The projective-schemes tag has no wiki summary.
4
votes
1answer
25 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
85 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
67 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
54 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
104 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
84 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 ...
0
votes
1answer
109 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
66 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
48 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
48 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 ...
5
votes
2answers
207 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
77 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 ...
3
votes
1answer
102 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 ...
8
votes
1answer
217 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
57 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
68 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
101 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
121 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
129 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
67 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
178 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 ...
2
votes
2answers
385 views
dimensionality reduction, projection points on a hyperplane
Imagine a N dimensional vector space. In it there are several points. I pick $m<N$ of them. They form a hyperplane in the N dimensional vector space. For example N = 10, m = 2. Now i want to ...
0
votes
3answers
181 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
278 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 ...
1
vote
1answer
214 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} ...
5
votes
1answer
212 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 ...
4
votes
1answer
161 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 ...
3
votes
2answers
242 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
338 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 ...
