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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Motivation for equivalence of Tautological Line Bundle and ${\operatorname{Bl} _0}\mathbb{A}_k^{n + 1}$

I currently reading a book on Complex Algebraic Geometry and both the Tautological Line Bundle $L \to \mathbb{P}_k^n$ and the blow up ${\operatorname{Bl} _0}\mathbb{A}_k^{n + 1}$ are defined set ...
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45 views

Obtaining a nice map to a curve by using blowups

Let $X$ be a smooth and projective variety over a finite field (separated, finite type, integral). Then after performing a number of blowups I should be able to find a proper surjective map from $X$ ...
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61 views

How to resolve the singularity of $xy+z^4=0$?

This singularity can not be resolved by one time blow-up. I don't know how to blow up the singularity of the "variety" obtained by the first blow-up, in other words, I am confused with how to do the ...
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36 views

Blowing up a model for a plane curve

Let $R = \mathbb{C}[[t]]$ and let $\mathcal{X} \hookrightarrow \mathbb{P}_R^2$ be the arithmetic surface defined by the equation $$(X^2 - 2Y^2 + Z^2)(X^2 - Z^2) + tY^3Z = 0.$$ The generic fiber ...
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26 views

How to prove the minimal resolution of double rational points can be achieved by iterated blow-ups

In the Slodowy's survey on Kleinian singularities, there is a statement that the minimal resolution of Kleinian singularities can be obtained by iterated blow-ups, I want to know the detail of the ...
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1answer
48 views

Blow up of an ideal in $\Bbb C^2$

As described in these notes, I am trying to compute the blow up of $\mathbb C^2=\text{ Maxspec }\mathbb C[x,y]$ along the subvariety corresponding to the ideal $\langle\ x^2,y\ \rangle$ but I am ...
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42 views

Cremona Transformation and isomorfism

Let $P_1=(1:0:0),P_2=(0:1:0),P_3=(0:0:1) \in \mathbb{P}_2$ (over an algebraic closed field). Denote $U=\mathbb{P}^2 \setminus \{P_1,P_2,P_3 \}$ and consider the map $$ f:U \rightarrow \mathbb{P}^2, (...
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52 views

When is the blow up at a point “as large as possible”

The blow up, $\widetilde{X}$ of an affine variety $X \subset \mathbb A^n$ at $f_1,...,f_m$ is the closure of $\Gamma_f = \{(x,f(x)) : x \in U \}$ in $X \times \mathbb P^{m-1}$ (where $U = X \backslash ...
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70 views

Resolution of the $E_8$ singularity with a weighted blowup

I am reading Miles Reid's notes on weighted projective spaces, and I'm a little confused about a particular paragraph (notes here, page 8): A famous case is the $E_8$ singularity $X: (x^2+y^3+z^5=...
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1answer
30 views

Relation between symplectic blow-up of a compact manifold and fibre bundles over same manifold

The symplectic blow-up of a compact symplectic manifold $(X,\omega)$ along a compact symplectically embedded submanifold $(M,\sigma)$ results in another compact manifold $(\tilde{X},\tilde{\omega})$ ...
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54 views

Pull back of canonical line bundle under a blow up

The question I am asking comes from Ravi Vakil's notes, specifically Exercise 22.4.S. Let $S$ be a surface over a field $k$ and let $p\in S$ be a smooth point $k$-point. Let $B$ be the blow-up of $S$ ...
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1answer
74 views

Explicit computation of a blowup

I am working on a computation of a blowup and got stuck at a point, I hope there is somebody to help me. Consider $V=V(y^2-x^3-x^2) \subseteq \mathbb{A}^2_k$ for some algebraic closed field $k$. I ...
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1answer
62 views

Algebraic proof of a result on blow-ups in Griffiths--Harris

In chapter 6, section 4 of Griffiths--Harris, Principles of Algebraic Geometry, on page 604 (at least in my version) one reads in point 4: As the reader may check by the same method as used in the ...
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33 views

Extension of projection from a point to a Blow Up

I feel like there's something obvious I'm missing here, and I'm not looking for a whole answer, but rather just a pointer in the right direction. Suppose you have the projection from a point $\mathbb{...
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1answer
62 views

Blowing up an affine scheme at a regular point

I am reading Liu's Algebraic Geometry and Arithmetic Curves and get stuck at Lemma 8.1.2: Let $A$ be a Noetherian ring an define for an ideal $I \subset A$ the $A$-algebra $$\tilde{A}:=\bigoplus_{d\...
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38 views

A very general question about blow-up for experienced symplectic topologists and algebraic geometers

I am trying to understand the process of symplectic blow-up of compact symplectic manifolds $M^{2n}$ along compact symplectic submanifolds $X$ on a deeper level, and I am also searching for relevant ...
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1answer
73 views

Blow-up and singularities

in Hartshorne, Section 4 we have the description of the blow-up of $y^2=x^2(x+1)$ at the origin, that curve have two singularities at $(0,0)$ and $(0,-2/3)$. But the equations of the blow-up defines ...
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139 views

Computations of Blow up

Let $V=Z(y^5-x^3(x+1))$ be an irreducible variety in $\mathbb A^2$. The partial derivatives of the equation $y^5-x^3(x+1)$ at the point $p=(0,0)$ are: $ \frac{{\partial f}}{{\partial x}}=-3x^2-4x^3$,...
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45 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{...
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37 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}^...
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56 views

Griffiths & Harris Notation

I am currently reading the section on blowing up in Griffiths & Harris Principles of Algebraic Geometry and am just confused by some notation in this section. They want to define the blow up of a ...
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1answer
26 views

How to calculate convolution with logarithm numerically?

I'm trying to compute an optimisation problem, which has a cost function involving $$I=\int_0^1\log|x-y|\rho(y)dy$$ where $x\in[0,1]$ and $\rho$ is a probability density. Eventually, I will want to ...
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46 views

How can I blow-up a smooth projective surface with certain conditions?

Let $X$ be a smooth projective surface and $K$ a canonical divisor on $X$. Suppose $V$ and $W$ are subspaces of $H^0(X, \mathcal{O}_X(nK))$ (for $n$ large). Q: How can we blow-up $X$ to obtain $\pi: ...
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170 views

Secant variety and tangent lines (Harris, Algebraic Geometry: A First Course)

Given a (smooth) projective variety $X\subset \mathbb{P}^n$, we can define a rational map $s:X\times X\rightarrow G(1,n)$ that takes a pair $(p,q)\in (X\times X)\setminus \Delta$ not on the diagonal ...
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1answer
79 views

Inverse limit of blow up

Suppose $X_{0} = X$ is a complex space of dimension 2 with divisor $p_{0} \in X_{0}$. We can construct the blow-up, $X_{1}$ of $X$, which comes with a blow-down map $X_{1} \to X_{0}$. Suppose that ...
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157 views

Weighted blow-ups

I would like to understand what's a weighted blow-up in a very simple case: $\mathbb{C}^2$ blown-up in the origin with weights $(a,b)$. In found some notes online saying that this is the surface $X$ ...
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2answers
210 views

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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85 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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65 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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102 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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64 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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1answer
232 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 here:...
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1answer
108 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 $\{((x,y),[x_{...
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111 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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77 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
55 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
78 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} \hspace{0.3cm}...
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1answer
115 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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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
80 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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54 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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182 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 $(x,z)=...
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88 views

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
201 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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1answer
66 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
201 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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306 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 C^...
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101 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 ...
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1answer
300 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 ...