All Questions
Tagged with blowup singularity-theory
18 questions
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29
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Does maximal contact hypersurface (m.c.h.) exist for any curve over a field of arbitrary characteristic, and if so, for what definition of m.c.h.?
Does a maximal contact hypersurface always exist for any curve over a field of arbitrary characteristic, and if so, for what definition of maximal contact hypersurface?
3
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0
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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
1
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57
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Transversality of strict transforms
When reading a book, I encountered the following assertion: let $k$ be a field (maybe perfect if it makes things easier), $X$ a smooth $k$-variety and $Y$ a closed irreducible subvariety, also smooth. ...
- 2,249
2
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1
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118
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How to show $x_0^2+x_1^2+x_2^2=0 \subset \mathbb{CP}^2 \iff \mathbb{CP}^1$
I am currently trying to blow-up an $A_n$ singularity defined by the hypersurface equation:
\begin{equation}
z_1^2+z_2^2+z_3^{n+1}=0 \subset \mathbb{C}^3
\end{equation}
Let $x_i, i=0,1,2$ denote the ...
- 23
0
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1
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424
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Resolution of ADE- singularities: The second blow-up for the A_2 singularity
I try to resolve the surface singularity $X\subset \mathbb{A}^3$ of type $A_2$. It is defined by $x^2+y^2+z^3=0$. I expect to need at least two blow-ups. But after the first blow-up of $X$ the strict ...
- 2,383
1
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1
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704
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Resolution of singularities of analytic spaces
It seems to me that the following resolution of singularities theorem (or a modification) is known to specialists but I have trouble finding references.
Let $X$ be a complex analytic space, then there ...
- 346
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1
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114
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Connectedness of exceptional divisors
Let $X$ be a quasi-projective variety over $\mathbb{C}$. Let $I$ be an ideal sheaf supported at a closed point on $X$. Is the exceptional divisor for the blow-up of $X$ along the ideal sheaf $I$, ...
- 467
0
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1
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136
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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
1
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270
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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 ...
- 31
1
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1
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635
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Exceptional divisor of the blow-up of affine cone at the vertex
Let $f(x) \in \mathbb{C}[X_1,...,X_n]$ be a homogeneous polynomial in $n$ variables such that the zero locus $V$ of $f$ in $\mathbb{C}^n$ is singular only at the origin. Denote by $\pi:\widetilde{V} \...
- 397
2
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273
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Blowing up lines in projective space
In $\mathbb{A}^3$ we can transform the surface $x^2=y^2z$ to a smooth surface by blowing up the z-axis.
What if we want to resolve this surface everywhere in $\mathbb{P}^3$? If we take projective ...
- 21
2
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1
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710
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Intuitively, why does there need to be an exceptional divisor?
This question is a bit vague, but I hope its still good enough for this site.
When resolving a singularity of, say, a curve by blowing up a point, we get new variety that besides the strict transform ...
- 7,405
2
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46
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Gloabal and Local data on Singular Varieties
Suppose we have a subvariety (codimension one) inside a smooth variety with a fixed divisor class D. However, let it be a singular one. I'm just wondering how much the usual techniques (such as ...
- 113
6
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2
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2k
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How to resolve the singularity of $xy+z^4=0$?
This singularity can not be resolved by one time blow-up. I don't know how to blow up the singularity of the "variety" obtained by the first blow-up, in other words, I am confused with how to do the ...
- 2,277
2
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1
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273
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How to prove the minimal resolution of double rational points can be achieved by iterated blow-ups
In the Slodowy's survey on Kleinian singularities, there is a statement that the minimal resolution of Kleinian singularities can be obtained by iterated blow-ups, I want to know the detail of the ...
- 2,277
3
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1k
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Is exceptional divisor always a Projective bundle over the centre?
Let $f:\tilde X \rightarrow X$ be a blow up at center Z. Is $f^{-1}(z)=\mathbb{P}^{k}$ ?for some $k$, $\forall$ $z\in Z$
In the case of blow-up of $\mathbb{A}^{n}$ at origin it is very clear that the ...
- 5,209
4
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0
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319
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Is it true that blowing up a quasi-affine variety at a nonsingular point never introduces new singularities?
If we let $M$ be a quasi-affine variety, is it true in general that the blowup of $M$ at a non-singular point $p$ does not introduce new singularities? I came across this statement in my reading, but ...
- 41
166
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1
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5k
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What is the Picard group of $z^3=y(y^2-x^2)(x-1)$?
I'm actually doing much more with this affine surface than just looking for the Picard group. I have already proved many things about this surface, and have many more things to look at it, but the ...
- 1,693