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How to increase the second cohomology group of the structure sheaf?

We know that $H^2(\mathcal{O}_{\mathbb{P}^3})=0$. I am looking for blow-ups $$\pi:X \to \mathbb{P}^3$$ such that $X$ is non-singular and $H^2(\mathcal{O}_X)>0$. Of course, if we blow-up along ...
user45397's user avatar
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3 votes
1 answer
122 views

A question related to the strong Oda conjecture

A fan is a collection of strongly convex rational polyhedral cones in $\mathbb Z^n$, which we often think of as contained in $\mathbb Q^n$ or $\mathbb R^n$ for purposes of visualizing it. The defining ...
Hugh Thomas's user avatar
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1 vote
0 answers
129 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 ...
kvicente's user avatar
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3 votes
1 answer
207 views

"General position" on $\mathbb{P}^1\times\mathbb{P}^1$

On $\mathbb{P}^2$ we have the notion of general positions: no 3 points on a line, no 6 on a conic, etc. In particular, blowing up points (up to 8) in general positions give ample anti-canonical class, ...
fp1's user avatar
  • 101
3 votes
1 answer
172 views

PDE: compactness vs blowup

There are, of course, plenty of approaches on how to solve a non-linear PDE. Two of them are the following: Solve (easier) approximate problems, show some form of compactness for the approximate ...
Sebastian Bechtel's user avatar
1 vote
0 answers
108 views

Global existence for large data in $H^{-1/2}(\mathbb R)$ of viscous Burgers' equation with external forcing

First, a quick summary of what to know about viscous (or dissipative) Burgers' equation $$ u_t-u_{xx}=(u^2)_x. \tag{1}\label{1}$$ Recall that $\dot H^{-1/2}(\mathbb R)$ is a scaling-critical Sobolev ...
Lorenzo Pompili's user avatar
12 votes
2 answers
517 views

Coordinate ring of universal centralizer (BFM space)

In the paper titled Equivariant (K-)homology of affine Grassmannian and Toda lattice, the authors, Roman Bezrukavnikov, Michael Finkelberg, and Ivan Mirković, derived the coordinate ring of each ...
Yunsong WEI's user avatar
1 vote
1 answer
301 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 $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(n))$....
Yromed's user avatar
  • 63
1 vote
0 answers
87 views

Adjunction correspondence for Blow up of double point

Let $C$ a curve over an algebr closed field $k$ with a singular double point singularity at $x$ and $\pi: C' \to C$ the blowup in $x$ and let $x_1,x_2 \in C'$ be the two points over $x$. Why holds for ...
user267839's user avatar
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51 views

Study of the properties of a non-local ODE

I am studying the following non-local ODE $$\dot p(x) \nu_{\varepsilon, \alpha}(x) + \int_{x}^{2x_0}\frac{\dot p(s)}{s + \varepsilon} ds = c \quad \text{for } x \in [0,2x_0].$$ The number $x_0$ can ...
Falcon's user avatar
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1 vote
0 answers
72 views

Factorising a multivariate polynomial, in terms of products of linear polynomials, using blowups

I am considering multivariate polynomials of the form $$f(x,y)=x^a\,y^b\,p(x,y)^c$$ (and similarly for higher dimensions). I am trying to transform these polynomials into the generic form $$\widetilde{...
Giulio Crisanti's user avatar
1 vote
0 answers
160 views

Cohomology of a stratified projective bundle

Let $S$ be a smooth algebraic variety, and suppose $X\to S$ is a smooth morphism of schemes such that the geometric fibers are all projective spaces. Let us suppose that the dimension of the fibers is ...
IMeasy's user avatar
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3 votes
0 answers
160 views

Cotangent complex of a blowup

Let $X$ be a nonsingular variety over an algebraically closed field $k$, and let $Y \subset X$ be a nonsingular subvariety. Consider the blowup $p: \tilde{X} \to X$ of $X$ along $Y$, with exceptional ...
John Nolan's user avatar
3 votes
1 answer
231 views

