# Tagged Questions

91 views

### Simple example of isolated critical point with non-semisimple monodromy

Consider a polynomial map $f :\mathbb{C}^{n+1} \rightarrow \mathbb{C}$ with $f(0)=0$ (no constant term) and with isolated critical point at $0 \in \mathbb{C}^{n+1}$. We can choose a disc $D$ of some ...
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### Topological degree and polynomial degree

Let $F:\mathbb{C}^n\to \mathbb{C}^n$ be a homeomorphism homogeneous of degree 1 (i.e., $F(tx)=tF(x)$, $t>0$) and $g:\mathbb{C}^n\to \mathbb{C}$ a homogeneous polynomial of degree $k$. Let $L$ ...
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### Bertini's Theorem

Let $p_1,...,p_n\in\mathbb{P}^{N}$ be general points. Consider the linear system $|L|$ of hypersurfaces of degree $d$ in $\mathbb{P}^{N}$ with prescribed multiplicities $m_1,...,m_n$ at $p_1,...,p_n$. ...
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### Properties of singularities that are preserved by categorical quotients

Let $G$ be a reductive group acting on an affine singular variety $X$, and let $X/G$ be the categorical quotient. I know that if $X$ has rational singularities, then so does $X/G$ ...
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### Blowdown and contraction

I am sorry, my question is very naive. 2nd Edit: Let us suppose that $V$ is a smooth complex projective variety, and $Y\subset V$ is a smooth divisor and has an ample conormal line bundle. We would ...
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### Do level sets always correspond to even graphs?

Suppose I have a level set of some function $f\colon\mathbb{R}^n\rightarrow\mathbb{R}^m$, say $L:=\{x:f(x)=c\}$. Let $S$ denote the points in $L$ at which $L$ is locally diffeomorphic to an open ...
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### Do there exist double points on an algebraic surface in $\mathbb{P}_{\mathbb{C}}^3$ that are not rational?

The title explains it all. I'm familiar with the du val singularities on surfaces, also known as rational double points. In http://homepages.warwick.ac.uk/~masda/surf/more/DuVal.pdf, 2.1, they are ...
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### singularities of the dual variety of a surface

I am looking for a proof/reference of the following simple fact, which I think it holds true. Let $S\subset \mathbb{P}^n$ be a surface embedded by a very ample linear system. Then I know that the ...
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### Which isolated surface singularity comes from a -5 curve?

Define the surface $X$ to be the total space of $\mathcal{O}_{\mathbb{P}^1}(-5)$. By contracting the exceptional curve in $X$, we get a surface with an isolated singularity. I am looking for the ...
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### Normal form for a holomorphic Morse function

Similarly to Morse lemma, a holomorphic Morse function can be written, near a critical point, as $W_1^2+W_2^2+...+W_n^2+C$ , for well chosen coordinates $W_1,W_2,...,W_n$. I want to cite this result ...
214 views

### on a characterisation of regular D-modules

Let $X$ be a smooth variety over a field of characteristic zero. Let $M$ be in the derived category of holonomic $\mathcal{D}_{X}$-modules, $D^{b}_{h}(\mathcal{D}_{X})$. We know that if we assume ...
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### Hypersurfaces with Gorenstein singular loci

Recall that a hypersurface $D$ in a complex manifold $X$ is called a free divisor if the Lie algebroid $\mathcal{T}_X(-\log D)$ of vector fields tangent to $D$ is locally free. This condition is ...
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### on Neron defect of smoothness for groups schemes

Let $G$ a semisimple simply connected group over $\mathbb{C}$. Let $\gamma\in G(\mathbb{C}[[t]])$ such that $\gamma$ is regular semisimple on $G(\mathbb{C}((t)))$. We consider $I_{\gamma}$ the group ...
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### cotangent complex for finite flat morphism and different ideal

Let a ring $A$, $F=A[X_{1},\dots X_{n}]$ and $B:= F/J$. We suppose that we have a finite flat lci morphism $f:Spec(B)\rightarrow Spec(A)$. To mesure the singularities of this map, Gabber-Ramero ...
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### semicontinuity of the conductor defined by Temkin

We say a principal pair $(X,\mathcal{I})$ where $X=Spec(A)$ is affine scheme and $\mathcal{I}=\tilde{I}$ where $I\subset A$ is a principal ideal generated by $\pi$ wich is a non zero divisor. For a ...
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### Surfaces in $\mathbb P^3$ with many simple isolated singularities

Could anybody help me with examples of surfaces $X\subset\mathbb P^3$ (projective, over $\mathbb C$) having many isolated singularities of the type $A_1$ ($x^2+y^2+z^2=0$) or $A_2$ ($x^2+y^2+z^3=0$) ...
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### Factoriality of one-nodal Calabi-Yau threefolds

Let $X$ be a projective Calabi-Yau threefold with a single ordinary double point at $x \in X$, and smooth elsewhere. Is $X$ necessarily factorial? I suspect that the answer is "yes", for the ...
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### Is it obvious that the defining conditions to obtain a particular singularity are well defined on the quotient space?

