The blow-ups tag has no wiki summary.
4
votes
1answer
140 views
Blow-up of $\mathbb{P}^4$ along a quadric surface
Let $Q\subset\mathbb{P}^3\subset\mathbb{P}^4$ be a smooth quadric surface, and let $X = Bl_Q\mathbb{P}^4$ the blow-up of $\mathbb{P}^4$ along $Q$. Let $H$ be the pull-back of the hyperplane section of ...
7
votes
1answer
233 views
Top self-intersection of exceptional divisors
Let $Y\subset\mathbb{P}^n$ be a smooth variety of codimension two. Consider the blow-up $X = Bl_Y\mathbb{P}^n$ of $\mathbb{P}^n$ along $Y$, and let $E$ be the exceptional divisor over $Y$. Then $E$ ...
1
vote
0answers
74 views
Can discrepancy change after pseudo isomorphisms?
Let $X$ and $Y$ be two projective irreductive algebraic varieties of dimension $3$ and let $f:X\dashrightarrow Y$ be a pseudo isomorphism, i.e. a birational map which restricts to an isomorphism ...
2
votes
2answers
133 views
blow-up along non-pure dimensional subvarieties
When speaking about blow-ups, I've seen that everybody says "let $Y \subset X$ be a smooth closed subvariety of codimension $r$..."
What happens when one want to blow-up subvarieties which are not of ...
0
votes
1answer
56 views
is the exceptional divisor of an equivariant blow-up linearized?
I hope someone can help me with this.
Let $X$ be a smooth projective variety, say over the complex numbers and let $Z$ be a closed subvariety of $X$. Assume that $X$ is acted upon by a finite group ...
3
votes
1answer
117 views
Blowing up rational singularities
Let $X$ be a projective surface embedded into $\mathbb{P}^n_{\mathbb{C}}$ having at most rational singularities. Let $\tilde{X} \to X$ be the minimal resolution of $X$. Is it possible to embed ...
5
votes
1answer
265 views
which varieties can appear as exceptional divisors?
Let $\pi$ be the blow up of $\mathbb{A}^n$ at an ideal
$I_{p}$ which is supported at a close point $p$; let $E$ be the associated exceptional divisor. What are the possible varieties $Y$ that can ...
3
votes
1answer
132 views
Is there a blow-up formula for the derived category of a singular ambient variety?
For a nonsingular variety sitting inside a nonsingular ambient variety there is a semi-orthogonal decomposition of the derived category of the blow-up (with center that subvariety).
What can be said ...
2
votes
1answer
218 views
Cohomology of tangent bundles
Let $X$ be a smooth scheme and $Z\subset X$ a smooth subscheme. Consider the blow-up
$$\pi:\widetilde{X}:=Bl_{Z}X\rightarrow X$$
of $X$ along $Z$.
What is the relation between the cohomology of the ...
5
votes
1answer
376 views
Is blowing down functorial?
Let $f: X \to Y$ be a morphism of smooth surfaces and $C \subset Y$ be a contractible exceptional curve on $Y$. Assume that $f^{-1}(C)$ is a contractible exceptional curve as well. Denote by $X'$ ...
3
votes
2answers
316 views
Terminology for blowups in algebraic geometry
This is a partial duplicate of this Stack Exchange question which unfortunately got no answer.
All schemes are Noetherian and of finite type, although they need not be normal.
With $Z \subset X$ a ...
1
vote
1answer
197 views
blow-up along singular variety
Can somebody give me a nice example of blow-up of a smooth algebraic variety along a singular subvariety? Something I can do some exercise on and check the differences with a smooth blow-up. Thanks!
1
vote
0answers
150 views
Contractibility of union of smooth rational curves in famillies
Let $\pi:\mathcal{X} \to S$ be a family of smooth surfaces containing a subfamily $\mathcal{Y} \subset \mathcal{X}$ proper, flat over $S$ parametrizing contractible curves in $\mathcal{X}$ satisfying ...
6
votes
0answers
165 views
blowups and group actions
Let $X$ be a smooth projective variety over the complex numbers and assume that $X$ is equipped with the action of a finite group $G$.
Denote by $Z$ the closed subscheme of fixed points of $G$ and ...
1
vote
0answers
221 views
cohomology of a blowup: reference needed
Does anybody know a reference in which the computation of the cohomology of a blow-up is made in detail?
0
votes
2answers
234 views
blow-ups and singularities
Let $X$ be a smooth and projective variety over a field of characteristic zero. Let $Y$ be a normal variety, with finite quotient singularities (an orbifold!) and let $\pi: Y \to X$ be a finite ...
