All Questions
Tagged with blowup birational-geometry
37 questions
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Blow up's and contracting Exceptional divisor along a relevant chart (Elementary transformations)
Sorry this is a question due to lack of understanding of blow ups and birationally geometry in general.
Consider the hypersurface in $\mathbb{A}^3$ given by:
$$X =xy - z^2 = 0,$$
which has a ...
- 155
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1
answer
138
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Are projective birational morphisms blowups?
This question seems to claim that a projective birational maps $f: X \to Y$ between varieties $X,Y$ over $\mathbb C$ is a blow-up. What is a reference for that?
According to Wikipedia,
The "Weak ...
- 10.4k
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153
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Is every normalization a blowup?
Is the normalization of a variety always a blowup along some coherent ideal sheaf? If not, I would like to see a concrete counter-example.
Let $Y \to X$ be the normalization. The answer is positive in ...
- 168
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1
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251
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Self-intersection of exceptional divisor of blowing-up along a singular point
Let $X$ be an $n$-dimensional projective variety with a simple singularity $p \in X$ which can be resolved by a blow-up $\pi\colon \tilde X \to X$ along $p$. The examples that I am considering are ...
- 103
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668
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Self intersection of exceptional divisor in blow-up of $\mathbb{P}^3$ along a curve
We work over the complex numbers. Let $X$ be the blow-up of $\mathbb{P}^3$ along a curve of genus $10$, which is the complete intersection of two cubics surfaces. The variety $X$ lives in $\mathbb{P}^...
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187
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Left exactness of the cotangent sequence
I am reading this note of Blickle on motivic integration and I am stuck at a technical point. At the beginning of the appendix, we are given a proper birational morphism $f: X' \longrightarrow X$ ...
- 2,249
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135
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Blowing up of a singular subvariety
I'm reading Kollár Mori Chapter 2.3. And they state the following lemma:
Lemma 2.29. Let $X$ be a smooth variety and $\Delta = \sum a_iD_i$ a sum of distinct prime divisors. Let $Z\subseteq X$ be a ...
- 747
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170
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Relation between exceptional divisor and tangent directions
Consider the hypersurface $X$ in $\mathbb{P}^2(\mathbb{C})\times \mathbb{P}^2(\mathbb{C})$ defined as the zero locus of
$$ X:Z(f)= (y_1y_2+y_0^2)x_0+y_1^2x_1+y_2^2x_2=0$$
with $(x_0,x_1,x_2;y_0,y_1,...
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Blowing Up the Indeterminancy Locus - Why is this sheaf invertible?
The following is explained in Hartshorne, chapter 2.7. I will be considering varieties instead of schemes. Let $X$ be a variety over $k$, and let $L$ be a line bundle on $X$. Let $s_0,\dots,s_n$ be ...
- 589
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Desingularization of the standard Cremona involution of $\mathbb P^2$.
Consider the birational map $\chi$ given by the blow up of the points $(1:0:0)$, $(0:1:0)$, and $(0:0:1)$ of $\mathbb P^2$, followed by the contraction of the strict transforms of the three lines ...
user878342
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165
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Lift rational map to the Blow-up by explicite construction
We work over complex numbers. Let $X \subset \mathbb{P}^n$ a complex projective subvariety
associated to homogeneous ideal $I(X) \subset \mathbb{C}[X_0,...,X_n]$.
Assume $X$ contains point $p:= (1,0,.....
- 8,449
2
votes
1
answer
121
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Arrow-reversing Proj and blow-up
Let $X$ be a normal projective variety over the field of complex numbers. Let $Y$ be a subvariety of $X$, and let $I_Y$ be the ideal sheaf of $Y$. From what I know, I can define the blow-up of $X$ ...
- 35
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Open neighborhoods on the blow-up of projective space.
Does every point of blow-up of the projective space along some smooth subvariety, have a neighborhood that is an open subset of affine space (i.e. $\mathbb{A}^n$)? I'm interested in char $p$ case.
- 1,364
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Are the numbers calculated from a log resolution birational invariants?
Let $C = V(x^2 - y^3)$ be the cuspidate cubic which sits inside $\mathbb{C}^2$. Let $\pi: \text{Bl}_0(\mathbb{C}) \to \mathbb{C}^2$ be the blow up of the origin. The reduction of the total transform ...
user506388
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339
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Ideal sheaf of a birational morphism arising from successive blowups
Let $L\subset \mathbb C^3$ be the line defined by $x=y=0$, and $p\in L$ the point defined by $x=y=z=0$. Let's consider the blowup of $\mathbb C^3$ at $p$ and then blow up the strict transform of $L$ ...
- 4,804
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answer
220
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Pencil of cubics with infinitely many exceptional curves
In Beauville's Complex Algebraic Surface, exercise V.21(5), we must find a surface with infinitely many exceptional curves.
He gives the following hint (I'm paraphrasing):
let $P$ be a pencil of ...
- 10.5k
1
vote
1
answer
914
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Dimension and blow-ups
Consider a projective variety $X$, and let $Y$ be a closed subvariety. Consider the blow-up of $X$ along Y: we obtain a new variety $\tilde{X}\subset X\times \mathbb{P}^{\dim Y}$, together with a ...
- 13
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1
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546
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Blowup along the fundamental locus of a rational map, II
This question is succeeding a question raised in this post. Let $$f:X\dashrightarrow Y$$ be a rational map between smooth projective varieties over $\mathbb C$ with smooth fundamental locus $B$ and $\...
- 4,804
2
votes
1
answer
685
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Multiple blow-ups
Suppose I consider, inside $\mathbb{P}^6$, the subvarieties $Y=V(x_2,\ldots,x_6)\simeq \mathbb{P}^1$ and $Z=V(x_0,\ldots,x_3)\simeq \mathbb{P}^2 $.
