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3
votes
3answers
239 views

Contractibility of curves and embedding into projective space

Let $f:X \to Y$ be a proper surjective morphism of projective surfaces such that there exists a curve $C \subset X$ for which $f|_{X\backslash C}$ is an isomorphism and $f(C)$ is a set of points. ...
2
votes
1answer
119 views

minimality/universality of the Springer resolution of a determinantal variety

Let $X\subset P^n$ be a singular determinantal variety and $S\to X$ its Springer resolution. Let $X'\to X$ another resolution of singularities (say, a blow-up). Does $S$ have some minimality/...
3
votes
0answers
209 views

References for resolutions of ordinary singular points

Let $X$ be a $n$-dimensional complex projective algebraic variety, let us suppose that $X$ has only isolated singularities. Edit: Let us say that an ordinary $m$-ple singular point is an isolated ...
8
votes
2answers
490 views

Can we foliate the punctured space by tori?

Is it possible to have a 2 dimensional foliation of $\mathbb{R}^{3}-\{0\}$ such that each leaf is homeomorphic to the torus? what algebraic topological obstruction exist? Another question: is there ...
3
votes
2answers
255 views

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 ...
3
votes
1answer
408 views

Is it possible to resolve singularities using only normal varieties?

In characteristic 0, is it possible to have a resolution of singularities where the algebraic varieties at every step of the desingularization process are normal. To be more precise, I would like a ...
3
votes
0answers
250 views

equivariant resolution of singularities

I am looking for some references on equivariant resolution of singularities. In most references quoted on mathoverflow (for instance : Reference on an equivariant resolution of singularities), they ...
3
votes
1answer
328 views

Small resolution of a non-isolated singularity?

Consider the the Hypersurface singularity given by the equation $$xyz+st=0 \subset \mathbb{C}^5.$$ How would you describe a (nice!=symmetric) small-resolution of this singularity?
1
vote
1answer
253 views

Serre's conditions under blow-ups, Blowup and normalization

Suppose $X = \mathbb{Z}[x, y, z]/(f,g)$ is a 2-dimensional Cohen-Macaulay surface. In particular, $X$ satisfies Serre's condition $S_2$. Suppose it is irreducible, reduced but not normal. $\bf{...
9
votes
2answers
639 views

Finite generation and Henselization

Now I understand the answers. I am trying to understand Henselian Weierstrass Theorem in Hironaka's Idealistic exponents of singularity, page 76 - 77. A glance at the paper. At some point there is ...
1
vote
1answer
227 views

Importance of Denjoy-Carleman classes as a class.

Denjoy-Carleman classes of differentiable functions, say in Roumieu's form: Given a log-convex sequence $M_n$ of positive number denote by $C_M=C_M(\mathbb{R}^n,0)$ the ring of germs of $C^\infty(...
3
votes
0answers
317 views

“Step-by-Step” toric resolution process?

WLOG the fan $\Sigma$ of our toric variety $X_{\Sigma}$ is simplicial. (So $X_{\Sigma}$ has at worst orbifold singularities and all cones $\sigma \in \Sigma$ are simplicial). The classical toric ...
3
votes
0answers
116 views

(semi-)Small resolutions of Peterson varieties

Peterson varieties (in type A) can be described as the subvarieties of the full flag variety $$\{(F_{i})\;|\; F_{i}\subset \mathbb{C}^{n}, \; \dim F_{i} =i,\; N(F_{i})\subset F_{i+1}\}$$ where $N$ ...
0
votes
0answers
96 views

solve the singularities of parabolic orbits of schubert cells

Let G a semsisimple connect'ed group over $k$, $B$ a Borel and $P$ a parabolic subgroup of $G$ with Weyl group W_{P}. For $w\in W_{P}\backslash W/W_{P}$, how can we solve the singularities of $X_{w}=\...
3
votes
1answer
194 views

Is $\pi_1(\widetilde{X/G})$ always finite if $\pi_1(X)$ is finite?

