Questions tagged [blow-ups]
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2
votes
0answers
107 views
Is this construction with stacks a blow-up?
Let $X$ be the stack of rank $1$ degree $b$ coherent sheaves $E$ with torsion of length at most 1 on an elliptic curve $C$. Let $Y$ be the stack of pairs $E^{'} \subset E$ such that $E \in X$ and $E/E^...
1
vote
1answer
104 views
Blow up of a Projective scheme along a projective-subscheme using Macaulay2
Given a closed-subscheme (corresponding to the homogeneous ideal $I$ inside $K[x_0,...,x_n]$) of a projective scheme $\mathbb{P}^n$, how can one find out the blow up using Macaulay2? ...
2
votes
1answer
153 views
Counter example about blow-up solution of DEs
Let $f(\cdot)$ be a continuously differentiable function over $\mathbb{R}$, and $u\in L^2_{loc}(0,\infty)$, $a\in \mathbb{R}$, and $x(t)$ solves the integral of $$\dot{x}(t)=ax(t)+f(x(t))+u(t), \quad ...
8
votes
0answers
147 views
Very ample divisors on blow ups of the projective plane
Let $X$ be $\mathbb{P}^2$ blown up at $k$ points in general position. The Picard group of $X$ is just $\mathbb{Z}^{k+1}$ and one knows the intersection product explicitly. If $D$ is an ample divisor, ...
2
votes
0answers
59 views
Blowing up the base of an elliptically fibered (non Weierstrass) threefold
Suppose $X$ is an elliptically fibered smooth threefold, with a nontrivial Mordel-Weil group. Lets call the sections $\sigma_i$ ,$i=1 \dots n$. None of these "sections" are honestly a section, they ...
3
votes
1answer
126 views
Gradient blowup for the 1-dim heat equation near an irregular boundary point
I'm trying to estimate the rate of boundary gradient blow-up for the 1-dim heat equation near an irregular boundary point.
Let $b(t) := (1-t)^\alpha$, $\alpha < 1/2$. Let $u(t,x)$ solve the ...
3
votes
0answers
170 views
The Chow ring of a blow-up along a badly embedded subscheme
Let $X$ be a variety and $i:Y\hookrightarrow X$ be a regularly embedded closed subvariety, and write $\tilde X$ for $Bl_Y X$ the blow-up of $X$ along $Y$. Let $A^*(X)$ refer to the Fulton-Macpherson ...
2
votes
1answer
194 views
An explicit computation of the blow-up of curve over $\mathbb{F}_3$ at two points
I would like to work through computing the blow-up of a particular curve along a subvariety consisting of just two points, both of which are ordinary double points.
Let $$F(X,Y,Z) = X^4 + Y^4 - X Y^2 ...
3
votes
2answers
363 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, $\widetilde{S}$ ...
1
vote
0answers
40 views
How can I describe in explicit geometric terms the (in general non-complete) linear system?
Let $\varphi\!: S \to S^\prime$ be a birational morphism of projective non-singular irreducible surfaces over a (algebraically closed) field $k$ and let $D \in \mathrm{Div}(S)$. Also, let $\big(\...
3
votes
0answers
149 views
When is the strict transform of very ample divisor ample?
Let $X$ be a projective variety and $Y \subset X$ is a very ample divisor on $X$. Let $Z \subset X$ be a regular subvariety (but $X$ need not be regular along $Z$) of codimension $2$ and $\pi:\tilde{X}...
1
vote
1answer
88 views
Gysin map for projective sub-bundles of exceptional divisors
Let $X$ be a smooth, projective variety, $Y \subset X$ a smooth, projective subvariety of codimension $3$. Denote by $\pi:\tilde{X} \to X$ the blow-up of $X$ along $Y$ and by $E$ the exceptional ...
1
vote
0answers
203 views
Cotangent sheaf of a blow up
It's well known that for a nonsingular variety $X$ and a nonsingular subvariety $Y$ with $\text{codim}(Y,X) \ge 2$, we can compute the canonical sheaf of $\widetilde{X}$ (the blow up of $X$ along $Y$) ...