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 ...
Hephaistos's user avatar
0 votes
0 answers
69 views

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 ...
Hephaistos's user avatar
2 votes
0 answers
289 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 ...
Serge the Toaster's user avatar
7 votes
0 answers
258 views

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 ...
Leo Herr's user avatar
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1 vote
0 answers
44 views

Points in a blowing up determined by an infinitesimal curve not in the centre

Let $\Bbbk$ be a field. Let $A$ be a $\Bbbk$-algebra (commutative and unital). All ring maps are $\Bbbk$-algebra homomorphisms. Let $I\subseteq A$ be an ideal. Let $X:=\mathrm{Spec}(A)$ and $Z:=\...
Display Name's user avatar
2 votes
1 answer
334 views

How to compute the transfer maps for G-theory of Noetherian schemes

Let $k$ be a field and $R$ be the ring $k[x,xy,xy^2,xy^3]$. Let $X$ be $\operatorname{Spec}(R)$ and $\tilde{X}$ be the blow-up of $X$ along the maximal ideal $I$ of $R$ generated by $x,xy,xy^2,xy^3$.I ...
Boris's user avatar
  • 549
0 votes
1 answer
254 views

Calculate blowup of a pencil of cubics "by hand"

I have one more question about the Example (I.5.1) on page 7 from Rick Miranda's the basic theory of elliptic surfaces: Let $C_1$ be a smooth cubic curve in $\mathbb{P^2}$ and let $C_2$ be any other ...
user267839's user avatar
  • 6,034
1 vote
1 answer
311 views

Blow-up of a pencil of cubic curves (from Miranda's basic theory of elliptic surfaces)

Rick Miranda's "The basic theory of elliptic surfaces" the Example (I.5.1) see page 7 on a pencil of plane curves contains an argument that I do not understand yet. Let $C_1$ be a smooth ...
user267839's user avatar
  • 6,034
2 votes
0 answers
200 views

Is this blow-up a line bundle over the projective line

Let $R$ be the ring $\mathbb{C}[a,b,c,d]/(ac-b^2,bd-c^2,ad-bc)$. Let $I$ be the ideal of $R$ generated by $a,d$. Let $X=\operatorname{Spec}(R)$ and $\tilde{X}$ be the blow-up of the affine scheme $X$ ...
Boris's user avatar
  • 549
4 votes
2 answers
631 views

Blowing up of a singular subvariety

I ask the same question on MathStackExchange but receive no answer. 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 ...
Hydrogen's user avatar
  • 313
1 vote
0 answers
156 views

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 $X$?
Andrey Ryabichev's user avatar
1 vote
0 answers
114 views

How to compute the G-theory groups of a blow-up of Noetherian schemes

Suppose that $k$ is an algebraically closed field and $R$ is a finitely generated $k$-algebra such that if $X$ denotes Spec$R$, then the only closed, singular point of $X$ is the origin. Let $\tilde{X}...
Boris's user avatar
  • 549
2 votes
0 answers
189 views

Blow-up at locally closed center

Let $X$ be a scheme, and $Z\subset X$ is a locally closed subscheme. I am wondering that is there a reasonable definition of blow-up of $X$ at $Z$? And of course, this blow-up $Bl_Z X$ should satisfy ...
Kim's user avatar
  • 505
1 vote
0 answers
145 views

Blow up singularities on curves

Let $p$ be a prime number and let $\bar{\mathbb F}_p$ be an algebraic closure of $\mathbb F_p$. Let $C$ be an irreducible singular projective curve over $\bar{\mathbb F}_p$. Let $P$ be a singularity ...
Yachen Liu's user avatar
2 votes
1 answer
174 views

0-framed smoothly slice knot that can be obtained by blowing down successively a link of unknots

A knot in $S^3$ is called a smoothly slice knot if it bounds a smoothly embedded 2-disk in $D^4$. Every ribbon knot is known to be a smoothly slice knot, and there are known some nontrivial smoothly ...
user302934's user avatar
0 votes
0 answers
252 views

