Questions tagged [singularity-theory]

Question 1

In singularity theory, one defines an intrinsic derivative for a vector bunle homomorphism $\phi: E \rightarrow F$ where $E\xrightarrow{\pi_E}B$ and $F\xrightarrow{\pi_F}B$ are vector bundles with ...

Question 2

In order to state my question, I will begin with some definitions. Let $\mathbb{K}$ be an algebraically closed field of characteristic zero. Suppose we have a singular irreducible plane curve defined ...

Question 3

Let $A \colon= {\Bbb C}[X_1,\cdots,X_n]$ be a polynomial ring over ${\Bbb C}$. Let us consider the action $\sigma \colon X_i \mapsto \zeta^{e_i} X_i$ for $i = 1,\cdots,n$, where $\zeta \colon= \zeta_n$...

Question 4

I am working with some homogeneous polynomials $f\in \mathbb C[x_1, \ldots, x_n]$ belonging to families $\mathcal F$ over $\mathbb C^k$, i.e. I am considering $$f=\sum a_Ix^I, \qquad\qquad a_I\in \...

Question 5

I am working on exercise 2.3.12 in Shafarevich's Basic Algebraic Geometry I. The problem is to classify double singular points of plain algebraic curves over an algebraically closed field $k$ of ...

Question 6

I'm working with the Milnor number for a polynomial $f$ and am puzzled as different authors compute it in different contexts. There appears to be an implicit assumption that the Milnor number will be ...

Question 7

My question concerns the smoothing of continuous vector fields V on a Riemannian manifold. Specifically, how can one approximate a continuous vector field using smooth vector fields? The approximation ...

Question 8

Let $X$ be a complex manifold, and let $Y\subset X$ be an irreducible hypersurface (i.e. analytic subset of codimension 1). Lemma 2.3.22 in Huybrechts' Complex Geometry shows the sheaves $\mathcal{O}(-...

Question 9

I would like to find a simple proof or counterexample to the following claim, which has come up in some work we are doing related to curves in surfaces which bound immersed disks. Let $F$ be any ...

Question 10

Let $R$ be a Noetherian normal domain of Krull dimension $2$. By Serre's criteria, if $\mathfrak p$ is a prime ideal such that $R_{\mathfrak p}$ is not regular, then $\mathfrak p$ must have height $2$,...

Question 11

We are given $p(z,u)$, a nonzero polynomial with real coefficients. Suppose we know that: (a) there is a generating function $g(z)$ that solves $p(z,g(z))=0$; (b) $g(z)$ has nonnegative coefficients; (...

Question 12

Smooth manifolds like $\mathbb{R}^n$ and $S^n$ behave nicely under slicing by coordinate hyperplanes — for example, slicing $\mathbb{R}^3$ along $x_3 = c$ yields $\mathbb{R}^2$, which is again smooth. ...

Question 13

An example of a distribution whose singular support has measure zero is the delta function $δ(x)$, whose singular support is just $\{0\}$. I don't know of any examples of distributions whose singular ...

Question 14

I want to study the behavior of a curve $\mathcal{C}$ implicitly defined as the zero of function $\mathbf{F}: \mathbb{R}^n\rightarrow\mathbb{R}^{(n-1)}$, $\mathcal{C}:\{\mathbf{x}\in \mathbb{R}^n \...

Question 15

I'm working with projective rational curves over an algebraically closed field $k$ (for simplicity, $k = \mathbb C$. More specifically, they are projective curves $C\subseteq \mathbb P^n$ with a ...

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Questions tagged [singularity-theory]

Intrinsic derivative

Local Milnor Number Using Gröbner Bases

Calculating difficulty for quotient singularities.

Statements about dimension of singularity loci in families of polynomials

Exercise 2.3.12 in Shafarevich's Basic Algebraic Geometry I [duplicate]

Milnor number Independent on the domain of its definition

Question about Smoothing of Vector Fields

Codimension of singular points of hypersurface $Y\subset X$

Can an immersion from the disk to a surface "double up" on its boundary?

Singular locus of Noetherian normal domain of Krull dimension 2

Value of algebraic generating function at $z=1$

Slicing pseudomanifolds and getting sub-pseudomanifolds?

Is the singular support of a distribution always measure zero?

Puiseux series for solution of polynomial equations

Is the the preimage of a cusp in a projective rational curve always a singleton?

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