4
votes
1answer
132 views
$f^{-1}\mathcal I \cdot \mathcal O_X$ vs $f^\ast \mathcal I$
Let $X$ ad $Y$ be (noetherian) schemes and let $\mathcal I \subseteq \mathcal O_Y$ be a sheaf of ideals on $Y$. Let $f \colon X \to Y$ be a morphism of schemes. In general the sheaf $f^\ast \mathcal ...
3
votes
1answer
239 views
blow-up and normal crossings
Hi friends,
Can anybody help me with the following question?
I start with a projective smooth variety $X$ over a field $k$ of a characteristic 0 and $D$ a normal crossing divisor on $X$, with ...
0
votes
0answers
101 views
Is the modification a rational map?
Good morning,
I would like to ask the following question concerning the desingularisation, but I'm not familiar at all with these notions.
We have the following theorem of Hironaka: Let $X\subset ...
3
votes
0answers
271 views
Blowing up a projective surface
Let $X$ be a smooth degree $d$ ($d>5$) surface in $\mathbb{P}^3$. We now blow up $X$ at a point, embed it in some projective space, and and consider a projection of it into $\mathbb{P}^3$. The ...
2
votes
3answers
311 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.
...
5
votes
1answer
231 views
universal property of blow up for stacks?
I will use as a reference Hartshorne Prop. II.7.14, the universal property of blow-up. $\tilde{X}$ is the blow up of $X$ along a sheaf of ideals. $Z\to X$ is the morphism that is to be lifted to ...
3
votes
2answers
176 views
blow up of segre primal and $\mathcal{M}_{0,6}$
The segre cubic primal $X\subset P^4$ is the GIT quotient of 6 points on $P^1$. Let $M_{0,6}$ the DM compactification of the moduli of 6-pointed rational curves. The Segre primal $X$ is a cubic 3-fold ...
0
votes
1answer
520 views
Functorial properties of blow-up
Let $X, Y$ be projective algebraic surfaces with isolated singularities. Suppose they are diffeomorphic to each other. Denote by $\phi$ the diffeomorphism from $X$ to $Y$.
Then does there exists a ...
3
votes
2answers
301 views
Embedding a surface in a projective space
Let $X$ be a smooth degree $d$ ($d >5$) surface in $\mathbb{P}^3$. Let $\pi:\tilde{X} \to X$ be a blow-up of $X$ at a point. When is it possible to embed $\tilde{X}$ into $\mathbb{P}^3$?
In ...
2
votes
0answers
109 views
Global minimal model over a non-affine base
Remark 10.1.8 in Liu's book (AG and Arithmetic curves) says that over a non-affine base (base is always assumed to be a Dedekind scheme of dim 1), the minimal regular model of a (smooth projective) ...
14
votes
0answers
597 views
Horrible sets and blowups in Hubbard's Teichmuller theory
Edit: I can rephrase this question this way: When blowing up every point in the $x$-axis in $\mathbb{C}^2$ by means of an inverse limit of finite blowups, how can anything be 'left over'? The horrible ...
4
votes
1answer
267 views
kapranov's realization of $\overline{M}_{0,n}$ over other fields
Kapranov gave a very nice desciption, over $\mathbb{C}$ of the moduli space of stable pointed rational curves $\overline{M}_{0,n}$ as a series of blow-ups of $P^{n-3}$. Does this, or a similar result, ...
10
votes
2answers
507 views
When do blowups ''commute''?
Let $M$ be a manifold (variety, scheme, your favorite object) and let $N_1,N_2$ be two submanifolds (subvarieties, closed subschemes, ideal sheafes, etc.) such that $N_1 \cap N_2 \neq \emptyset$. ...
3
votes
2answers
435 views
Blowing down -1 curves
After a good look for anything in the way of an explicit example, and harassing the algebraic geometers in the department, I still am unable to find an answer to my question, so any light will be very ...
4
votes
1answer
242 views
When do blow-up and quotient commute?
Let a finite group $G\subset SL(n,\mathbb{C})$ act on $\mathbb{C}^{n}$ in a natural way. Assume there is a crepant resolution of $f:X\rightarrow \mathbb{C}^{n}/G$. When is it possible to write $X$ as ...
4
votes
1answer
372 views
Relation between blowup and normalization
Let $X$ be a variety over an algebraically closed field with null characteristic. Let $C$ be a smooth subvariety of $X$ of dimension 1, and let $x$ be a point of $C$. We assume that $X$ is ...
6
votes
0answers
470 views
Is the Springer resolution a blow-up?
Let's consider the Springer resolution of the nilpotent cone $\mathcal{N}$ of a complex semisimple Lie algebra $\mathfrak{g}$, which is
$$
\widetilde{\mathcal{N}}=T^*\mathcal{B}\rightarrow ...