6
votes
1answer
152 views
Blowing-up an ideal generated by squares
Let $f_1,\dots,f_r$ be regular functions on a smooth projective variety $X$, and consider the ideals $I = (f_1^2,\dots,f_r^2)$ and $J = (f_1,\dots,f_r)$. Let $Y = Z(I)$ and $W = Z(J)$ be the ...
13
votes
0answers
316 views
Cohomology of a blow-up of a real algebraic variety
Let $X$ be a complex algebraic variety, $Z \subset X$ a closed subvariety, $\mathrm{Bl}_Z X$ the blow-up and $E$ the exceptional divisor. There is an isomorphism of cohomology groups
$$ H^k(X(\mathbf ...
2
votes
0answers
95 views
Blow-up, strict transform and gysin map
Let $X$ be a non-singular, projective variety and $Y \subset X$ non-singular, projective subvariety. Denote by $\pi:W \to X$ the blow-up of $X$ along $Y$. Let $Y_1 \subset W$ be a non-singular ...
4
votes
2answers
202 views
Is this divisorial contraction a blow-up?
Let $C$ be a curve in a smooth $3$-fold $X$ with an ordinary node $p\in X$. Blow-up $p$ let $E$ be the exceptional divisor, and $\widetilde{C}$ the strict transform of $C$. Furthermore let $L$ be the ...
4
votes
1answer
176 views
Gysin map and blow up
Let $X$ be a smooth projective variety and $W \subset X$ a smooth, projective subvariety. Let $\pi:\tilde{X} \to X$ be the blow-up of $X$ along $W$. Let $E$ be the exceptional divisor of $\pi$ and $i:...
10
votes
1answer
540 views
Blow-ups, pullbacks and proper transforms
Let $X$ be a smooth projective variety, $Z$ a smooth subvariety of $X$, and let $f:\widetilde{X}\to X$ be the blow-up of $X$ along $Z$. Then for a subvariety $V\subset X$, we have two cohomology ...
0
votes
1answer
96 views
Critical gKdV - tutorial
I'm looking for a good introduction to the critical generelized KdV equation
$$u_t +u_{xxx}+5u^4u_x = 0 \, , $$
$$ u(t=0,x) = u_0 (x) \, , \qquad x\in \mathbb{R} \, ,$$
and its blowup solutions. There ...
2
votes
0answers
153 views
Blow-up along singularity of the degeneracy locus
Let $X$ be a smooth, projective variety, $E$ a rank $r$ locally free sheaf on $X$. Fix a closed embedding $i:X \hookrightarrow \mathbb{P}^N$ and denote by $\mathcal{O}_X(m)=i^*\mathcal{O}_{\mathbb{P}^...
2
votes
0answers
94 views
Quantitative Approach to Existence of Minimal-Mass Blowup Solutions to NLS
Consider the mass-critical defocusing NLS in dimension $d\geq 1$:
$$iu_{t}+\Delta u = |u|^{4/d}u, \quad (t,x) \in I\times\mathbb{R}^{2}$$
Define the mass $M(u)$ and scattering size $S(u)$ of the ...
6
votes
1answer
391 views
Resolution of an isolated cyclic quotient singularity
I am looking for a reference to the following fact which seems to be true and which is probably well-known (at least to experts in resolution of singularities):
Consider an isolated cyclic quotient ...
2
votes
1answer
147 views
blow up in finite time of hyperbolic system of conservation law
Let say I have a hyperbolic system of conservation law. How do I show there is a blow up in finite time? For a single conservation law, I think, I could just show that there is collision of ...
1
vote
0answers
134 views
On simultaneous resolution of singularities in certain flat families
Let $X$ be a smooth projective variety (over the complex numbers) of dimension at least $2$, $B$ a finite set of closed points. Consider the closed subscheme $E:=B \times X + \Delta \subset X \times X$...