Let $~f:\mathbb{C}^2 \rightarrow \mathbb{C}$ be a holomorphic function vanishing at the origin, with the following properties:  f_{00}, ~f_{10}, ~f_{01}, ~f_{20}, ~f_{11} =0,~~f_{20} \neq 0 ...
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### Is Gorenstein singularity a locally analytic property

Let $X$ be a variety over $\mathbb{C}$, and $X^{an}$ be the analytic space associated to $X$. $X$ has Gorenstein singularity at $x \in X$ iff the local ring $\mathcal{O}_{X,x}$ is a Gorenstein ring. ...
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### Finite construction of lacunary functions using algebraic and certain analytic operations

Algebraic functions have a discrete set of singularities. Lacunary functions, e.g. $f(z)=\sum_{n=0}^\infty z^{2^n}$, have a continuum of singularities at every point of the boundary of their disk of ...
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### Is the space of degree $d$ curves with marked smooth points dense inside the space of curves with marked points?

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of nonzero homogeneous degree $d$ polynomials in three variables upto scaling, where $\delta_d = \frac{d(d+3)}{2}$ (basically degree ...
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### Is there an analogous concept for the degree of a map, when the spaces are singular?

Let $M$ and $N$ be two smooth compact, oriented manifolds and $X\subset M$ an oriented submanifold of $M$ of dimension $k$ (not necessarily closed). Suppose in addition that $\bar{X}-X$ is contained ...
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### Are there general position results in singular algebraic sets?

Let $X$ be a real algebraic set, and let $Y \subset X$ be its singular set. In this question I'll focus on the analytic topology, so we can just imagine that $X$ is the zero set, in $\mathbb{R}^n$, ...
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### When is the intersection of an isolated normal singularity with a generic linear subspace through that singularity normal?

Suppose I have an affine subvariety $A \subset {\mathbb C}^N$ of dimension $n \geq 3$ which has an isolated singularity at $0$ (lets say for the sake of simplicity that it is non-singular everywhere ...
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### How do I check whether an orbifold admits deformations?

(Cross-post from math.stackexchange, where it has received no attention.) Orbifolds $\mathbb{C}^2/\mathbb{Z}_n$, given by the action $(x, y) \mapsto (\zeta x, \zeta^{-1} y)$ with $\zeta$ a primitive ...
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### Whitney stratifications

Many results on characteristic classes of singular varieties (as well as other singularity-theoretic constructions) make use of a so-called "Whitney stratification" of the variety under consideration, ...
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### Can I compute the cohomology of the complement of a log canonical divisor as if it were normal crossings?

Let $X$ be a smooth projective variety and $D$ a log-canonical divisor and let $U = X \setminus D$. I have heard the slogan "log-canonical is just as good as normal crossings for Hodge theory". This ...
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### Bounds for the milnor number of a hypersurface singularity

I am having a hard time in finding an upper bound in terms of the degree and the dimension for the Milnor number of an isolated hypersurface singularity. I am mostly interested in surfaces on the ...
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### Producing $(-2)$ curves on a smooth surface

We know that blowing up a point on a surface produces a $(-1)$ curve. Is there any such standard techniques to produce $(-2)$ curves in a smooth surface?
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### Factoriality: local or global?

Let $X$ be an algebraic variety. I have read the following definitions: $X$ is factorial if every Weil divisor on $X$ is Cartier. $X$ is locally factorial if all its local rings are unique ...
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### Smoothing of a hyperquotient singularity

Let $f$ be a polynomial in $k$ complex variables, and suppose the affine variety $V$ given by $f = 0$ has an isolated singularity at the origin, but is otherwise smooth. Now assume that some cyclic ...
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### Blowing up a projective surface

Let $X$ be a smooth degree $d$ ($d>5$) surface in $\mathbb{P}^3$. We now blow up $X$ at a point, embed it in some projective space, and and consider a projection of it into $\mathbb{P}^3$. The ...
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### Linearization instability and singular points of algebraic varieties

In a well known 1973 paper, Fischer and Marsden pointed out (with similar, contemporary remarks made in the physics literature by Brill and Deser) that the space of solutions of some non-linear ...
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### “Arithmetic genus” of a plane curve singularity.

I believe that the following questions are very basic, but I don't know how to get a reference. Consider a curve in the plane $C\in \mathbb C^2$ with a singularity at $0$ and suppose it is ...
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### Jacobian ideals reference

Suppose that $f : X \to V$ is a flat equidimensional (of dimension $h$) morphism of schemes of finite type and $V$ is excellent (or a variety) For this one can formulate something called the Jacobian ...
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### Factoriality vs $\mathbf{Q}$-factoriality for threefolds hypersurfaces with isolated singularities

Let $X \subset \mathbf{P}^4$ be a complex threefold hypersurface with isolated singularities. We denote as usual by $\textrm{Cl}(X)$ the group of Weil divisors modulo linear equivalence and by ...
How many lines in $\mathbf{P}^5$ passing through a fixed point $p$ meet in at least two points a fixed smooth surface $S$ given by the intersection of three quadrics? Or equivalently, calling $T$ the ...
Let $X\subset \mathbb{C} \mathbb{P}^{N}$ be a smooth, algebraic (locally closed) complex submanifold of $\mathbb{C} \mathbb{P}^N$ of complex dimension $k$. More concretely, $X$ is of the ...