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12
votes
0answers
423 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)^{\...
12
votes
0answers
362 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 ...
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 ...
3
votes
0answers
119 views

Resolving quotient singularities without the quotient

This has been asked on MSE here, but has not had much traffic so I will ask a similar question here as many people here enjoy topics around resolving singularities. Let $X$ be a complex threefold ...
3
votes
0answers
131 views

Singularities in mixed characteristic

Let $R$ be a regular local ring in mixed characteristic. Moreover, I assume that $R$ is the local ring of a point on a smooth $\mathbb Z_p$-scheme and that $R/pR$ is regular. ($\mathbb Z_p$ is the ...
3
votes
0answers
155 views

big and small resolutions of singularities of a 4-fold

Suppose we have a projective 4fold hypersurface $X\subset P^n$ with ordinary singularities along a smooth curve $C$, and suppose that there exist a projective small resolution $s:Y\to X$. let us ...
3
votes
0answers
208 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 ...
3
votes
0answers
241 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
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 $N$ ...
2
votes
0answers
160 views

Ricci curvature in resolution of singularities

Let $X$ and $X'$ are Kahler variety and $f: (X',\omega')\to (X,\omega)$ be the resolution of singularities of $X$ then from $K_X=f^*K_X'+E$ how can we find the relation between $Ric(\omega)$ and $...
2
votes
0answers
83 views

Resolution of indeterminacies for a map to Grassmannian of planes

Let $X\subset \mathbb P_k^N$ be a $n$-dimensional smooth projective variety ($n\geq 2$) and $\phi_l:X^l\dashrightarrow Gr(l,N+1)$ ($l\leq n+1$) be the natural rational map which associates to a ...
2
votes
0answers
120 views

Does the invariant from resolution of singularities provide a Whitney stratification?

The topic of Whitney stratifications came up in a lecture, and the general procedure in the examples was to decompose the singular locus of the variety into the strata starting with the "worst" ones. ...
2
votes
0answers
139 views

A question on resolution of singularities

I am wondering if it could be possible in particular cases to resolve a singularity of dimension $n$ by blowing-up a locus of dimension smaller than $n$. For instance consider a cubic surface $S\...
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
0answers
340 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_{...
1
vote
0answers
88 views

Chern classes of a resolution of singularities

Let $j:X\subset \mathbb P_{\mathbb C}^n$ ($n\geq 3$) be a hypersurface, defined by a section of a very ample line bundle $\mathcal L$, with a ordinary double point $P$ as the only singularity and $\...
1
vote
0answers
261 views

Projective tangent cones, ordinary singularities and blow-ups

Let $X\subset\mathbb{P}^n$ be a projective variety and let $Y\subset X$ be the singular locus of $X$. Assume that $Y$ is smooth. I would like to know if the following are equivalent: $X$ has an ...
1
vote
0answers
139 views

Resolution of singularities of projective varieties

Let $X\subset\mathbb{P}^n$ be an irreducible variety, and let $Sing(X)$ be its singular locus. Let $Y$ be the blow-up of $\mathbb{P}^n$ along $Sing(X)$. Assume that we know that the strict transform ...
1
vote
0answers
260 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,...
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 $...
0
votes
0answers
134 views

A definition of arithmetic divisor with conic singularities?

I have a question related to the preprint "Heights and metrics with logarithmic singularities" by G. Freixas i Montplet. Let $X$ be an arithmetic variety with arithmetic divisor $D$ how can we ...
0
votes
0answers
218 views

Is dimension invariant under blow-ups?

Let $X'\rightarrow X$ be a blow-up of a finitely dimensional scheme $X$ in a center $D$. Under which assumptions one has $\dim X'=\dim X$? Do you know a proof or a reference for a proof? Do you know ...
0
votes
0answers
69 views

A simple question about a resolution of a conifers singularity

Let $X$ be a conifold defined by the equation $xy-zw=0$ in $\mathbb{C}^4$ and $\tilde{X}$ its crepant resolution, which is isomorphic to $\mathcal{O}_{\mathbb{P^1}}(-1)^{\oplus 2}$. Then there is a ...
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}=\...
0
votes
0answers
113 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 \...