# 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.

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### Blow up at point is finite?

Let $X$ be an affine algebraic curve with $0 \in X$ and $\tilde{X}$ the strict transform of $X$ w.r.t the blowup of $X$ at $0$. How to prove that $\pi \colon \tilde{X} \to X$ is finite? Is it even ...
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### Why is this construction a complex manifold?

I am beginning to study blow-ups, and in the development of the blow-up of $\mathbb C ^2$ in the origin, the author claims without further clarification that the following construction yields a ...
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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 ...
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### rational map on curves coincide iff tangent

The following is an exercise from Shafarevich. Let $\varphi: \mathbb{P}^2\rightarrow\mathbb{P}^4$ be the rational map defined by $$\varphi(x_0:x_1:x_2) = (x_1x_2:x_0x_2:x_0x_1).$$ Consider the ...
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### Canonical bundle of blow up at singular point

Let $X$ be a complex variety/ manifold with one singular point $x_0\in X$. If we blow up $X$ at $x_0$, we obtain a smoot variety/manifold with exceptional divisor $Y$. How can we calculate the ...
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### Euler Characteristic of a blowing up

I'm trying to compute the Euler Characteristic of the blowing up of $\mathbb{C}\mathbb{P}^2$ at $n$ points. Does anyone know how could I do this?
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### Is it possible to blow-up in codimension one?

In the context of complex manifolds, one can consider a blow-up along a complex submanifold. For a linear subspace of $\mathbb{C}^n$ there is a general procedure to perform such a blow-up: for a ...
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### Exceptional divisor as projectivization of the tangent space

Let $X\subseteq \mathbb{P}^n$ be a projective variety, and let $p\in X$. I define the blow up of $X$ at $p$ as the closure $\Gamma$ in $X\times \mathbb{P}^{n-1}$ of the graph of the projection $\phi$ ...