3
votes
0answers
23 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
48 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 ...
4
votes
0answers
68 views

A morphism from $\mathbb P^1_\mathbb C$ to $\mathbb P^1_\mathbb C$

Consider the projective scheme $\mathbb P^1_\mathbb C$ that is different from the projective line $\mathbb P^1(\mathbb C)$. Now look at the following lemma: I don't understand what is a ...
2
votes
0answers
37 views

Does $\operatorname{Proj}(\sigma)$ fix some points?

Consider a subfield $K\subseteq\mathbb C$, then by some properties of the fibered product of schemes we have that: $$\mathbb P^1_\mathbb C\cong\mathbb ...
2
votes
1answer
53 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
43 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
46 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
49 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
32 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
44 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
40 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
43 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
102 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
21 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
156 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
70 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
63 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
171 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 ...
7
votes
1answer
218 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
197 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 ...
5
votes
1answer
128 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
61 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
118 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
119 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
102 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
142 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
109 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
132 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
102 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
56 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 ...
7
votes
2answers
368 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
103 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
121 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
253 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 ...
2
votes
0answers
65 views

Isomorphism of first infinitesimal neighborhoods

Take an abelian variety $X$ over a field and consider $Z$ to be the first infinitesimal neighborhood of the diagonal in $X\times X$. Let furthermore $Y$ be the first infinitesimal neighborhood of the ...
1
vote
0answers
81 views

Morphisms of abelian variety and torus in additive group

let $A$ be an abelian variety and $T$ be an algebraic torus over a field $k$; furthermore denote with $\mathbb G_a$ the additive group over $k$, i.e. just the affine space. Why does then hold (i) ...
3
votes
1answer
136 views

What does “Biextension of Abelian Varieties” mean?

If I have two schemes $X$ and $Y$, which are such that my question makes sense (I guess, they should be abelian varieties over a field $k$, so assume this). Then I have often read, but nowhere found ...
1
vote
1answer
139 views

Ample divisor on abelian variety

just a short question: if one has an abelian variety $X$ over a field $k$ and an ample irreducible divisor $D$ on $X$, then why is $H^1(X-D,\mathcal O_X)$ zero? Should it be that $X-D$ is affine? ...
3
votes
1answer
142 views

Geometry of abelian varieties

if $X$ and $Y$ are abelian varieties over a field $k$ and $f:X\rightarrow Y$ is a homomorphism of abelian varieties, are then the following true: 1) ...
3
votes
1answer
82 views

Smooth ample hypersurface on variety

I read the following fact which wasn't explained further and wonder how you exactly get it. Maybe you can give me some hint. Start with a smooth projective variety $X$ over a $k$. Then the author ...
3
votes
1answer
225 views

If $f^*(D)$ is a Cartier divisor, is $D$ Cartier also?

Let $f:X\to Y$ be a finite, surjective morphism of normal algebraic varieties and let $D$ be a Weil divisor on $Y$. In this case, one can pull back to get a Weil divisor $f^*D$ on $X$ associated to ...
0
votes
3answers
256 views

Proj of graded rings

my question actually concerns an exercise II5.13 in Hartshorne. You have a graded ring $S=\oplus S_n$ with $n\ge0$ generated as $S_o$-Algebra by $S_1$ and you set $S^{(d)}=\oplus S_{dn}$ for a ...
2
votes
0answers
411 views

Very ample line bundles and global sections

my question concerns a very ample line bundle $L$ on a projective k-scheme. It gives a closed immersion $i:X \rightarrow P^{N-1}_{k}$ to some projective space over k. It can be viewed as induced by ...
2
votes
1answer
241 views

Proj construction and ample dualizing sheaf

my question concerns a smooth projective variety $X$ with dualizing sheaf $\omega_X$: if I have that this dualizing sheaf is ample, then I have read you can conclude that $X\simeq Proj(\oplus_{k} ...
6
votes
1answer
248 views

Is the graph of morphism of projective varieties $X \rightarrow Y$ closed in $X \times Y$?

The graph of a morphism $X \rightarrow Y$ is closed in $X \times Y$ if $X$ and $Y$ are affine varieties. What if $X$ and $Y$ are projective varieties? I am still not quite familiar with projective ...
6
votes
1answer
268 views

graded ring homomorphism induces isomorphism on Proj

I have a simple question about Exercise II.2.14(c) in Hartshorne's book. The claim is that if $\varphi : S \to T$ is a graded homomorphism which induces isomorphisms on all homogeneous pieces of ...
4
votes
2answers
274 views

Stalks on Projective Scheme

Let $k$ be an algebraic closed field. Let $x$ be a point in $X=P_k^1$. What is $O_{X,x}$? For example, if I have $x=(t-a)\in \text{Spec }k[t]$. Looking $x$ inside $P_k^1$, does ...
4
votes
1answer
411 views

Classifying Quasi-coherent Sheaves on Projective Schemes

I know some references where I can find this, but they seem tedious(Both Hartshorne and Ueno cover this). I am wondering if there is an elegant way to describe these. If this task is too difficult in ...