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# Questions tagged [blow-ups]

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### How to compute direct images for a blowing up?

Let $X$ be a smooth algebraic variety, $Z\subset X$ a smooth closed subvariety, and $\pi:\tilde{X}\to X$ the blowing up of $X$ along $Z$. Let $E\subset\tilde{X}$ be the exceptional divisor, and $n$ an ...
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### Direct image of sheaves for blowing ups [duplicate]

Let $X$ be a smooth algebraic variety, $Z\subset X$ a smooth closed subvariety, and $\pi:\tilde{X}\to X$ the blowing up of $X$ along $Z$. Let $E\subset\tilde{X}$ be the exceptional divisor, and $n$ an ...
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### Blow up at an ordinary double point

Let $X \subset \mathbb{C}^n$ be a complex complete intersection surface with only ordinary double point singularities. Let $o$ be such an ordinary double point. Let $\tilde{X}$ be the strict transform ...
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### Cohomology of fibers of a morphism of a blowup of affine space

Consider $\mathbb A^n$ and let $\Sigma$ be a subdivision of its toric fan $\mathbb R^n_{\geq 0}$. This induces a toric blowup $\pi : Y \to \mathbb A^n$. Let $X \subseteq Y$ be the preimage of the ...
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### How to show the inclusion of the exceptional divisor is the zero section of the line bundle

Let $k$ be a field and $R$ be the ring $k[x,xy,xy^2,xy^3]$. Let $I$ be the ideal of $R$ generated by $x,xy,xy^2,xy^3$.Let $X$=Spec$(R)$ and $\tilde{X}$ be the blow-up of $X$ along $I$.I managed to ...
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### Do blowups generate the birational equivalence relation?

Suppose $X$ and $Y$ are birational varieties (projective, irreducible, over $\mathbb C$) of the same dimention. Is there a sequence of blow-ups and blow-downs that makes $Y$ from $X$?
1 vote
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### On intersections of exceptional divisors

Let $X$ be a smooth, projective variety of dimension $n \ge 2$, $L$ a very ample line bundle on $X$ and $\pi: \widetilde{X} \to X$ be the blow-up along a closed subvariety of codimension at least $2$. ...
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### Problem in calculating the global sections of $\mathcal{O}_{\mathbb{P}^3}(d)\otimes \mathcal{I}_Z$

This is an additional question to the one I posed in Equivalence of sequences of blowups of $\mathbb{P}^3$ Let $[x_1,x_2,x_3,x_4]$ be coordinates of $\mathbb{P}^3$ and $Z\subset \mathbb{P}^3$ the ...
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### Equivalence of sequences of blowups of $\mathbb{P}^3$

Let $[x_1,x_2,x_3,x_4]$ be coordinates of $\mathbb{P}^3$ and $Z\subset \mathbb{P}^3$ the subscheme given by the ideal $$I_Z=(x_1,x_2,x_3^2) \subset \mathbb{C}[x_1,x_2,x_3,x_4]$$ i.e. $Z$ is a double ...
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Let $\mathbb P(V)$ be a projective space containing $Y$ as a subvariety. Let $Z$ be the blow-up of $\mathbb P(V)$ along $Y$. Clearly there exists a divisorial contraction given by $b: Z \to \mathbb P(... 9 votes 1 answer 703 views ### Inverse Kirby knot Given an (oriented framed) knot$K$in the 3-sphere$S^3$, we can perform a surgery along$K$to get another 3-manifold$M$. From$M$, we can perform the inverse surgery back to$S^3$. However, the ... 9 votes 0 answers 298 views ### Proof of Artin–Rees / Krull intersection motivated by universal property of blowup I was very confused by the proof of Artin–Rees / Krull intersection theorem when I was younger. Now that I learnt about blow up— I saw the Rees algebra again and I want to now gain a better ... 4 votes 0 answers 156 views ### How to use blow-up to prove the boundary regularity for a harmonic function While reading the book Regularity Theory of Elliptic PDE I’m confused with a theorem: Thm. 2.30. Let$\alpha \in (0,1)$and$k \in N$with$k \leq 2$, and let$\Omega$be a bounded$C^{k, \alpha}$... 2 votes 0 answers 273 views ### On the exceptional divisor of the resolution of indeterminacy locus of rational map Let$f:X \dashrightarrow Y$be a rational map between smooth, projective varieties over$\mathbb{C}$. We know that there is a resolution of the indeterminacy locus using which we obtain a smooth, ... 2 votes 1 answer 208 views ### Blow-ups of surfaces over a field Let$S$be a smooth projective surface of Picard rank$\rho(S)$over a field$K$, and$\overline{S}$its algebraic closure. Take a point$p\in\overline{S}$and denote by$\overline{X}$be blow-up of$\... 243 views

### Which projective varieties cannot appear as an exceptional divisor of a blow-up of the projective space

I am looking for examples of projective varieties (over $\mathbb{C}$) of dimension, say $n$ which cannot appear as an exceptional divisor of a blow-up of $\mathbb{P}^{n+1}$ along some closed subscheme....
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### Base locus of a linear system after blow-up

I have a smooth proejective fourfold $X$ with an effective divisor $D$ on it. The base locus ${\rm Bs}|D|$ of the linear system $|D|$ is a smooth rational curve $C$ and a generic member of $|D|$ has ...
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### Help about "Varieties with small Dual Varieties" by L.Ein

I'm studying the paper "Varieties with small Dual Varieties" by L.Ein and in the construction he gives about the $10-$dimensional spinor variety $S_4 \subset \mathbb{P}^{15}$ I'm finding ...
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### General conditions for normality of blow-up

Let $X$ be an integral, affine, normal complex surface. I am looking for conditions on zero-dimensional closed subschemes $Z$ in $X$ such that the reduced scheme associated to the blow-up of $X$ along ...
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### Relative canonical class of blowing-up a flag ideal

Let $X$ be a smooth complex projective variety of dimension $n$. Consider a flag ideal $I$ on $X\times \mathbb{P}^1$, namely, $$I=I_0+I_1t+I_2t^2+\cdots+I_{N-1}t^{N-1}+(t^N)\,,$$ where $t$ is the ...
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### Serre vanishing on one-point blow-ups

This is basically the last step of problem 5.3.7 in Huybrechts' Complex Geometry. Let $X$ be a complex manifold, $x \in X$, $E$ a holomorphic vector bundle on $X$ and $L$ a positive line bundle. ...
Let $R$ be a discrete valuation ring with fraction field $K$ and $C, D$ two nodal (=prestable) curves over $\operatorname{Spec} R$. If I have a map $C_K \to D_K$ between the restriction of the curves ...