A technique in geometry (especially algebraic and differential, and by extension to study of pseudo-differential operators) for resolution of singularities. Not to be confused with the formation of singularities in solutions of ordinary or partial differential equations.
7
votes
2answers
50 views
A question about the strict transform on blow-ups
I arrived at the following phrase at a material that I'm reading:
Let $\pi :N'\rightarrow N$ be the blow-up of center $P$. For a given
$a\in\mathcal{O}$ and $P'\in\pi^{-1}(P)$, the strict ...
4
votes
1answer
63 views
Classifying a Branched Covering Space
This question comes from the proof of proposition 2.2 in Henry Laufer's 'Normal Two-Dimensional Singularities" text. I am excerpting the part I don't understand, and I think it's a self-contained ...
1
vote
1answer
52 views
Divisor class group on blowup of nodal surface
All varieties will be over $\mathbb{C}$ and projective unless stated otherwise.
In Beauville - complex algebraic surfaces, the following is described: Let $S$ be a smooth surface and $p \in S$ a ...
2
votes
1answer
58 views
Algebraic definition of blow-ups
Let $X$ be a scheme. Choose $C\subset X$ be a subscheme of $X$ and let $\mathcal{I}\subset \mathcal{O}$ be the corresponding ideal sheaf. Then $\mathcal{B}=\oplus_{d\ge0}\mathcal{I}^d$ is a sheaf of ...
1
vote
1answer
315 views
What is a blow-up?
Can anyone explain to me what a blow-up is? If would be great if someone could provide a definition and some examples. Any free introductory texts are welcome too. Thanks!
1
vote
0answers
64 views
Blow up of a subvariety
I have a problem understanding blowing up a subvariety. I've had some experience blowing up singular points on curves. I suppose the best way to address the question is to pose an example. Take the ...
1
vote
0answers
39 views
Descent through blow-up
Let $X$ be a variety with $Y \subsetneq X$ a proper closed subvariety. Let $Z$ denote the blow-up of $X$ along $Y$. Let $f: Z \rightarrow X$ be the canonical map. Suppose that we have a coherent sheaf ...
2
votes
0answers
70 views
Degree 1 elements in a graded ring from a blow-up perspective
This may be an elementary question but I hope this question will benefit others as much as myself.
Let $k^4 = Spec \; k[x_1, x_2, x_3, x_4]$.
Writing $Bl_{\mathcal{I}}(k^4)$ as $R = ...
2
votes
1answer
95 views
Blow-up along an ideal sheaf
Let $k^2=\operatorname{Spec} \; k[x,y]$ where $k$ is an algebraically closed field. Let $\mathcal{I}$ be the ideal sheaf defined by $(x,y)$. Then
$$
Bl_{\mathcal{I}}k^2
$$
is covered by two open ...
2
votes
1answer
148 views
blowing up $\mathbb{A}^n$ at a point.
I would like to get an informal and very visualizable explaination of the concept blowing up. I read " In blowing up $\mathbb{A}^n$ at a point $p$, the idea is to leave $\mathbb{A}^n$ unaltered except ...
