Questions tagged [blowup]

A technique in geometry (especially algebraic and differential, and, by extension, the study of pseudo-differential operators) for resolution of singularities. Not to be confused with the formation of singularities in solutions of ordinary or partial differential equations.

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Why a polynomial factors in the coordinate ring of an affine chart of the blow up of $\mathbb{A}^{n+1}$ (as a toric variety) at the origin?

I am studying a theorem by Ishi in which she proves there is a resolution of the singularity of a hypersurface defined on de affine space considered as the toric variety given by the fan $\sum_i \...
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Smooth hypersurfaces of the blow-up.

Let's assume $X$ is the blow up of $\mathbb{P}^n$ along a smooth subvariety $Z$. Especially $X$ is smooth. I was wondering what the hypersurfaces of $X$ look like? The hypersurfaces should give an ...
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Blow up of toric variety corresponds to subdivision of cone

Look at the lattice $N= \mathbb{Z}^3/\mathbb{Z}(1,1,2)$, let $u_0,u_1,u_2$ be the images of the standard basis elements of $\mathbb{Z}^3$ and consider the cone $\sigma = \text{Cone}(u_0,u_1)$. Then it ...
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Example of a Cartier divisor that cannot be written as the difference of effective divisors.

Suppose $X$ is a variety and $D$ is a Cartier divisor on $X$. Fulton argues in his Intersection theory, that if $\pi \colon \tilde X \to X$ is the blow-up of $X$ with respect to the ideal of ...
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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 ...
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Understanding blow up of curves

I am trying to desingularize the curve $V=V(y^n-x^d)$, where $n>d$ and $gcd(n,d)=g$. It is singular at $P(0,0)$ so I am trying a blow-up. My intuition is that a blow-up is locally like $(x,y)\...
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Find the ideal of a birational map as 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 $L$ ...
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Blowing up of the curve $x^4+y^4-xyz^2$ in $\mathbb{P}^2$

[Problem] (Fulton's Algebraic curves Problems 7.9) Let $C=V(x^4+y^4-xyz^2)$. Write down equations for a nonsingular curve $X$ in some $\mathbb{P}^N$ that is birationally equivalent to $C$. (Use the ...
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Exercise in Beauville, about the blowup of $P^2$ at seven points

I am trying to solve: (this is an exercise in Beauville's book, Complex Algebraic Surfaces, page 52.) Consider $7$ points of $P^2$ in general position. Let $P_7$ denote the blowup of $P^2$ at these ...
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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 ...
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Fano surface vs. del Pezzo surface

This article from wikipedia defines a Fano variety as a complete variety whose anticanonical bundle is ample. It also states that: A Fano surface is also called a del Pezzo surface. Every del Pezzo ...
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Map between two Blow-ups

Let $\widetilde{X}$ denote the blow-up of a scheme $X$ with respect to a sheaf of ideals $\mathcal{I}$. Let $Y$ be a closed subscheme of $X$, such that $\mathcal{I}\mathcal{O}_Y$ (the inverse image ...
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Section of blowup map of schemes

Let $X$ be a scheme. Let $\mathcal{I} \subseteq \mathcal{O}_X$ be a quasi-coherent sheaf of ideals on $X$, and let $Z \subseteq X$ be the closed subscheme corresponding to $\mathcal{I}$. Let $X' := \...
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There are no sections of blowup $Bl_0(\mathbb{A}^2)$ of affine plane at the origin

Let $p:Bl_0(\mathbb{A}^2) \to \mathbb{A}^2$ be blowup of plane at the origin. Is there a geometric reason why there are no sections $s: \mathbb{A^2} \to Bl_0(\mathbb{A^2})$, that is no maps to ...
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Blowing down of a Quadric Variety

Let $ Y = V(xy-uv) \subset \mathbb{C}^4$ be a variety and consider the blowup $\pi: \tilde{X} \rightarrow Y$ at the point $(0,0,0,0)$. The exception set $Q$ is the projective quadric $V(xy-uv)\subset \...
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Identifying bad fibers of an elliptic surface given by a pencil of cubics

I know that every rational elliptic surface is given by the blowup of the $9$ points of intersection of two cubics in $\Bbb{P}^2$ (at least one of them being smooth). Take for example the cubics $C_1,...
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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 ...
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Blow up $\operatorname{Pic}^3(C)$ along $C$.