I want to understand what is the blow up of $\mathbb{...
- 488
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Integral of a blow up of $\mathbb P^2$
In Demailly's icm2006 p21, there is a statement:
If $X$ is the surface obtained by blowing-up $\mathbb P^2$ in one point, then the exceptional divisor $E ≃ \mathbb P^1$ has a cohomology class {$\...
- 691
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180
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How to show the blowing down a ruled surface is projective
Let $X$ be an irreducible projective threefold, let $S$ be a ruled surface over a curve $C$, where $C$ has finitely many singular points. Let $\pi:X\rightarrow Y$ be a map such that $\pi(S)=C\subset Y$...
- 151
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1
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342
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When is the projection from a point on the variety smooth?
Let $X\subset\mathbb{P}^n$ be a smooth irreducible variety (over $\mathbb{C}$) and $p\in\mathbb{P}^n$ a point. Let $\pi:\mathbb{P}^n\setminus\{p\}\to\mathbb{P}^{n-1}$ be the linear projection with ...
- 3,700
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2
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847
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Why the exceptional divisor of blowup of $\{x^2+yt=0\}$ has multiplicity one (but not two)?
Let $X$ be the affine surface $\{x^2+yt=0\}\subseteq \mathbb C^3$, then $X$ has an $A_1$ singularity at $0$. Consider $X$ as a family of curves via the projection to the last coordiate $$\pi:X\to \...
- 4,804
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0
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783
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$P^2$ blow up nine points
I quote the following paragraph form Kollar-Mori on page 22:
Let $X$ be obtained from $P^2$
by blowing up at the nine
base points of a pencil of cubic curves, all of whose members are
irreducible. ...
- 759
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Why is the blow-up of 9 points an elliptic surface?
One example of elliptic fibration is obtained as follows:
Let $Z(F),Z(G)\subset\Bbb{P}^2$ be two non-singular cubics intersecting in distinct points $P_1,...,P_9$ and take the rational map
\...
- 10.5k
4
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1
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364
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Sections of the exceptional divisor of a blowup
Let $C$ be a smooth curve in a smooth threefold $X$. Denote by $Y$ the blowup of $X$ along $C$ with exceptional divisor $E$. Then $E \rightarrow C$ is a $\mathbb{P}^1$-bundle over $C$.
Is it true ...
- 43
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492
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Blow up the elliptic singularity $x^2+y^3+z^6=0$
It is mentioned in this question https://mathoverflow.net/questions/148826/do-there-exist-double-points-on-an-algebraic-surface-in-mathbbp-mathbbc that $X=\{x^2+y^3+z^6=0\}\subset \mathbb C^3$ defines ...
- 4,804
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2
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Blowup along the fundamental locus of a rational map
Assume $f:X\dashrightarrow Y$ is a rational map between varieties, where $X$ is normal and $Y$ is complete. Then, the fundamental locus the $f$ (which means cannot extend the definition of $f$ on it), ...
- 1,111
1
vote
1
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801
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Birational invariant Hodge numbers
Consider the Hodge numbers of smooth projective varieties over $\mathbb C$. I am aware of the fact that only the outer Hodge numbers (essentially $h^{p,0}$) are invariant under birational equivalence. ...
- 3,330
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0
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66
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blow up of affine reducible variety at point has isomorphic total ring of fractions?
Assume $X$ is an reducible affine variety over $\mathbb{C}$ with coordinate ring $A$. Let $\pi \colon \tilde{X} \to X$ be the blowing up at a point $p \in X$ (which can be included in more than one ...
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Blow Up of a Surface
I have a question a step in the example demonstating the blowing up in Liu's "Algebraic Geometry and Arithmetic Curves" in the excerpt below (or look up at page 320):
We blow up the surface $X= ...
- 8,449
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116
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Blowup of non-singular varieties
How to construct a blowup $f$: X $\rightarrow$ Y between non-singular quasi-projective varieties?
This is an exercise from Shafarevich "Basic Algebraic Geometry 1".
I want to construct for any $n \...
4
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1
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358
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Blow up of one point is isomorphic to $\mathbb{P}(\mathcal{O} \oplus \mathcal{O}(1))$
My question comes from an example in Hartshorne (Example V.2.11.4) which I'm having trouble following. It is claimed that the Blow up of a point $p \in \mathbb{P}^n$ is isomorphic to $\mathbb{P}(\...
- 1,320
2
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2
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770
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blow-ups of subvarieties $Y_1,Y_2$ are disjoint in the blow-up at the ideal $I(Y_1)+I(Y_2)$
Exercise 9.19 in Gathmann's notes reads:
Let $X\subset \Bbb A^n$ be an affine variety, and let $Y_1,Y_2$ be irreducible closed subsets of $X$ , none contained in the other. We blow-up X along the ...
- 3,557
4
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0
answers
85
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Obtaining a nice map to a curve by using blowups
Let $X$ be a smooth and projective variety over a finite field (separated, finite type, integral). Then after performing a number of blowups I should be able to find a proper surjective map from $X$ ...
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1
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Blow-up of $\mathbb P^2$ in 2 points is isomorphic to blowup of quadric in 1 point
How to show that blow-up of $\mathbb P^2$ in 2 points is isomorphic to blowup of quadric in 1 point? I think it is a standard fact, but Google can not help me with it.
Update: I found an answer here:...
- 3,861
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610
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Factoring a birational morphism through blowup
Let $X,Y$ be smooth, proper varieties, and $f: X \to Y$ be a proper birational morphism. Suppose $E$ is a smooth, irreducible exceptional divisor, with the image $f(E)$ also smooth. Let $I$ be the ...
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