# Questions tagged [resolution-of-singularities]

The tag has no usage guidance.

232 questions
Filter by
Sorted by
Tagged with
163 views

### Image, upto direct summands, of derived push-forward of resolution of singularities

Let $\mathcal C$ be a full subcategory (closed under isomorphism also) of an additive category $\mathcal A$. Then, $\text{add}(\mathcal C)$ is the full subcategory of $\mathcal A$ consisting of all ...
• 365
241 views

### Question about surface singularities

Throughout, $X$ will be a projective surface. I am looking for examples of the following surface singularities, I) A rational singularity that is not quotient. Obviously, it has to be non-Gorenstein, ...
• 233
95 views

### Singularities of curves over DVRs with non-reduced special fibre

Let $R$ be a complete DVR of mixed characteristic with fraction field $K$ of characteristic $0$ and residue field $k$ of characteristic $p>0$. Suppose that $\mathcal{X}$ is a normal $R$-curve such ...
180 views

1 vote
40 views

### Verdier (w) condition implies the $w_f$ condition when the restriction of $f$ in each stratum is a submersion?

Let $X\subset\mathbb{R}^n$ be and let $\Theta=(X_\beta)_{\beta\in I}$ a Verdier stratification for X. Let $f:X\rightarrow\mathbb{R}$ be a polynomial function, such that $f_{|_{X_\beta}}$ is submersion ...
1 vote
54 views

### How can I calculate the derivative of an integral with respect to a parameter if Leibniz's formula gives a divergent integral?

We are working on the problem related to a magnetic field in an axially symmetric magnetic plasma trap. Let's express the vector potential through the magnetic flux function \begin{gather} \label{1:01}...
117 views

### Semi-stable model over a totally ramified extension

Notation: Let $R$ be a DVR, $K=\text{Frac}(R)$ and $k=R/\mathfrak{m}$. Given an $R$-scheme $X$, write $X_K=X\times_{R} K$ for the generic fiber and $X_k=X\times_R k$ for the special fiber. Suppose $k$ ...
303 views

### Krull dimension of the smooth locus

Let $R$ be a normal complete local domain of dimension $n \geq 4$. Does there exist a prime ideal $\mathfrak{p}$ of height $\dim(R) - 1$ such that $R_{\mathfrak{p}}$ is a regular local ring? In ...
1 vote
130 views

### Relative compactification without resolutions of singularities

Let $Y$ be a smooth proper variety over a field $k$, let $X$ be a smooth variety over $k$, $U\hookrightarrow X$ the complement of a strict normal crossing divisor and $\phi\colon U\to Y$ a map. By ...
• 181
175 views

### Extending étale covers from the regular locus to a resolution of singularities

Let $X$ be a normal proper variety with rational singularities (or terminal if that is necessary) and $X_{\text{reg}} \to X$ the regular locus. Let $\pi : \tilde{X} \to X$ be a resolution of ...
• 3,301
326 views

### Lindelöf paper on meromorphic singularities

Does anyone know a digital link to the following paper, written by Ernst Lindelöf: "Mémoire sur certaines inégalités dans la théorie des fonctions monogènes, et sur quelques propriétés nouvelles ...
• 232
264 views

### Normal forms of ADE singularities

Given a surface $X:f(x,y,z)=0\subset \mathbb{A}^{3}_{\mathbb{C}}$ with only ADE singularities, how does one determine the correct singularity type of $X$ by computing the normal forms? Does a similar ...
• 471
108 views

### Crepant resolution of quotient singularities

Let $G$ be a finite subgroup of $U(m)$ such that $G$ acts freely on $\mathbb C^m \setminus \{0\}$. If $\mathbb C^m/G$ has a crepant resolution, can we necessarily derive that $G \subset SU(m)$?
• 1,361
128 views

### Inverse image Weil divisor on a toric variety as a Cartier divisor

Let $X$ be a normal toric variety over an algebraically closed field and let $D$ be a torus invariant (prime) divisor. Assume $\pi\colon \tilde{X}\rightarrow X$ is a toric resolution of singularities ...
• 337
276 views

### Minimal resolution of singularities of surfaces

Let $X$ be a normal projective irreducible surface over an algebraically closed field $k$. Let $\pi\colon Y\to X$ be a birational morphism, such that $Y$ is a smooth projective surface, and assume ...
• 7,652
1 vote
126 views

### Question about the definition of variety in Kollár's book on resolution of singularities

In Kollár's book "Lectures on Resolution of Singularities" it is claimed in 3.8 page 125: "Our resolution is strong and functorial with respect to smooth morphisms" I would like to ...
• 595
232 views

### Motives of resolutions of singularities

Suppose $X'$ is a resolution of singularities of a projective variety $X$ over a field $k$ of characteristic 0 that is functorial for smooth morphisms. How are the (mixed) motives of $X$ related to (...
• 21
69 views

### How to determine the singlarity type (up to local analytic isomorphism) of a hypersurface surface singularity

Given a polynomial f(x,y,z), it defines a hypersurface in $\mathbb C^3$. I guess there is a classification of hypersurface singularity like Arnold normal form. I wonder given an explicite example of f,...
• 613
1 vote
121 views

1 vote
297 views

### Beauville Exercise VII.7 (3)-A proof that $\kappa(X)\geq \kappa(Y)$ for $f\colon X\to Y$ surjective morphism of smooth projective varieties

Here $\kappa(X)$ denotes the Kodaira dimension of a smooth projective variety $X$. Question 1: I would like to solve Exercise VII.7 (3) from the Beauville book "Complex Algebraic Surfaces": ...
98 views

### Deformation to normal cone of the exception divisor of a log-resolution

I am reading the paper Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink due to G. Guibert, F. Loeser, and M. Merle. The main tool, like a lot of papers in ...
• 726
113 views

• 263
198 views

### Searching for resolutions of generalized determinental varieties

I'm interested in studying a certain generalization of determinental varieties as defined here: https://en.wikipedia.org/wiki/Determinantal_variety To be more specific, I must first lay out a few ...
1 vote
144 views

### Blow up singularities on curves

Let $p$ be a prime number and let $\bar{\mathbb F}_p$ be an algebraic closure of $\mathbb F_p$. Let $C$ be an irreducible singular projective curve over $\bar{\mathbb F}_p$. Let $P$ be a singularity ...
228 views

### Birationally equivalent elliptic curves and singularities

I got the following cubic elliptic curve from some physical problem $$E_c(\mathbb{C}): w^2=4 z^3-zG_2-G_3,$$ where $G_2=3 \alpha ^2+\gamma$ and $G_3=\alpha ^3-\alpha \gamma -\beta ^2$ for known ...
• 211
332 views

### Do there exist linear relations between exceptional divisors

Let $X$ be an isolated, Gorenstein singularity of dimension at least $2$ and $\pi: \widetilde{X} \to X$ be a resolution of singularities. Let $E$ be the exceptional divisor and $E_1,...,E_r$ be the ...
• 2,022
343 views

### References on Namikawa-Weyl group

What are the most reasonable references on the definition of the Namikawa-Weyl groups and the first results about them? In particular, are there more recent (or more educational) texts than the ...
• 133
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 ...