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.

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Blow up of reduced scheme is reduced

Why is the blow up of a reduced scheme reduced? This is in Vakil's notes (22.2.C) right after he gives the universal property of the blow up involving Cartier divisors, but before the explicit ...
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Blow up solution of a Riccati's equation.

Consider the Cauchy problem $$ \left\{ \begin{array}{l} \dot x=x(t)^2+t\\ x(0)=0 \end{array} \right. $$ Show that its solution is not defined in $[0,3]$.
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134 views

Blow Up: Resolution of Singularity

For blow ups, I have worked only in $\mathbb{CP}^2$. Once I locate the base-point, say $[x,y,z]=[0,1,0]$, I go back to $\mathbb{C}^2$ by considering the chart $y=1$. I then proceed to blow up ...
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extraction of relative picard number one of an irreducible divisor

my question is the following: suppose an algebraic variety $X$ is given, along with a divisor $E$ corresponding to an algebraic valuation of the rational functions field $K(X)$. I know that one can ...
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Factoring a birational morphism through blowup

Let $X,Y$ be smooth, proper varieties, and $f: X \to Y$ be a proper birational morphism. Suppose $E$ is a smooth, irreducible exceptional divisor, with the image $f(E)$ also smooth. Let $I$ be the ...
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156 views

Projection of a hypersurface from a point

Let $k$ be an algebraically closed field. We consider the projective space $\mathbb P_n$ over defined over $k$, the point $Q=(0:\dots:1)$, the hyperplane $H=\{X_n=0\}$ and a hypersurface $X$. We want ...
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34 views

Blow up of a solution of the p-Laplace equation at a boundary point

My doubt is about a proof from the lemma 2.8 of the article 18 of this homepage: http://www.math.kth.se/~henriksh/Henriks_page/publications.html The result that I said is: Lemma: Let $D_1$ and $D_2$ ...
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64 views

Blow-up in Projective Space: Choosing the Appropriate Chart

Consider $x^2+y+\alpha=0$ (made-up example), where $(x,y)\in\mathbb{C}^2$ and $\alpha\in\mathbb C$ is a free parameter. The equation defines a family of curves. The curves have no common points in ...
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133 views

Blow-ups in Projective Space

This is in regards to a question (no solutions or comments thus far :-() I asked earlier in regards to the blow-up of an elliptic curve: Question Let $f(x,y)=y^2-4x^3+ax+b$, where $(x,y)\in\mathbb ...
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Blowing up a Singular Point More Than Once.

I am trying to understand how $I_n$-fibres appear in an elliptic surface by performing a sequence of blow-ups. To be concrete, I am looking at the following elliptic surface given in Weierstrass ...
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Resolution of Singularities: Base Point

Consider the curve $y^2=4x^3-ax-b$, where $a$ is a fixed constant and $b$ is a free constant. For each value of $b$ we get a family of curves. Part 1: Show that the family of curves intersect at ...
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Projective line bundles and blowing up

I'm trying to understand some facts about Chern classes. Looking up for some special examples I found that seems the total space of the projective line bundle $P(M\oplus M^{-1})\to \mathbb{CP}$ must ...
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82 views

How to understand blowing up a submanifold

I am trying to understand the idea of blowing up a submanifold of a smooth real manifold. The definition I know is replacing the submanifold by its unit tangent bundle (however, in the place I read ...
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45 views

Degree of blow up of a smooth projective surface

Let $X$ be a smooth projective hypersurface in $\mathbb{P}^3$ and $\{x_i\}_{i \in I}$ be a finite set of closed points in $X$. Let $X'$ be the blow up of $X$ at these points. Then, $1)$ Is there a ...
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Associated graded ring of a Fermat cubic

Let $R$ be a graded Fermat cubic, i.e. $R$ is a graded ring given by $$ R=\mathbb{C}[x,y,z]/(x^3+y^3+z^3), $$ with a standard grading $\operatorname{deg}(x)= ...
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Blowup of $\mathbb{P}^3$ along the ideal $(w^3 + x^3 + y^3 + z^3, w^4 + \alpha wxyz)$ for fixed $\alpha \in k$

I want to compute the blowup of $\mathbb{P}^3$ along the ideal $(w^3 + x^3 + y^3 + z^3, w^4 + \alpha wxyz)$ for fixed $\alpha \in k$. I've been working with blowups for a couple weeks, but this seems ...
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blow up of algebraic variety

If we have an Affine algebraic variety $V$, is its blow up necessarily another algebraic variety? Is it possible to extend a holomorphic function on a complex manifold to the blow-up of the manifold? ...
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Is it true that blowing up a quasi-affine variety at a nonsingular point never introduces new singularities?

If we let $M$ be a quasi-affine variety, is it true in general that the blowup of $M$ at a non-singular point $p$ does not introduce new singularities? I came across this statement in my reading, but ...
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Blowup of $\mathbb{P}^n$ at a point is irreducible

The blowup of $\mathbb{P}^n$ at a point is irreducible. This seems clear intuitively, but I'm not sure how to prove it. Thoughts?
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Linear system of degree $d$ curves passing $m$ times through $P$ in the blow-up at $P$.

Given a point $P$ in $\mathbb{P}^2$ and a natural number $m$ we consider the linear system $\mathcal{L}$ of curves of degree $d$ passing $m$ times through $P$. If $H$ is the line class of the plane, ...
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Blow up commute with base change

Let $X$ be an irreducible Noetherian scheme and $X_{red}$ be the reduced subscheme associated to it. This induces a natural morphism from $X_{red}$ to $X$. Fix a point $p$ in $X$ (by a point we mean ...
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Terminology for blow-ups in algebraic geometry

Definition (from Eisenbud-Harris' Geometry of Schemes): Let $X$ be any scheme, $Y \subset X$ a subscheme. We say that $Y$ is a Cartier subscheme in $X$ if it is locally the zero locus of a single ...
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158 views

Blowing up at a point

Let $C:=\{y^2=x^3\}$ be a curve in $\mathbb C^2$ and $\pi:X\to \mathbb C^2$ the blowing up of $\mathbb C^2$ at the origin $o:=(0,0)$. The dimension of $X$? Take another blowing $\pi':X'\to X$ at the ...
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Can one calculate a blowup étale locally?

In what follows, I assume all schemes are Noetherian and of finite type. It follows from the universal property that one can calculate the blowup of a coherent ideal sheaf $\mathscr{I}$ on a scheme ...
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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 ...
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help with a blowup

I'm currently learning the basics of blowups and I find that a bit hard. I would like to work out the following example. Could you help me? Let $k$ be a field and ...
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170 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 ...
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103 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 ...
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140 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 ...
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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 ...
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801 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!
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113 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 ...
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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 ...
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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 = ...
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169 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 ...
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260 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 ...