Let $X$ be a smooth complex manifold with finite fundamental group. Suppose that a finite group $G$ acts on $X$ and let $\widetilde{X/G}$ be a resolution of singularities. Is $\pi_1(\widetilde{X/G})$ ...
1
vote
0answers
262 views

How to Construct a ''Nice'' Birational Model in Characteristic $p>0$?

Let $X={\rm Spec} A$ be a normal affine variety of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Also, assume that $S\subset X$ is a prime Weil-divisor on $X$. Now,...
0
votes
0answers
114 views

Is the modification a rational map?

Good morning, I would like to ask the following question concerning the desingularisation, but I'm not familiar at all with these notions. We have the following theorem of Hironaka: Let $X\subset \...
4
votes
1answer
326 views

Which schemes can be presented as limits of smooth varieties?

I can prove a certain statement for any scheme that can be presented as the limit of an essentially affine (filtering) projective system of smooth varieties over a perfect field such the connecting ...
12
votes
0answers
431 views

Is a flop on Calabi-Yau threefolds always Atiyah flop?

Is it true that any flop on a Calabi-Yau threefold is given by the Atiyah flop? That is, there always exists a rigid rational curve $\mathbb{P}^1$ with normal bundle $\mathcal{O}_{\mathbb{P}^1}(-1)^{\...
3
votes
1answer
181 views

Normal bundle of exceptional locus of the conifold

Let us consider the conifold singularity $xy-zw=0$ in $\mathbb{C}^4$. By blowing up along the divisor defined by $x=z=0$, we have a small resolution of the conifold with $\mathbb{P}^1$ as the ...
2
votes
2answers
745 views

Embedded resolution of singularities

I'd like to check with my colleagues whether I have correctly understood "embedded resolution of singularities". Let $X$ be a nonsingular projective variety over $\mathbf C$ and let $D$ be a "nice" ...
2
votes
0answers
115 views

Global minimal model over a non-affine base

Remark 10.1.8 in Liu's book (AG and Arithmetic curves) says that over a non-affine base (base is always assumed to be a Dedekind scheme of dim 1), the minimal regular model of a (smooth projective) ...
2
votes
1answer
174 views

resolution of strata of the affine grassmanian

Let G a semisimple simply connected group over an algebraically closed field. Let $Gr:= G(k((t))/G(k[[t]])$ be the affine grassmanian. It admits a stratification indexed by the dominant cocaracter $...
2
votes
2answers
269 views

springer resolution over $\wedge^3 \mathbb{C}^6$

The action of $GL_6$ on $P(\wedge^3 \mathbb{C}^6)=P^{19}$ has 4 orbits (of dim 19, 18, 14, 9). Can you describe how the springer resolution applies to each of these orbits? It should have positive ...
2
votes
0answers
343 views

Crepant resolutions of cDV singularities?

Compound Du Val 3-fold singularities form a good class of singularities in 3-fold singularity theory. I would like to know which singularities admit crepant resolutions. If I remember correctly, $cA_{...
8
votes
1answer
448 views

Schubert varieties which admit small resolutions of singularities

I am looking for an (incomplete) list of partial flag varieties for which all Schubert cells admit small resolutions of singularities. This is interesting, for many reasons. My motivation is, that a ...
5
votes
1answer
304 views

When do blow-up and quotient commute?

