Questions tagged [resolution-of-singularities]
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262 questions
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Equiresolvable stratification of families of hypersurfaces
Roughly speaking, my question is: given any hypersurface $X$ with $i$ ordinary double point singularities, can I find a family of hypersurfaces parametrized by a disc such that every fiber over the ...
- 2,647
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Relation between Du Bois singularity and cotangent complex
I am currently going through the notes Hodge theory and singularities by M. Popa. By reading the definition of Deligne Du Bois complex, I am wondering if there exists any relation between this with ...
- 1,619
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Inequalities among exponents in the unit part of the resultant of a quasi-ordinary polynomial
Following my previous question here, and after checking several references and carrying out explicit computations, I was able to substantially simplify the problem. I therefore formulate below a ...
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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}...
- 315
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Resolution of isolated quasi-homogeneous singularities via weighted blow-ups
Background and setup
Let $k$ be a field of any characteristic, and let $f \in k[x_1, \ldots, x_n]$ be a quasi-homogeneous polynomial of degree $d$ with weight $(d_1, \ldots, d_n)$, meaning: $$f(t^{d_1}...
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Is there a finite group $G$ with nontrivial Bogomolov multiplier such that $\mathbb{C}^n/G$ admits a crepant resolution?
Let $G$ be a finite group acting linearly on $\mathbb{C}^n$.
A crepant resolution of $\mathbb{C}^n/G$ is roughly one which does not affect the canonical class. See here for a nice discussion of ...
- 111
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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. ...
- 1,146
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Descent of vector bundles on Quot schemes?
Let $S$ be a scheme and $T : M \to N$ a morphism of locally free rank $n$ sheaves on $S$. Let $p : Q(T , r) \to S$ where $Q(T , r) := \mathrm{Quot}(\mathrm{CoKer}\,T , r)$ for $0 \leq r \leq$ minimal ...
- 526
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The leading order in the expansion of complex oscillatory integral with degenerate critical point
Consider the following complex oscillatory integral on a neighborhood of zero $U\subset \mathbb{R}^n$:
\begin{equation}
\mathscr{I}(a)=\int_{U}e^{a S(x)}\varphi(x)\, d^nx
\end{equation}
The functions $...
- 159
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Push-pull of canonical bundle for a resolution of singularities
Let $\pi : Y \to X$ be a resolution of singularities. Is there a formula for $\mathscr{F} = \mathrm{im}(\pi^* \pi_* \omega_Y \to \omega_Y)$ perhaps in terms of the discrepancies of the exceptional ...
- 4,480
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Computing the relative canonical divisor of the cone of a plane curve
Let $C \subset \mathbf{P}^2$ be a smooth projective plane curve of degree $d$ and $V \subset \mathbf{A}^3$ its associated affine cone. Let $\mu : X \longrightarrow V$ be the resolution given by ...
- 187
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Gabber’s $\ell'$-alteration theorem and varieties over function fields acquiring regular models after base change
I have seen it claimed that one can use Gabber’s $\ell'$-alteration theorem (theorem X.2.4 here) to prove the following statement: if $X$ is a smooth projective variety over a global function field $K$...
- 1,146
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Resolution in codimension $2$ in positive characteristic
Let $X$ be a normal quasi-excellent noetherian scheme in characteristic $p$ which is of f.t. over a perfect field. Is it possible to find a birational surjective map $X'\to X$, which is bijective on ...
- 183
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Origin of Hamburger-Noether expansions
The Hamburger-Noether expansion is a technical tool to "parametrize" in terms of a "coordinate" function, and subsequently desingularize curve germs. It plays a role similar to ...
- 5,534
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Rational singularities vs rationally smooth
Let $X$ be a normal algebraic variety over $\mathbb{C}$. What is the relationship between:
-$X$ has rational singularities, i.e. for any resolution of singularities $f:\tilde{X} \to X$, we have $R^*f_*...
- 1,135