All Questions
21 questions
1
vote
2
answers
85
views
Computing $K_{X'/\mathbb{A}^2}$
Let $k$ be an algebraically closed field of characteristic zero, $X=\mathbb{A}^2_k$ and $X'=\operatorname{Bl}_{(x,y)} X$. I'm looking to compute $K_{X'/X}$.
Note that $$X'=\operatorname{Proj} k[x,y,...
- 265
1
vote
1
answer
62
views
Projection Morphism of Blowup
I'm currently reading the article A Short Course on Geometric Motivic Integration, by Manuel Blickle. In his proof of Theorem 3.3, the author considers the following.
Let $X'=\operatorname{Bl}_{0}\...
- 265
3
votes
0
answers
153
views
Is every normalization a blowup?
Is the normalization of a variety always a blowup along some coherent ideal sheaf? If not, I would like to see a concrete counter-example.
Let $Y \to X$ be the normalization. The answer is positive in ...
- 168
0
votes
0
answers
116
views
$A_1$ Du Val singularity and blow ups
Blowing up $\operatorname{Spec}K[x,y]=\mathbb{A}^2$ along $\operatorname{Spec}K[x,y]/(x^2,y)$ in the $U_B$ chart gives a singular point $U_B=\text{Spec}K[x,y][a]/(ya-x^2)$ at the origin. This ...
- 2,451
3
votes
0
answers
125
views
Blowing Up along Reduced vs Non-Reduced Subschemes
In Eisenbud's book, 'The Geometry of Schemes' (see Proposition IV-40), he demonstrates a connection between blowing up schemes along reduced and non-reduced subschemes. Specifically, he illustrates ...
- 2,451
2
votes
1
answer
87
views
Unicity in Hartshorne Corollary II.7.15
What am I missing in the following:
Let $X=\mathbb{A}^2_k$ be the affine plane over an algebraically closed field $k$, and let $O$ be the origin. Let $\tilde{X}$ be the blow-up of $O$. If $\mathcal{I}$...
- 2,839
1
vote
1
answer
215
views
Dependence of strict transform on the subscheme along which we blow up
It is stated in https://stacks.math.columbia.edu/tag/080C that
Note that taking the strict transform along a blowup depends on the closed subscheme used for the blowup (and not just on the morphism $...
- 168
3
votes
1
answer
139
views
Blowup extends a regular map to $\mathbb{P}^{N+1}$
Let $(X_0,X_1,...,X_n)$ homogeneous coordinates of $\mathbb{P}^n$ and let assume that
$X^r \subset \mathbb{P}^n$ is a complex variety where
$x:= (1,0,...,0) \in X$ and $X$ isn't a cone with vertex $x$....
- 8,449
2
votes
0
answers
52
views
Blowup of $\Bbb{A}^2$ at the intersection of a cubic with a triple line
I'm familiar with blowups in classical algebraic geometry but I'm still learning about blowup of schemes.
For now I'm trying to be as concrete as possible, because the formal definition is ...
- 10.5k
2
votes
0
answers
82
views
Map between two Blow-ups
Let $\widetilde{X}$ denote the blow-up of a scheme $X$ with respect to a sheaf of ideals $\mathcal{I}$. Let $Y$ be a closed subscheme of $X$, such that $\mathcal{I}\mathcal{O}_Y$ (the inverse image ...
- 63
3
votes
1
answer
148
views
Section of blowup map of schemes
Let $X$ be a scheme. Let $\mathcal{I} \subseteq \mathcal{O}_X$ be a quasi-coherent sheaf of ideals on $X$, and let $Z \subseteq X$ be the closed subscheme corresponding to $\mathcal{I}$. Let $X' := \...
- 5,234
2
votes
1
answer
134
views
Question concerning blowups: Closed immersion $\operatorname{Proj}(A+\mathfrak m+\mathfrak m^2+...)\rightarrow \operatorname{Spec}(A[T_1,...,T_n])$
I am looking at the following exercise:
If $A$ is a ring with maximal Ideal $\mathfrak{m}$, define $\tilde{A}:= A\oplus\mathfrak{m}\oplus\mathfrak{m}^{2}\oplus...$. Assume $\mathfrak{m}$ is generated ...
- 1,572
0
votes
1
answer
253
views
Closure of the pre-image of the closed subscheme of a blow-up gives the whole blow-up?
Let $X$ be a Noetherian scheme and $Z$ be a closed subscheme of $X$. Let $\pi: \tilde X\to X$ be the blowup along $Z$ so that $\pi|_{ \pi^{-1}(X\setminus Z)} : \pi^{-1}(X\setminus Z)\to X\setminus Z$ ...
- 515
5
votes
0
answers
106
views
Are these types of blow-ups affine?
When reading Stacks Project I encountered an argument that shows some blow-up is affine. It seems that the situation can be generalised, but I am not very sure if I am missing something, so I would ...
- 16.8k
1
vote
0
answers
360
views
Higher Direct Images of Birational Morphisms between Regular Schemes
Let $X$ be a normal, of finite type over a field of characteristic zero and regular scheme. Let $s \in X$ be a closed point. Assume we have a proper birational map
$f: Y \to X$
with the property ...
- 8,449
1
vote
1
answer
339
views
Blow Up of Integral Curve
My questions refer to some arguments used in Liu's "Algebraic Geometry and Arithmetic Curves" in Chapter 8.1.4 (p 330). Here the excerpt:
We consider the blow-up (I'm working with the definition ...
- 8,449
1
vote
1
answer
877
views
Blow-up of the affine plane in the origin, using schemes.
I am working through section IV.2 about blow-ups of the book Geometry of schemes by Eisenbud and Harris.
I am having some trouble understanding the details of the following example, which they discuss ...
- 329
3
votes
1
answer
474
views
Is $\Bbb P^1_k$ simply the blow up of $\operatorname{Spec}k$?
I was playing around with the definition of a blow up when I encountered something interesting.
Theorem IV-23 Eisenbud & Harris
Let $X$ be a scheme and $Y\subset X$ a closed subscheme. Let $\...
2
votes
2
answers
482
views
Hartshorne Example II.7.17.3
Let $X$ be a smooth projective variety over $\mathbb{C}$ of dimension $n$. Let $L$ be a line bundle on $X$, and let $V\subset H^0(X,L)$ be a subspace of sections. Suppose that $s_1,..,s_{l+1}$ is a ...
- 3,442
1
vote
0
answers
328
views
Description of Blow up via Ress Algebra
Let $X=\mathrm{Spec}(R)$ be an affine scheme and $Z=\mathrm{Spec}(R/I)$ be a closed subscheme of $X$ defined by an ideal $I$ of $R$. The blow up of $X$ along $Z$ is (say)$\mathrm{Bl}_{Z}(X)=\mathrm{...
- 5,209
2
votes
0
answers
101
views
Fibers of the Blow up of $\mathbb{A}^6$ over the center defined by ideal defined by vanishing of rank two minors.
Let us consider set of all matrices \begin{pmatrix}{}
x & u & v \\
u & y & w \\
v & w & z \end{pmatrix} where $x,y,z,u,v,w\in \mathbb{R}$ which is equivalent to $\mathbb{A}^...
- 5,209