# Questions tagged [singularity-theory]

Singularities in algebraic/complex/differential geometry and analysis of ODEs/PDEs. Singular spaces, vector fields, etc.

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### Cohen-Macaulyness of Milnor algebra

Denote by $R = \mathbb{C}\{x_1, \dots, x_n\}$ the ring of germs of analytics maps at the origin in $n$ variables and let $f \in R$ such that $Sing(V(f))=V(x_1, \dots, x_{n-1})$ as sets. In addition, ...
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### Surfaces with rational double points

Let $S\rightarrow \mathbb{P}^1$ a surface fibered in conics over a field. Assume that $S$ has a single non reduced fiber $F$ with two points of type $A_1$ on it. Blowing-up the two points and ...
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### Singularities of surfaces fibered in rational curves

Let $S$ be a projective surface with a morphism $S\rightarrow\mathbb{P}^1$ whose fibers are either smooth $\mathbb{P}^1$'s or the union of two smooth $\mathbb{P}^1$'s intersecting in a point. ...
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### Is a function looking like a cubic cusp globally equivalent to the cubic cusp?

Let's consider a family of smooth odd functions $\phi_v(u)\colon \mathbb{R}^2\to\mathbb{R}$, which ,looks like' a family of functions $f_y(x)=x^3-yx$ in the vicinity of $(0,0)$: $\phi_v(u)$ has no ...
1 vote
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### Homogeneous deformation of isolated singularities

Let $f\in \mathbb{C}[x_1, \dots,x_n]$ be a homogeneous polynomial of degree $p$ and let $F \in \mathbb{C}[x_1, \dots,x_n,t]$ be a polynomial such that $F(x_1, \dots, x_n,0)=f$, for every $t_0$ we have ...
165 views

### Deformation of isolated singularities and non zero divisors

Consider $f \in \mathbb{C}\{x_1,\dots,x_n\}$ such that $(V(f),0)$ has an isolated singularity. Let $F \in \mathbb{C}\{x_1,\dots,x_n,t\}$ be a deformation of $f$ such that there exists some integer $m$...
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### Maximal number of certain types of isolated singularities on $3$-folds

Let $X$ be an algebraic surface with only canonical singularities and such that $K_X$ is nef. Then Miyaoka's "The Maximal Number of Quotient Singularities on Surfaces with Given Numerical ...
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### Singularities of Painlevé II

It is known that Painlevé II has only simple poles $t_n$ as singularities and these poles have non-movable property, i.e., they depend only on the equation rather than initial conditions. One can try ...
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I am trying to understand the article Reduction of Singularities of the Differential Equation $Ady=Bdx$ by Arno van den Essen. Let $A,B\in k[[X,Y]]$ be formal power series and $D$ the vector field $A\... 3 votes 1 answer 94 views ### du Val singularities in Magma Is there any way to decide whether a singularity of a surface embedded in$\mathbb{P}^5(\mathbb{Q})$is a du Val/rational double point in Magma? Any help is much appreciated. 3 votes 0 answers 139 views ### Stability of nodal hypersurfaces We denote by$\Pi_{n,d}$the space of homogeneous polynomials of degree$d$in$n+1$variables$x_0,\ldots,x_n$, i.e.$\Pi_{n,d}=\Gamma(\mathbb{P}^n(\mathbb{C}),\mathcal{O}(d))$. The group$G=SL(n+1)$... 4 votes 2 answers 437 views ### Smoothness of fibers over finite fields Let$f:X\rightarrow Y$be a morphism of smooth projective varieties over a finite field of characteristic different from$2$. Is there any result on the existence of a point$y\in Y$such that$X_y = ... 172 views

### Preimage by birational maps

I am looking for an example (I guess that in complex projective space $\mathbb{P}^{n}$ is good) such that satisfy the following condition (in non trivial case, for this assume $X \neq \tilde{X}$): Let ...
1 vote
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I'm looking for good (as simple as it is possible) reference for the local discriminant variety. I need it in the following situation: I have an unfolding $F: (\mathbb{K}^n \times \mathbb{K}^p, 0) \to ... 2 votes 0 answers 131 views ### plumbing description of resolution of ADE singularities Let$G$be a finite subgroup of$SU(2)$and consider the quotient of the unit ball$B\subset \mathbb{C}^{2}$by$G$. The result, denoted by$V$, has a boundary$S^{3}/G$and has an ADE singularity at$...
Let $X_{2,3}\subset\mathbb{P}^n$, with $n\geq 5$, be a complete intersection of a quadric $X_2$ and a cubic $X_3$ containing a $2$-plane $H$. Assume $X_2$ and $X_3$ to be general among the ...