I want to solve this problem from E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris (auth.) - "Geometry of Algebraic Curves Volume I" I can show that the fibers of $\{(K_C-P) \in \...
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Using blowups to desingularise curves - understanding a simple example

I'm learning about blowups in Algebraic Geometry, and am having trouble understanding how to apply them to desingularise varieties. To illustrate my confusion, I will use the first example from these ...
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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 $\...
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Handle Blow-Ups in differential equations

Hello interested helpers, my actual problem is that I want to define a solution operator to a "nice" initial value problem, say $$ \begin{cases} \dot y = f(y),\\ y(0) = y_0, \end{cases} $$ ...
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Formula $m_p(C\cap D)=\sum_xm_x(C)m_x(D)$, for $x$ infinitely near $p$, applied to a concrete example

I'm reading this wikipedia article about infinitely near points. In the section "Applications", the article says: If $C,D$ are irreducible curves on a smooth surface $S$ which intersect in a ...
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Blow-Up of Quasiprojective Variety and Conormal Space

My question is with regards to understanding blowing up a variety, $X$, along an arbitrary subvariety, $Y$, with respect to the conormal bundle. I have been told that a blow up in essence is replacing ...
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Why does the blowup of a curve at a singular point decreases the arithmetic genus?

In Beauville's Complex Algebraic Surfaces, problem II.20 we are asked to show that an irreducible curve $C$ in a smooth surface $S$ becomes smooth after a finite number of blowups. He says that a way ...
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Direct image of rank 2 vector bunldes by blow ups

I am reading ON THE DIFFEOMORPHISM TYPES OF CERTAIN ALGEBRAIC SURFACES. II of Friedman and Morgan. In the section 5. says Let $\bar{V}$ be a rank 2 vector bundle on $\bar{Y}$ with $c1(\bar{V}) = 0$. ...
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What is the intuition behind blowup of affine space?

I learnt that the dimension of the blowup fibers “explodes” in the center of the blowup and my imagination for now goes as follows: every line on the affine plane is sent identically to a line in the ...
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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{...
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Vakil 9.3 F: Fibers over Blow-Up Map

This in in regards to Vakil 9.3 F. I am trying to compute the fiber over any closed point $p$ of $\mathbb P^1_k$ of the map $$g: \operatorname{Bl}_{(0, 0)} \mathbb A^2_k \to \mathbb P^1_k$$ The ...
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Fulton intersection Theory Example 4.3.9

I could use some help properly understanding Example 4.3.9 of Fulton's Intersection Theory. The setup is that For a variety $X$ of dimension at least 2, a simple point P in the variety we have the ...
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Trying to understand exceptional divisors and Du Val singularities.

I'v been trying to understand how Du Val singularities are resolved and what the exceptional divisors look like so I can work out their dynkin diagrams. A basic example I tried in the interest of ...
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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 {$\...
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Projective bundle and blow up

Consider $\mathbb{P}^2$ and a point $p\in\mathbb{P}^2$. We can construct the blow-up of $\mathbb{P}^2$ at $p$ (we denote it by $\tilde{\mathbb{P}}^2$) as the closed subvariety $$\tilde{\mathbb{P}}^2\...
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Elements of the Picard group which have self intersection number of -1

Define $\tilde S_m$ to be the blowup of $\mathbb P^2$ at $m$ general points, I am trying the find all elements $[D]$ of $\operatorname{Pic} ( \tilde S_m)$ such that $[D]^2 = -1$ Attempt At Solution: I ...
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Complex and Symplectic Blow-ups

I am reading about complex and symplectic blow-ups from McDuff-Salamon's Introduction to Symplectic Topology and have some troubling extracting the main point from all the details. If I understand ...
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Question concerning blowups: Closed immersion $\operatorname{Proj}(A+\mathfrak m+\mathfrak m^2+…)\rightarrow \operatorname{Spec}(A[T_1,…,T_n])$

