1
vote
0answers
32 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
47 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 ...
6
votes
1answer
72 views

Base change for a projective variety

Consider a projective variety over $\mathbb C$, $X=\textrm{Proj}\frac{\mathbb C[T_0,\ldots,T_n]}{(f_1,\ldots,f_m)}$ and a field automorphism $\sigma\in \text{Aut}(\mathbb C)$. Now we want to ...
3
votes
0answers
27 views

About the functor between varieties over $k$ and $k$-schemes

Consider an algebraically closed field $k$, Thanks to Hartshorne II(2.6) we know that there exists an equivalence of categories $$F:\textrm{Sch}(k)\longrightarrow\textrm{Var}(k)$$ Where ...
2
votes
1answer
50 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
39 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
78 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
31 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} ...
1
vote
0answers
66 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
41 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
44 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
34 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
0answers
50 views

Projective schemes and fiber product

Consider the graded $K$-algebra $A=\frac{K[T_1,\ldots,T_n]}{(f_1,\ldots,f_m)}$ and construct the projective $K$-scheme $X=\operatorname{Proj} A $ (i.e. isomorphic to a closed subscheme of $\mathbb ...
4
votes
1answer
40 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
36 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
53 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
31 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
61 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
119 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 ...
1
vote
1answer
53 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
58 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
54 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 ...
1
vote
0answers
42 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
52 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
54 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
53 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
votes
1answer
108 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 ...
0
votes
0answers
22 views

Closed subset of projective schemes

I must prove that if $R$ is a graded ring finitely generated over a ring $A=R_0$, $Proj R$ is isomorphic to a closed subscheme of some projective space $\mathbb{P}_A^r$. I've problems in proving this ...
2
votes
3answers
165 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 ...
1
vote
0answers
89 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 ...
1
vote
0answers
68 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 ...
4
votes
1answer
233 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 ...
8
votes
1answer
241 views

Computing Kähler differentials on $\mathbb P^1_A$ (What did Liu intend?)

Let $A$ be a ring and $X=\mathbb P^1_A$. On $D_+(T_0)\cap D_+(T_1)$, we have $$T^2_0d(T_1/T_0)= -T^2_1d(T_0/T_1).$$ How does one formally compute this transition? Of course this is ...
8
votes
0answers
211 views

Why do these two constructions of a sheaf associated to a module on a projective scheme agree?

Let $B$ be a graded ring an $M$ be a $\mathbb Z$-graded $B$-module. We can associate to $M$ a sheaf of modules on $\operatorname{Proj} B$ by defining the sheaf on the principal open sets $D_+(f)$ to ...
6
votes
1answer
151 views

Why are projective spaces over a ring of different dimensions non-isomorphic?

Let $A$ be a nonzero commutative ring with unit. Define $\mathbb P_A^n$ to be the scheme $\operatorname {Proj} A[T_0,\dots,T_n]$, where the grading on the polynomial ring is by degree. Why is it true ...
4
votes
1answer
65 views

Radical of prime ideal in homogeneous localization is prime

Let $B$ be a graded ring, $B=\oplus_{d\ge 0} B_d$. If $f\in B$ is homogeneous, we let $B_{(f)}$ denote the subring of $B_f$ made up of elements of the form $af^{-N}$, $N>0$, where $a$ is a ...
5
votes
3answers
129 views

Compactness of the complex points of a $\mathbb{C}$-variety

Let $X$ be a $\mathbb{C}$-variety and let $X_{cx}$ denote the topological space formed by its $\mathbb{C}$-points with the complex topology (i.e. the associated analytic space). If $X$ is projective, ...
2
votes
1answer
128 views

Showing that intersection multiplicity at a point is finite for prime divisors

My question has two parts two it: one vaguely more elementary, one perhaps less so. In Beauville (Complex Algebraic Surfaces), we define the multiplicity of intersection of two (irreducible, no ...
2
votes
1answer
127 views

A proof that every projective morphism is proper?

I am currently working my way through Q. Liu's book "Algebraic Geometry and Arithmetic Curves". I'm puzzled by the proof that every projective morphism is proper, see below I understand that ...
4
votes
1answer
164 views

Does the singular locus of a conical variety (or scheme) determine the singular locus of its projectivization?

Lets say $X$ is a conical affine algebraic variety (conical meaning $X$ is the zero set of homogeneous polynomials of positive degree, equivalenty $X \subsetneq k^n$ and $x \in X \Rightarrow ax \in X$ ...
5
votes
1answer
121 views

Dimension of the irreducible components of an affine open in $\mathbb{P}^n_k$.

I was doing some exercises in Liu's book on Algebraic Geometry. I am currently trying to solve a problem by showing the following: Let $U \subset \mathbb{P}^n_k$, k a field, be an affine open ...
1
vote
1answer
142 views

Proj construction and fibered products

How to show, that $Proj \, A[x_0,...,x_n] = Proj \, \mathbb{Z}[x_0,...,x_n] \times_\mathbb{Z} Spec \, A$? It is used in Hartshorne, Algebraic geometry, section 2.7.
1
vote
1answer
115 views

Are smooth relative curves over an arbitrary base normal?

Let $X/S$ be a smooth, projective scheme of relative dimension one over a scheme $S$ (which we may assume is affine Noetherian, but need not be reduced nor irreducible nor even connected). For $s \in ...
1
vote
0answers
30 views

Understanding $Bl_{\mathcal{I}}(k^4)/S_2$ where $\mathcal{I}$ is defined by $(x_1-x_2,x_3-x_4)$

Let $k_4=Spec(k[x_1, x_2, x_3,x_4])$ and $\mathcal{I}$ is the ideal sheaf defined by $(x_1-x_2,x_3-x_4)$. Then $$ Bl_{\mathcal{I}}(k^4) = Proj (\oplus_{i\geq 0} I^i t^i) $$ where ...
0
votes
1answer
53 views

The variety associated to a polynomial ring with a particular grading

We impose various grading on an algebra or a module to understand its homological properties. So I made up the following problem and I would like to understand its correspondence as a variety. ...
2
votes
1answer
57 views

If the reduction is smooth and projective, can I conclude the same about the scheme

Let $X$ be a $R$-scheme, where $R$ is a dvr. Suppose that the reduction of $X$ (over the closed point of $\mathrm{Spec} \ R$) is smooth and projective. Does this imply that $X$ is smooth and ...
8
votes
2answers
420 views

Classification of automorphisms of projective space

Let $k$ be a field, n a positive integer. Vakil's notes, 17.4.B: Show that all the automorphisms of the projective scheme $P_k^n$ correspond to $(n+1)\times(n+1)$ invertible matrices over k, modulo ...
1
vote
1answer
112 views

Can a non-proper variety contain a proper curve

Let $f:X\to S$ be a finite type, separated but non-proper morphism of schemes. Can there be a projective curve $g:C\to S$ and a closed immersion $C\to X$ over $S$? Just to be clear: A projective ...
4
votes
1answer
122 views

Connections on line bundles on product of varieties

Let $X$ and $Y$ be smooth projective varieties over $\mathbb C$. Suppose I have given a line bundle $L$ on $X\times Y$ with a connection relative to $Y$, i.e. $\nabla: L \rightarrow ...
9
votes
1answer
271 views

Tangent space in a point and First Ext group

Let $X$ be an abelian variety over an algebraically closed field $k$. I have read that one has for an arbitrary closed point $x$ on $X$ a canonical identification $$T_x(X)\simeq ...