 views

What does the notation $\mathbb{P}V$ mean for a vectorspace $V$?

In algebraic geometry, I keep seeing the notation $\mathbb{P}V$ when $V$ is given as a vectorspace. My best guess is that $\mathbb{P}V$ is to mean the projective closure of $V$. But it would be nice ...
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0answers
122 views

Projective scheme

How can I prove that the two different construction of $\mathbb{P}_k^n$ (as $Proj K[x_0,x_1,...,x_n]$ and by gluing copies of $\mathbb{A}_k^n$) agree? And how can I prove that if $A$ is reduced also ...
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0answers
90 views

Proj description of successive blowups

I am attempting to understand the global Proj description of a blowup. The following example is giving me difficulty. Start by taking $\mathbb{A}^2_{\mathbb{C}} = \text{Spec}(\mathbb{C}[x,y])$ and ...
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2answers
147 views

How does Liu intend his readers to compute $H^0(X,\Omega^1_{X/k})$ of this scheme?

In the chapter on duality theory in Liu's Algebraic Geometry and Arithmetic Curves, there is the following exercise. I'm having trouble seeing how one can apply the duality theory developed in Liu to ...
4
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1answer
341 views

When are complete intersections also local complete intersections?

First, let us recall some definitions. Let $P=\mathbb P^d_k$ be a projective space over a field $k$. Let $X$ be a closed subvariety of $P$ of dimension $r$. We say that $X$ is a complete ...