Let a finite group $G\subset SL(n,\mathbb{C})$ act on $\mathbb{C}^{n}$ in a natural way. Assume there is a crepant resolution of $f:X\rightarrow \mathbb{C}^{n}/G$. When is it possible to write $X$ as $...
3
votes
2answers
683 views

Crepant resolution of isolated fourfold singularity

I stumbled upon this isolated singularity of a Calabi-Yau fourfold: \begin{equation} x_1x_2+x_3x_4+x_5^2=0 \end{equation} as a hypersurface in $\mathbb{C}^5$. Clearly, I can resolve this by a simple ...
4
votes
2answers
293 views

affinization of T^*CP^n

Is there an elementary description of the affinization of the algebraic cotangent bundle of $CP^n$? I know that it can be described as some sort coadjoint orbits, but I am interested in a translation ...
1
vote
0answers
123 views

compatible resolutions of singularities

Let $U$ and $V$ be complex vector spaces with an action of a finite group $G$. Denote by $P_{G,d}$ the space of $G$-equivariant polynomial maps $U\longrightarrow V$ with degree less than or equal to $...
5
votes
1answer
476 views

Hodge numbers of a Calabi-Yau 3-fold via deformation theory

In their paper "Calabi-Yau 3-folds and Moduli of Abelian Surfaces I", Gross and Popescu calculate the hodge numbers of smooth CY 3-folds obtained in the following way (see Remark 4.11 of their paper): ...
2
votes
2answers
281 views

Line bundles and rational singularities

Hi, I have some problem to understand the proof of lemma 3.2 of this article: http://www.ams.org/journals/jams/2001-14-03/S0894-0347-01-00368-X/. The lemma states the following: Let $X$ be a variety ...
5
votes
2answers
390 views

Crepant resolutions of ODP's on a 3-fold

It seems to be a well-known fact that if we have a 3-fold $X$ with only ODP singularities (ordinary double point) and a smooth Weil divisor $E$ passing through them, then by blowing-up $X$ along $E$ ...
3
votes
1answer
313 views

$A_{\infty}$ singularity

What kind of singularity is commonly meant by $A_{\infty}$?
5
votes
2answers
716 views

Do fixed point sets in equivariant crepant resolutions have the same cohomology? How about for the specific case of Nakajima quiver varieties?

A crepant resolution $f:Y\to X$ is a resolution of singularities with $f^*(K_X)=K_Y$. Crepant resolutions do not always exist, and when they exist they may not be unique. However, different crepant ...
4
votes
0answers
238 views

Embedding of a smooth variety into a complete smooth variety.

Consider the following fact from algebraic geometry: Any (complex) smooth algebraic variety can be embedded into a complete smooth variety as a locally closed set. I know how to prove this fact ...
2
votes
2answers
180 views

commuting the resolution of 1-dim singular locus and 0-dim singularities in a non isolated singularity of a surface

Let $X$ be a surface with a non isolated singularity $C = Sing(X)$ such that the curve $C$ has singularities itself. We can solve $Sing(X)$ by blowing up close points and by normalizing. Indeed, we ...
1
vote
1answer
314 views

'Reference' request: Program to work with cyclic quotient singularities.

I'm looking for program code to deal with cyclic quotient singularities on normal surfaces. In particular, at the moment I need that given a singularity $p$ like $p=\frac{1}{n}(1,a)$ the code computes:...
8
votes
2answers
540 views

Resolution of singularities for flat families.

Is there a resolution of singularities for flat families? More precisely, if $X \rightarrow \mathbb{A} ^n$ is a flat map, does there exist a map $Y \rightarrow X$ such that, for every $p \in \mathbb{...
4
votes
1answer
306 views

Property of singularity

Let $X$ be an algebraic variety, $S \subset X$ its singular locus, and $x \in S$. Say that $x$ is good if for any neighborhood $U$ of $x$, any top differential form $\omega$ on $U \setminus S$ and ...
4
votes
2answers
560 views

Vanishing associated to a resolution of singularities

Let $\pi: V\to W$ be a resolution of singularities, let $E \subset V$ be the exceptional divisor, and let $F$ be a coherent sheaf such that $R^i\pi_*F=0$ for $i>0$. Can we conclude that $R^i\...
12
votes
0answers
365 views

When are the fibers of a resolution of singularities reduced?

I apologize if this is too much of a fishing expedition, but I've had bad luck searching for any literature on this subject, and I was hoping someone could tell me if it's too easy to be worth ...