All Questions
Tagged with blow-ups complex-geometry
15 questions
14
votes
1
answer
830
views
$\mathbb{Z}/2$ is to $\mathbb{Z}/3$ as K3 is to what?
I'd like to know "what" (say, in the classification of complex surfaces) the following complex manifold $X$ is:
Construction: Let $\Lambda$ be the hexagonal lattice in $\mathbb{C}$; that is, the ...
- 3,212
6
votes
1
answer
1k
views
Blowing-up an ordinary double point, then contracting the exceptional locus to a curve
Let $X\subset\mathbb P^4$ a projective hypersurface with an ordinary double point at $o\in X$.
Blow-up $\mathbb P^4$ at $o$ and let $E\simeq\mathbb P^3$ the exceptional divisor of this blow-up. ...
- 7,902
4
votes
2
answers
2k
views
Topology of the blowup of a surface at a point (connected sum)
Let $S$ be a complex algebraic (smooth) surface and $\widetilde{S}$ be the blowup of $S$ at a point $p\in S$.
I would like to understand the statement:
As a topological manifold, $\widetilde{S}$ ...
- 761
4
votes
3
answers
1k
views
Divisor class group on blowup of nodal surface
The following got no answer on mathstackexchange. I believe it not to be hard, but maybe it is a little specialized?
All varieties will be over $\mathbb{C}$ and projective unless stated otherwise.
...
- 469
4
votes
1
answer
584
views
What is the relation between holomorphic blow-up and symplectic blow-up?
McDuff has shown us exactly how the symplectic blow-up procedure along a symplectic submanifold affects the symplectic structure in the ambient space, i.e., if $\omega$ is the original symplectic form,...
- 191
4
votes
1
answer
378
views
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 ...
- 329
4
votes
0
answers
259
views
Blow-up of a stratified space
Let $X$ be a smooth projective variety over $\mathbb{C}$, and $D_1, \ldots, D_n$ be a collection of simple normal crossing divisors. The divisors induce a stratification $\mathcal{T}_X$ of $X$.
Let $...
- 283
4
votes
0
answers
101
views
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. ...
- 1,141
3
votes
1
answer
1k
views
Blow-ups and cohomology
Let $W\subset \mathbb P^4$ be a smooth surface and let $X$ be the blow-up of $\mathbb P^4$ along $W$ with exceptional divisor $E$. By Griffiths-Harris, we have $$A^2(X)=A^2(\mathbb P^4)\oplus A^1(W),$$...
- 587
2
votes
1
answer
199
views
Section 3 of Atiyah's "On analytic surfaces with double points" — some questions
I have some questions about section 3 of Atiyah's "On analytic surfaces with double points," a short 9 page paper. Section 3 is all dedicated to proving lemma 4.
Near the end of section 3, ...
- 129
2
votes
2
answers
564
views
Holomorphic vector fields on blow-ups of CP^2
On $X=CP^2\#k{(-CP^2)}$ in $k$ generic points, let $h^i=\dim H^i(T^{1,0}X)$, for $i\ge 0$. First, we know $h^i=0$ for $i\ge 2$. By Riemann–Roch formula, I obtain that $h^0-h^1 = 8-2k$. Would someone ...
- 516
2
votes
0
answers
339
views
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 ...
1
vote
1
answer
359
views
Three-dimensional analogues of Hirzebruch surfaces
There are several ways of describing a Hirzebruch surface, for example as the blow-up of $\mathbb{P}^2$ at one point or as $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(n))$....
- 183
1
vote
0
answers
149
views
Blowing up $\mathbb{CP}^2$ nine times and exactness of symplectic form
Consider the (symplectic) blow up $\operatorname{Bl}_k(\mathbb{CP}^2)$ of $\mathbb{CP}^2$ at $k$ points. I have heard that for $k=1,2,\ldots,8$ the size of the balls been blown up can be choosen in ...
- 191
0
votes
1
answer
166
views
Blow up and critical points of the projection map
Denote $Z=V(x_1, \dots, x_{n-1}) \subset \mathbb{C}^n$ and let $Bl_Z(\mathbb{C}^n)$ be the blow up of $\mathbb{C}^n$ along $Z$ together with the projection map $\pi \colon Bl_Z(\mathbb{C}^n) \to \...