Vector bundles on blowups

$\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\ker{ker}$Let $X$ be a (projective) variety, which is singular at a point $p$ of $X$ (we can also replace $p$ with some closed subvariety). Suppose $Y$...
taf's user avatar
  • 448
0 votes
1 answer
161 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 \...
Serge the Toaster's user avatar
2 votes
0 answers
196 views

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 $2$. ...
Jana's user avatar
  • 2,022
4 votes
0 answers
238 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 $...
calc's user avatar
  • 243
1 vote
0 answers
117 views

Representability of blow-up of stacks

Suppose $\mathcal{Z}\subseteq\mathcal{X}$ is a closed immersion of stacks. Is the blow-up morphism $p:Bl_{\mathcal{Z}}(\mathcal{X})\to\mathcal{X}$ representable? More precisely, if $C\to\mathcal{X}$ ...
Nanjun Yang's user avatar
6 votes
1 answer
429 views

If power of an ideal is locally free then it is locally free

Let $X$ be a noetherian scheme and $\mathcal{I} \subset \mathcal{O}_X$ a coherent sheaf of ideals. Suppose that $\mathcal{I}^d$ is locally-free for some power $d$. Then the blowing up $\mathrm{Bl}_{\...
Ben C's user avatar
  • 3,363
0 votes
0 answers
132 views

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 ...
gigi's user avatar
  • 1,333
2 votes
1 answer
224 views

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 ...
gigi's user avatar
  • 1,333
2 votes
1 answer
160 views

Mori fiber space contractions of the blow-up of a projective space

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(...
Bobech's user avatar
  • 381
9 votes
1 answer
721 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 ...
Student's user avatar
  • 5,038
9 votes
0 answers
335 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 ...
Andy's user avatar
  • 515
4 votes
0 answers
197 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 $C^{k, \alpha}$ ...
user734979's user avatar
2 votes
0 answers
435 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, ...
Jana's user avatar
  • 2,022
2 votes
1 answer
238 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 $\...
user avatar
5 votes
1 answer
277 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....
Ron's user avatar
  • 2,126
1 vote
0 answers
289 views

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 ...
Basics's user avatar
  • 1,821
12 votes
1 answer
2k views

What's the cohomology ring structure of a blow-up?

Let $X$ be a compact Kähler manifold, with $j_Z: Z\hookrightarrow X$ a submanifold of complex codimension $r$, $\tau: \widetilde{X} \to X$ the blow-up of $X$ along $Z$, with exceptional divisor $j: E \...
Lineer 's user avatar
  • 478
2 votes
0 answers
186 views

Higher cohomology of projective bundles

Let $C$ be a curve and $L$ be a line bundle with sufficiently large degree. Let $C_p$ denote the $p$-th symmetric product of $C$, which consists of all the effective divisors of degree $p$ on $C$. Let ...
Li Li's user avatar
  • 419
2 votes
0 answers
293 views

Intersections of strict transform and strict transform of intersections

Let $Z_1,Z_2$ and $Y$ be subvarieties of a locally complete intersection variety $X$ over $\mathbb C$. Consider the strict transforms of $Z_1$ and $Z_2$ in the blowup $Bl_YX$, the question is: when ...
Blow's user avatar
  • 21
2 votes
1 answer
272 views

Is the blowup of a toric variety corresponding to a subdivision normal?

Toroidal Embeddings 1 by KKMS say subdividing the fan of a toric variety yields the fan of a normalized blowup. How do I avoid normalization? Do I need to choose my subdivision carefully, or is it ...
Leo Herr's user avatar
  • 1,084
2 votes
0 answers
202 views

Surgery for algebraic varieties

I have a number of vague questions that I wasn't sure whether they are suitable to ask or not, but I decided to ask! According to this result, any two birational varieties can be constructed by a ...
user127776's user avatar
  • 5,851
1 vote
0 answers
180 views

Factorization of birational maps in char $p$

So I was reading about the factorization result that any birational map between smooth varieties is composition of blow-ups and blow-downs with smooth centers. It is apparently true only in ...
user127776's user avatar
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