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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.

Blowup is a technique in geometry (especially algebraic and differential, and, by extension, the study of pseudo-differential operators) for resolution of singularities. It should not be confused with the formation of singularities in solutions of ordinary or partial differential equations.

For more information, one should consult Wikipedia on singularity resolution.

380 questions
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What is the Picard group of $z^3=y(y^2-x^2)(x-1)$?

I'm actually doing much more with this affine surface than just looking for the Picard group. I have already proved many things about this surface, and have many more things to look at it, but the Picard group continues to elude me. One of the…
topspin1617
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Intuition for Blow-up.

If I blow up a complex manifold along a submanifold, can you give me a picture to have in mind for the blown-up manifold? Can you also tell me why this is the right picture?
nick
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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!
mick
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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 $X$ defined in $\mathbb{C}^2\times \mathbb{P}(a,b)$ by…
John Simis
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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 about it it is not specified how), and the…
Pandora
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Blowing up a point in projective n-space $\mathbb{P}^n$

I have clearly understood the blowing up of $\mathbb{A}^n$ at the origin and it is the zero locus of the polynomials $x_{i}y_{j} = x_{j}y_{i}$ in the mixed product space $\mathbb{A}^n \times \mathbb{P}^{n-1}$ where $(x_1,...x_n) \in \mathbb{A}^n$…
Tittu
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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 nonsingular variety), which is that you replace the…
Cass
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Understanding blowups at nonreduced loci

Blowing up at a reduced subscheme is geometrically clear to me: it is just like blowing up at a reduced point, and point on the exceptional divisor corresponds to different tangent lines through the base point. However, I always have problem…
User X
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Where do toric varieties appear naturally?

I'm reading Fulton's book. There's an awesome theorem that classifies all smooth toric surfaces as blowups at points starting from either $P^2$ or some Hirzebruch surface. I want to be more excited about the result rather than the proof, but I don't…
Andy
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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 transform of $a$ in $P'$ is the ideal $str(a;P')$ of…
Marra
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Direct image of structure sheaf under blow-up along non-singular subvariety

I'm trying to prove the following statement: Theorem A Let $X$ be a non-singular variety over a field $k$ and let $Y \subset X$ be a smooth subvariety. Consider the blow-up $f : \widetilde X = Bl_Y(X) \to X$. Then for $i > 0$: $$R^i f_* \mathcal…
Jędrzej Garnek
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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$, $ \frac{{\partial f}}{{\partial y}}=5y^4$, then…
Carlos I.
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Blowing up the whitney umbrella over the z-axis

My professor gave us the example of the Whitney Umbrella as an example of a non-trivial resolution of singularities. I'm aware that to resolve the singularities of the Whitney Umbrella, I need to first, blow up the Whitney Umbrella over the…
Rushabh Mehta
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Direct Image by a Blow up

Let $\pi : \widetilde{X} \longrightarrow X$ be the blow up morphism of $X$ a long of $Y \subset X$, with exceptional divisor $E$ and $\text{dim}Y > 0$, where $X$ and $Y$ are smooth projectives schemes. For an answer to a question asked here in…
Allain JF
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Universal property of blow up of complex analytic space

I know that there is a universal property of blow-ups in the algebraic setting (see Wikipedia). How does this translate to the case of complex geometry and holomorphic/bimeromorphic maps? I am particularly interested in the case of a blow up of a…
abdul
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