I am looking at the following exercise: If $A$ is a ring with maximal Ideal $\mathfrak{m}$, define $\tilde{A}:= A\oplus\mathfrak{m}\oplus\mathfrak{m}^{2}\oplus...$. Assume $\mathfrak{m}$ is generated ...
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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 $\pi(S)=C\subset Y$...
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Why is the blow up of a submanifold of $\mathbb{P}^n$ again projective

I saw somewhere that the blow up (at any point) of a submanifold of $\mathbb{P}^n$ is still projective. I have the feeling that this is a consequence of the Kodaira embedding theorem, any thoughts?
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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 ...
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If the blow up module $B_{\cal J} M$ is finitely generated, why is $M_n \oplus M_{n+1} \oplus \cdots$ generated by $M_n$ as a $B_I R$-module?

From Eisenbud's Commutative Algebra: I don't see how it's so obvious that $M_n \oplus M_{n+1} \oplus \cdots$ is generated by $M_n$ as a $B_I R$-module. For simplicity, let's look at a homogeneous ...
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Relation between tautological line bundle and blow up at the origin

We can define the projective $n$-space $\mathbb{P}^n$ as the quotient of $\mathbb{C}^{n+1}\setminus \{0\}$ by the action of $\mathbb{C}^*$ with all weights equal to $1$. Moreover we can define the ...
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Resolution of branch point singularity

Let $\pi:Y\to\mathbb{P}^2$ be a Galois cover of the projective plane which is branched along $r$ lines $L_1, L_2,...,L_r$ in $\mathbb{P}^2$. Suppose the lines $L_1, L_2,...,L_r$ all pass through the ...
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Closure of the pre-image of the closed subscheme of a blow-up gives the whole blow-up?

Let $X$ be a Noetherian scheme and $Z$ be a closed subscheme of $X$. Let $\pi: \tilde X\to X$ be the blowup along $Z$ so that $\pi|_{ \pi^{-1}(X\setminus Z)} : \pi^{-1}(X\setminus Z)\to X\setminus Z$ ...
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Why does a blowup of $\mathbb{A}^2$ about the origin result in the Möbius band?

I am new to learning about blowups, so I was searching for an easy example to wrap my mind around the concept. I came across this document https://homepage.univie.ac.at/herwig.hauser/Publications/...
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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 \...
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Exceptional Set vs Tangent Cone

Let $k$ be an algebraically closed field, $R=k[x_1,\ldots,x_r]/J$ for some ideal $J$, $X=Z(J)\subseteq\mathbb{A}^r$, and $I=(x_1,\ldots,x_r)$. I'm following Eisenbud's Commutative Algebra with a View ...
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Blowing up the whitney umbrella over the z-axis

My professor gave us the example of the Whitney Umbrella as an example of a non-trivial resolution of singularities. I'm aware that to resolve the singularities of the Whitney Umbrella, I need to ...
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133 views

Resolving indeterminacy of map on projective spaces induced by a linear map

Let $V$ and $W$ be two $k$-vector spaces, where $k$ can be assumed as $\mathbb{C}$, and $f:V\longrightarrow W$ a non null $k$-linear map with kernel $K$. The map $f$ naturally induces a rational ...
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algebraic de Rham cohomology of blowup of relative line

Let $$T = \mathbb{A}^1_k,\,\,\, Y = \mathbb{P}^1_T,\,\,\, X = \mathscr{B}(Y),$$ the blowup of $Y$ at a point. I am trying to compute the de Rham cohomology $H^1_{dR}(X/T)$, but I could use some help. ...
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58 views

Projection from Blowup is Isomorphism Away from Exceptional Set

I'm following Eisenbud's description of blowing up: let $X$ be an affine algebraic variety, $R$ the coordinate ring of $X$, and let $a_1,\ldots,a_r$ generate $R$ as a $k$-algebra. Let $Y\subseteq X$ ...
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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. ...

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