# Questions tagged [blowup]

A technique in geometry (especially algebraic and differential, and, by extension, the 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.

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### Orientability and blow-ups

Is it true that if I have a smooth (if you want algebraic) orientable surface (as a real manifold) then blowing up at a point will yield a non-orientable surface and vice versa (i.e. blowing up a non-...
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### Resolution of $D_n$ singularity and exceptional divisor

Consider a $D_n$ singularity. Let's do type $D_4$ explicitly here, by defining as 0 set of $x^2+y^2z+z^3$ in $\mathbb{A}^3$. Then do a blowup, and consider the chart given by $u=x/z$, $v=y/z$ and $z$. ...
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### About the integral closure of a DVR inside a galois extension of function field with two variables

Edit: I see my mistake now (murphy), w(X + Y) = min(w(X), w(Y)) only when the valuations are not the same. I have the following algebraic problem, which I encountred after thinking about blowups. My ...
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### Global sections of pullback under a blowup

Let $X$ be a normal projective variety and consider a blowup of $X$ $$Y \rightarrow X.$$ It is a theorem that global sections of $\mathcal{O}_Y = f^\star \mathcal{O}_X$ biject naturally (i.e. via ...
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### Left exactness of the cotangent sequence

I am reading this note of Blickle on motivic integration and I am stuck at a technical point. At the beginning of the appendix, we are given a proper birational morphism $f: X' \longrightarrow X$ ...
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### Blowing up of a singular subvariety

I'm reading Kollár Mori Chapter 2.3. And they state the following lemma: Lemma 2.29. Let $X$ be a smooth variety and $\Delta = \sum a_iD_i$ a sum of distinct prime divisors. Let $Z\subseteq X$ be a ...
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### An explicit example of the blowup of an algebraic curve

Consider the curve $x^3+y^4=0$ in $\mathbb{A}^2$, I am trying to retrieve the Dynkin diagram of $E_6$ by consecutive blowups at the origin, however I run into issues after performing just one of these ...
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### Total and strict transforms of blowups

I have a rather simple question about the nature of blowups: Let's say I have $\pi: X \rightarrow Y$, which is the blowup of a center $Z$ in $Y$. Denote the exceptional divisor by $E$. Is it correct, ...
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### Blow up points on real projective plane

What is the genus of the blow-up at $3$ non-colinear points of $\mathbb{R} \mathbb{P}^2$? This question occurred to me as I was trying to eliminate the sigular points at the vertices of a given ...
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### Sheaf of divisor divisor $rH+sE$ on blowup $\operatorname{Bl}_0(\mathbb{P}^2)$ has basepoints but is generated by global sections?

I work with toric varieties and took a look at the blowup $\pi:X=\operatorname{Bl}_0(\mathbb{P}^2)\to \mathbb{P}^2$ ($0=(0:0:1)$) with points represented as $(x_0:x_1:x_2),(y_0:y_1)$. Technically we ...
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### Arrow-reversing Proj and blow-up

Let $X$ be a normal projective variety over the field of complex numbers. Let $Y$ be a subvariety of $X$, and let $I_Y$ be the ideal sheaf of $Y$. From what I know, I can define the blow-up of $X$ ...
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### How to use blow-up to prove the boundary regularity for a harmonic function

While reading the book Regularity Theory of Elliptic PDE I’m confused with a theorem: Thm. 2.30. Let $\alpha \in (0,1)$ and $k \in N$ with $k \leq 2$, and let $\Omega$ be a bounded $C^{k, \alpha}$ ...
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### Can blow-up of a surface be a product of two curves? [closed]

Is there any smooth projective surface $S$ over $k=\bar{k}$, such that the blow up $\tilde{S}$ along some point $x\in S$ can be written as $\tilde{S}=C_1\times C_2$ for two curves $C_i$? 319 views

### Criteria for finite time blow up for two simple ODEs

I've got two simple questions on criteria for a finite time blow up of solutions of two simple ODEs: 1: If $u$ is a solution of $u'=f(u)\ge0$, $u(0)=u_0$, how do we see that $u$ blows up in finite ...
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### Blow-up of origin of $K^2$ with $K$ a non-archimedian local field

Let $K$ be a non-Archimedian local field. We can define $\mathbb{P}^1_K$ in the standard way and the blow up \text{Bl}_0 K^2=\left\{ ((x_1,x_2),[y_1,y_2])\in K^2\times \mathbb{P}^1_K \lvert x_1y_2=... 1 vote
I would like to know when we can "zig-zag-connect" blow-ups of the same base space, by which I mean the following: Let $X$ be a $k$-scheme and $Z_1,Z_2$ two closed subschemes and \$\text{Bl}_{...