Questions tagged [singularity-theory]
Singularities in algebraic/complex/differential geometry and analysis of ODEs/PDEs. Singular spaces, vector fields, etc.
579 questions
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Hyperplane sections of Gushel-Mukai threefolds
Let $X$ be a Gushel-Mukai threefold, and $S \in |\mathcal{O}_X(H)|$ a hyperplane section of $X$. How are possible hyperplane sections classified?
I know that if $S$ is a generic hyperplane section, ...
- 395
1
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1
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The normalization of a semi normal complex surface germ is a holomorphic immersion?
Given a reduced and irreducible complex surface germ $(X,0)$ which is seminormal, consider its normalization
$$
n : (\overline{X},0) \longrightarrow (X,0).
$$
Is the normalization map $n : \overline{X}...
- 135
2
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Commutativity of nearby cycle functors and closed immersions under transversality condition
Let $X$ be a smooth variety over $\mathbb{C}$,
$Z$ be a closed subvariety of $X$.
Let $f:X \to \mathbb{A}_{\mathbb{C}}^{1}$ be a regular function, $X_0 := f^{-1}(0)$.
Denote $Z_0 = X_0 \cap Z$ and $i :...
- 121
4
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1
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286
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Does the intersection of a variety with rational singularities with a smooth variety have rational singularities?
Let $X \subset \mathbb{P}^n$ be a complex projective variety with rational singularities, and $Y$ a smooth subvariety in $\mathbb{P}^n$ that meets $X$ transversally, that is to say their tangent ...
3
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1
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200
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The Intersection number on a complex noncompact variety
Let $X$ be a variety over $\mathbb{C}$ and $C \subset X$ be a proper curve. Let $L$ be a line bundle on $X$. Then, the intersection number $L \cdot C$ is given by $\deg L|_{C^\nu}$, where $C^\nu$ is ...
- 573
1
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0
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63
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Nodal varieties with simple resolutions
I am interested in classes of nodal varieties whose minimal resolutions (i.e. the blowups at all of the nodes) are of a well-understood form. For instance, the blowup of a nodal quartic surface is K3. ...
- 1,135
4
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225
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An infinite sequence of birational proper morphisms
Let $X$ be a projective normal variety and $X=X_1 \to X_2 \to \ldots$ be an infinite sequence of birational proper morphisms of normal proper varieties. Is it true that $X_i \to X_{i+1}$ is an ...
- 573
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116
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Multiplicity at a point of a parameterised algebraic variety
Let $f_1,\dots,f_{n},g_1,\dots,g_n\in\mathbb C[t_1,\dots,t_{n-1}]$ polynomials of degree $d$ such that $V(g_1,\dots,g_{n})=\emptyset$. Consider the map
$$\begin{array}{cccc}
\psi:& \mathbb A^{n-1} ...
- 83
1
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0
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193
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Examples of Poincaré spaces
Recall, an algebraic variety $X$ of dimension $n$, is called a Poincaré space if the associated cohomology groups satisfy Poincaré duality. I am looking for examples of Poincaré spaces. I found some ...
- 2,577
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237
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Existence of Lefschetz pencil in singular varieties
Given a smooth, projective variety $X$, we know that there exists a Lefschetz pencil parameterizing divisors in $X$ such that the fibers are either smooth or has at worst ordinary double point ...
- 2,577
0
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66
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Concordance group of stable maps from $S^1$ to $S^1$
It seems that the concordance group of stable maps from $S^1$ to $S^1$ is computed and is given by the topological degree of these maps, that is, is 𝑍. However I don't agree with this statement ...
- 31
4
votes
1
answer
267
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Rational smoothness and local intersection cohomology
I know that if the Kazhdan-Lusztig polynomial $P_{uv}=1$ for all $u<v$, then the Schubert variety $X_v$ is rationally smooth. If I understand correctly, the $KL$ polynomial can be written in terms ...
- 759
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221
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What is the "correct" definition of a local deformation?
Short version:
If a local deformation of $X$ is any flat morphism $\pi$ into the spectrum of a local ring together with an isomorphism of the special fiber and $X$, there is no bound on how "bad&...
6
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217
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Sufficient infinitesimal conditions to ensure that $f \colon \mathbb{R}^n \rightarrow \mathbb{R}^N$ is locally injective at $0$
Let $f \colon \mathbb{R}^n\rightarrow \mathbb{R}^N$ be smooth (assume it is analytic if it helps). I am looking for infinitesimal conditions on $f$ at $0$ (i.e. condition on the derivatives and ...
- 1,329
2
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What are all the possible Bernstein-Sato polynomials?
Recall that to any polynomial $f(x_1,\dots,x_n)$ in $n$-variables, one can associate a $1$-variable polynomial $b_f(s)$ called the Berntein-Sato polynomial. It is known that $b_f(-1) = 0$ and moreover ...
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