1
vote
2answers
214 views
Second Chern Class of Blow-up
Let X be a compact surface and $\tilde{X}$ its blowing-up, how can I show the formula
$c_2(\tilde{X})=c_2(X)+1$?
3
votes
1answer
395 views
Blowing up intersections of two divisors make them disjoint?
Let $D,E \subset \mathbb{C}^3$ be prime divisors where $D$ is smooth and $E$ is not necessarily smooth.
Assume that $D \cap E$ has SNC support and let
$D \cap E = \bigcup \Gamma_i$ be a ...
17
votes
1answer
939 views
Which functor does the blowing up represent?
Let $X$ be a scheme and $I \subseteq \mathcal{O}_X$ be a quasi-coherent ideal of finite type. The blowing up $\mathrm{Bl}_I(X)$ has the following universal property: It comes with a morphism $p : ...
2
votes
1answer
193 views
Intersection powers of the exceptional divisor (and the transform of a hyperplane)
In light of my previous question, I am interested in the following scenario: Let $\tilde Y$ be the blow-up of $Y=\mathbb{P}^n$ along a linear subvariety $X\subseteq Y$ of codimension $d$, i.e. ...
8
votes
2answers
430 views
Blow-up along a subscheme and along its associated reduced closed subscheme
Let $X$ be a noetherian scheme and let $Y$ be a closed subscheme of $X$.
What relation is there between $\mathrm{Bl} _ {Y}(X)$ and $\mathrm{Bl} _{ Y _{\mathrm{red}}}(X)$ ?
Thanks.
4
votes
2answers
246 views
Comparing blow-ups with comparable centres
Let $X$ a variety. Say $X=\operatorname{Spec} A$.
Consider two ideals of $A$, say $I$ and $J$, with equal radical ; and consider the blow-ups of X with centre $I$ and $J$, say $Y_I$ and $Y_J$. How ...
6
votes
1answer
441 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. ...
4
votes
0answers
268 views
degenerating surface II
In degenerating surface, Robert Bryant give us an example of a sequence of minimal immersions which converges (in $C^2$- topology) to $z\mapsto z^{2k+1}$ on the unit disc $\mathbb{D}$. My question is ...
6
votes
1answer
894 views
A characterization of the blow-up
It is well-known that there is a universal characterization of blow-ups. However, this one is not so easy to use. According to Hartshorne-Theorem II.8.24. The blow-up of a nonsingular $X$ along a ...
15
votes
2answers
645 views
When does the blow-up of $CP^2$ at N points embed in $CP^4$?
Write $X_N$ for this blow up. Place the N points in 'general position' as needed. Then $X_6$ embeds in $CP^2$ as a smooth cubic surface. (See, eg, Griffiths and Harris.) But there is no other ...
4
votes
2answers
647 views
degenerating surface
Hi,
i have a sequence of immersed disc $u_n: \mathbb{D} \rightarrow \mathbb{R}^3$ which converge to a singular cover of the disc: $z^k$ for $k\geq 2$, moreprecisely $u_n \rightarrow z^k$ in ...
11
votes
1answer
581 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 ...
5
votes
1answer
361 views
Completion of local rings in the exceptional divisor of a blow-up
Let $X=\mathrm{Spec}(A)$ be an affine variety, $Z\subseteq X$ a closed, reduced subscheme. Let
$$\beta:Y=\mathrm{Bl}_Z(X)\to X$$
be the blow-up of $X$ in $Z$. In other words, ...
1
vote
1answer
301 views
Blowing-up along few points and embedding
Hello all.
Let
$X_{p_0,\ldots,p_n}\subset\mathbb{P}^N=\mathbb{P}^N_{\mathbb{C}}$
be the sequence of blowing-ups
$\pi:\mathrm{Bl}_{p_0,\ldots,p_n}\longrightarrow\mathbb{P}^n$
along $n+1$ points of ...
1
vote
1answer
416 views
Self intersection of blown up points and the lines which they lie on
I'm currently trying to understand the process of blowing-up, and a few things strike me as a little difficult to get an intuitive understanding of what's happening.
The current problem is on self ...
2
votes
1answer
556 views
(Anti)Canonical divisor of a blow up
This question may be utterly trivial, or not, but as someone with hardly any knowledge of algebraic geometry I thought there could be a chance I get lucky.
Let X be a rational surface obtained by n ...
0
votes
2answers
257 views
Smooth blowup of a surface
Suppose I have a smooth algebraic surface $X$ and a subscheme $Z$ such that the blowup
$$Y = Bl(X,Z)$$
is smooth.
Now $Z$ does not have to be smooth, say if it is given by some power of the maximal ...
