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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Geometric Intuition Behind Blowing Up a Cusp on a Plane Curve?

I'm reading Hartshorne AG V.3 on monoidal transformations and embedded resolutions. I understand one sort of intuition behind blowing up a point on a surface (or more generally a subvariety of a ...
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52 views

Blowing up families of singular curves

I am stuck with a simple example, but I guess the more general question would be whether blow ups commute with restrictions to subsets (points) of the blow-up locus. Over $\mathbb{C}$, suppose that ...
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1answer
45 views

Blow-up of pair of intersecting lines

I have the reducible variety $X=\mathbb{V}(x_1x_2)\subset\mathbb{A}^2$, which is a pair of lines intersecting transversely, and I would like to compute the blow-up at the origin. The Rees ring of the ...
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48 views

Is exceptional divisor always a Projective bundle over the centre?

Let $f:\tilde X \rightarrow X$ be a blow up at center Z. Is $f^{-1}(z)=\mathbb{P}^{k}$ ?for some $k$, $\forall$ $z\in Z$ In the case of blow-up of $\mathbb{A}^{n}$ at origin it is very clear that the ...
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46 views

Blowup of six points in $\mathbb{P}^2$

I have been looking at sequential blowups of six points $P_i$ in $\mathbb{P}^2$. In the general case, we can identify the blowup with a nonsingular cubic surface in $\mathbb{P}^3$ and under this ...
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28 views

induced isomorphism on the blow up

Let $k$ be algebraic closed field and let $\mathbb{P}^2$ the projective space over $k$ of dimension 2. Consider the birational map $$f:\mathbb{P}^2 ---> \mathbb{P}^2, [x_0, x_1, x_2] \mapsto ...
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1answer
64 views

Blow-up of $\mathbb P^2$ in 2 points is isomorphic to blowup of quadric in 1 point

How to show that blow-up of $\mathbb P^2$ in 2 points is isomorphic to blowup of quadric in 1 point? I think it is a standard fact, but Google can not help me with it. Update: I found an answer ...
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1answer
40 views

Blowup of smooth subscheme of smooth scheme is smooth

In Vakil's notes (http://math.stanford.edu/~vakil/216blog/FOAGjan2915public.pdf), Theorem 22.3.10, he shows that, if $X\hookrightarrow Y$ is a closed embedding of smooth varieties over $k$, then ${\rm ...
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The 2 Charts of “Blowing up the Origin in $\mathbb{C}^2$ ”

Consider the algebraic curve $\mathcal{C}$ given by $f(x,y)=0$, where $(x,y)\in\mathbb{C}^2$. Suppose that the singular point of $f$ is $p=(x,y)=(0,0)$. The blow-up of $p$ is given by ...
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43 views

Understanding BlowUp Computation in Singular

Many of us might know that "Singular" is a computer algebra system for Algebraic Geometry, Commutative Algebra and Non-commutative algebra. This is a procedure in "Singular" for computing blowups. ...
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Family of curves with one base point and a blow-ups.

Suppose that a smooth complex surface $X$ is covered by a family of curves $\{C_\alpha\}_{\alpha\in\mathbb P^1}$. Suppose moreover that $\bigcap C_\alpha=p\in X$, and that these intersection are ...
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63 views

Strict transform and projective space bundle

This concerns Example 2.11.4 in chapter V of Hartshorne's Algebraic Geometry. $\mathscr{E} := \mathcal{O} \oplus \mathcal{O}(1)$ on $\mathbb{P}^n$. $P_0$ is the point $[1 : 0 : \ldots : 0]$ in ...
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1answer
19 views

blow up time for function $G(t)$

given $G:[0,\infty)\rightarrow \mathbb R$ $$ G'(t)\geq -\lambda G(t)+ f(G(t)) $$ where $\lambda >0$ and $f$ is a convex function. $G(0)>0$ is also given. Show that $G$ blows up at finite ...
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1answer
74 views

Submanifold associated to blow up.

I 'm trying to understand the classical blow up given by $$X=\{(x,[y])\in \mathbb{R}^n \times \mathbb{P}_N / \hspace{0.2cm} \exists \lambda \in \mathbb{R} \hspace{0.3cm} \text{such that} ...
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1answer
74 views

Strict Transform of a Line in a Blow Up

Consider the blow up $\pi:B \to \mathbb{A}^2$ of the origin in $\mathbb{A}^2$. Let $L=Z(ax+by)$ be a line through the origin in $\mathbb{A}^2$ and let $\widetilde{L}$ be the strict transform of $L$ ...
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24 views

What is the relation between modification and blow-up along the base locus?

Let $X \hookrightarrow \mathbb{P}^N$ be a compact sub manifold of dimension $n$. Let $\mathbb{P}(d, N)$ denote the projectivization of degree $d$ homogeneous polynomial on $\mathbb{P}^N$. Each ...
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1answer
49 views

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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44 views

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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160 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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1answer
174 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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52 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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1answer
142 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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235 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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60 views

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 ...
2
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1answer
272 views

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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1answer
179 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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1answer
75 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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1answer
90 views

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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40 views

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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1answer
112 views

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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1answer
285 views

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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1answer
174 views

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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187 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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1answer
120 views

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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364 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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1answer
111 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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1answer
219 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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1answer
143 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 ...
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1answer
1k 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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137 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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65 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 ...
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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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1answer
215 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 ...