# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $D_2$ ...
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 ...