3
votes
0answers
89 views
Which blow ups in the base of a conic bundle preserve the “standard” condition?
Assume we are given a nontrivial standard conic bundle $\pi: X\rightarrow S$, that is $X$ and $S$ are smooth projective varieties (say over $\mathbb{C}$), $\pi$ is flat and furthermore we have $Pic(X)=...
7
votes
1answer
2k views
Blow-up in family
Let $\pi \colon X \to T$ be a flat projective morphism, and let $Y$ be a closed sub-scheme of $X$ which is flat over $T$. We can assume that everything is defined over the complex numbers, and $T$ is ...
1
vote
1answer
70 views
Analytic conjugacy of vanishing holonomy groups implies analytic conjugacy of foliations
i am reading and trying to do some exercises and problems of the book Lectures on Analytic Differential Equations- Y. Ilyashenko, S. Yakovenko.
I can not solve the problem 11.6 that says
Consider ...
3
votes
1answer
165 views
Does $u_{t}=g(t)u_{x}^{2}$ blow-up for bounded positive g? What about $u_{t}=u_{xx}+g(t)u_{x}^{2}$?
My original problem is to see if the following pde develops blow-ups in $(-L,L)$
$$u_{t}=u_{xx}+g(t)(u_{x})^{2}$$
for periodic boundary $u_{0}(-L)=u_{0}(L)$, where $0<g(t)<1$; specifically $g(...
3
votes
1answer
289 views
What is the relation between holomorphic blow-up and symplectic blow-up?
McDuff has shown us exactly how the symplectic blow-up procedure along a symplectic submanifold affects the symplectic structure in the ambient space, i.e., if $\omega$ is the original symplectic form,...
5
votes
1answer
619 views
Resolution of the $E_8$ singularity with a weighted blowup
I am reading Miles Reid's notes on weighted projective spaces, and I'm a little confused about a particular paragraph (notes here, page 8):
A famous case is the $E_8$ singularity $X: (x^2+y^3+z^5=0)...
2
votes
1answer
138 views
What finite groups are stabilizers in Kirwan's desingularization construction?
Assume $X$ is a smooth projective curve of genus $g\geq 3$ over $\mathbb{C}$ and let $M$ be the (singular) moduli space of semistable rank two vector bundles with trivial determinant on $X$. Then ...
2
votes
0answers
55 views
Relation between symplectic blow-up of a compact manifold and fibre bundles over same manifold
The symplectic blow-up of a compact symplectic manifold $(X,ω)$ along a compact symplectically embedded submanifold $(M,σ)$ results in another compact manifold $(\tilde{X},\tilde{ω})$ given by
$$\...
6
votes
1answer
357 views
Failure of universal flatification
Raynaud and Gruson proved a beautiful "flatification" theorem (5.2.2): If $S$ is a quasicompact, quasiseparated scheme, and $X$ is a finitely presented $S$-scheme, $M$ is an $\mathcal O_X$-module of ...
2
votes
0answers
208 views
the normalized blowup
Let $X$ be a normal variety over $\mathbb{C}$ and $x\in X$ a singular point.
Let $f:Y^{\nu}\to X$ be the normalized blowup at $x\in X$. (i.e. $f$ is a composition of the blowup $Y:=Bl_xX\to X$ and ...
2
votes
0answers
284 views
Blow up along a section of a smooth morphism
Let $C$ be a ground locally notherian and quasiprojective scheme. Let $\pi:S\to B$ be a $C$-morphism of finite type $C$-schemes. We call $\pi$ a $C$-smooth morphism if the morphisms $S\to C$ and $B\to ...
7
votes
3answers
336 views
Existence of a morphism between two toric varieties
Does there exist a morphism between the blow-up of $\mathbb{P}^3$ in four general points and $\mathbb{P}^1\times\mathbb{P}^1$? If not why?
3
votes
2answers
244 views
Fibrations on blow-ups of $\mathbb{P}^2$
Let $X_n = Bl_{p_1,...,p_n}\mathbb{P}^2$ be the blow-up of $\mathbb{P}^2$ in $n$ general points $p_1,...,p_n\in\mathbb{P}^2$.
Let $f_i:\mathbb{P}^{2}\dashrightarrow\mathbb{P}^1$ be the linear ...
1
vote
0answers
171 views
Is any blow-up of smooth subvarieties always an extremal contraction?
Let $X$ be a smooth complex projective variety and $Z$ be a smooth subvariety of $X$.
Take the blow-up $\pi: Y \to X$ of $X$ along $Z$.
Then I want to know whether $\pi$ is the contraction of an ...
2
votes
1answer
453 views
On the number of irreducible components of an exceptional divisor
Let $X$ be a complex, affine variety and $Z\subseteq X$ a closed subset of $X$ (i.e. a closed, reduced subscheme). Let $E$ be the exceptional divisor of the blow-up $\pi:\tilde X\to X$ of $X$ with ...
4
votes
0answers
177 views
Nonlinear Schrödinger blow-up for non radial solutions
I am studying a paper of Frank Merle and Pierre Raphaël,
http://math.unice.fr/~praphael/Publications/blow-up-norme-critique.pdf.
The equations are
$$
i\partial_tu+\Delta u=-|u|^{p-1}u
$$
on $\mathbb{R}...
5
votes
0answers
223 views
Does integral closure commute with pushforward
Suppose that $\pi : Y \to X$ is a proper birational morphism between normal varieties (schemes, whatever). Suppose that $I$ is an ideal sheaf on $Y$.
One can form $\pi_* I$ and construct an ideal ...
2
votes
1answer
206 views
Iterated blow-ups above a point
Let $X = \mathbb A^2$. Say we blow up the origin $(0,0)$, and then blow up the intersection of $(x=0)$ with the exceptional divisor. The resulting space is the blow-up of $\mathbb A^2$ along what ...
1
vote
0answers
219 views
higher direct images of O(E)
I hope this is well known, I just could not work it out myself.
Say I have a variety X (smooth and projective over C is my usual setup) with a smooth subvariety Z. Let f: BL_Z(X) --> X be the blowup ...
2
votes
1answer
325 views
Blowing-up the Grassmannian at a point
Does anyone know what the blow-up of the Grassmannian at a point looks like? Consider $G=Gr(r,n)$ and $V\in G$. I want to understand more explicitly what $Bl_V(G)$ should mean.
Of course for affine ...
9
votes
1answer
870 views
Blowing-up a point in the singular locus
Let $X\subset\mathbb{P}^n$ be a variety singular along a smooth subvariety $Z\subset X$ of positive dimension. Let us assume that $X$ has ordinary singularities along $Z$. Now, let $\pi:Y\rightarrow \...
4
votes
1answer
419 views
Quadrics and Moduli Spaces
It is well known that $\overline{M}_{0,5}$, the moduli space of $5$-pointed rational curves, can be realized as the blow-up of $\mathbb{P}^2$ in four general points. Therefore, we may interpret $\...
2
votes
1answer
351 views
A question about an intersection number
Let $\pi:Y\rightarrow \mathbb{P}^3$ be the blow-up of two points $p,q\in\mathbb{P}^3$, and then of the strict transform of the line $L$ spanned by them. Now, Let $E_p,E_q, E_{p,q}$ be respectively the ...
4
votes
0answers
169 views
Blow-up of $\mathbb{P}^4$ along a smooth surface
Let $\pi \colon X\to \mathbb{P}^4$ be the blow-up of a smooth surface $S\subset \mathbb{P}^4$. Is there a formula to compute $(K_X)^4$ ? (which should be dependent on invariants of $S$).
In dimension ...
0
votes
1answer
228 views
Intersection Matrix of a resolution
Probably this is a very easy question. Let $f:X\rightarrow S$ be a resolution of a projective surface such that
$$K_X = f^{*}K_S+\sum_ia_iE_i$$
with $a_i>0$. By Grauert-Mumford theorem the ...
