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3
votes
4answers
361 views

How singular can the Stein factorization of a proper map between smooth varieties be?

A little bit of motivation (the question starts below the line): I am studying a proper, generically finite map of varieties $X \to Y$, with $X$ and $Y$ smooth. Since the map is proper, we can use the ...
0
votes
0answers
49 views

Are the fibers of this morphism geometrically regular?

Let $A\rightarrow B$ be a local morphism of complete noetherian rings making $B$ a formally smooth $A$-algebra. Does the induced morphism $\textrm{Spec}(B)\to\textrm{Spec}(A)$ have geometrically ...
8
votes
1answer
290 views

Blowing-up a point in the singular locus

Let $X\subset\mathbb{P}^n$ be a variety singular along a smooth subvariety $Z\subset X$ of positive dimension. Let us assume that $X$ has ordinary singularities along $Z$. Now, let $\pi:Y\rightarrow ...
0
votes
0answers
64 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
1answer
111 views

Intersection Matrix of a resolution

Probably this is a very easy question. Let $f:X\rightarrow S$ be a resolution of a projective surface such that $$K_X = f^{*}K_S+\sum_ia_iE_i$$ with $a_i>0$. By Grauert-Mumford theorem the ...
2
votes
0answers
90 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. ...
6
votes
4answers
670 views

Cone over the Veronese surface

Let $V\subset\mathbb{P}^5$ be the Veronese surface and let $X\subset\mathbb{P}^6$ be the cone over it. Since $X$ is $\mathbb{Q}$-factorial there are two integers $a,b$ such that $aK_X = ...
2
votes
1answer
143 views

Log Canonical pairs

Let $X$ be a normal scheme ad $D = \sum_id_iD_i\subset X$ be a $\mathbb{Q}$-divisor such thay $K_X+D$ is $\mathbb{Q}$-Cartier. Let $f:Y\rightarrow X$ be a log resolution of the pair $(X,D)$ and let us ...
1
vote
0answers
118 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
115 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 ...
2
votes
1answer
109 views

Resolving nodes of a quintic CY 3-fold

Let's consider the following quintic 3-fold $X$: \begin{equation} \{(x_i) \in \mathbb{P}^4 \ | \ x_1f(x)-x_2g(x)=0\} \end{equation} for generic homogeneous polynomials $f(x),g(x)$ of degree four. It ...
3
votes
0answers
114 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 ...
6
votes
3answers
274 views

Contracting a rational curve in a Calabi-Yau threefold

Let $X$ be a Calabi-Yau threefold and $C \subset X$ be a rational curve with $N_{C/X}\cong \mathcal{O}\oplus \mathcal{O}(-2)$. Can one contract the curve $C$? Assuming the answer is yes, what kind of ...
3
votes
1answer
127 views

A question on klt pairs

Let $D$ be a $\mathbb{Q}$-divisor in a smooth variety $X$. In Lazarsfeld book "Positivity in Algebraic Geometry 2" I found Proposition 9.5.13 saying that if for any $x\in D$ we have $mult_xD < 1$ ...
3
votes
1answer
124 views

Embedded resolution of curves on smooth varieties

As far as I understand, embedded resolution of singularities means the following: given a variety $X$ over an algebraically closed field, and a closed subvariety $Y$, there exists a birational map ...
4
votes
2answers
129 views

Invariant planes of a nilpotent matrix with two Jordan blocks of size two

Describe all the invariant 2-dimensional subspaces of $\mathbb{C}^4$ (or $\mathbb{R}^4$) of the nilpotent map $$ N = \begin{pmatrix} 0 & 1 & & \\ 0 & 0 & & \\ & & 0 ...
2
votes
0answers
126 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 ...
2
votes
1answer
169 views

Singularities of secant varieties of rational normal curves

Let $C\subset\mathbb{P}^n$ be a rational normal curve of degree $n$, and let $Sec_k(C)\subset\mathbb{P}^n$ be its $k$-th secant variety. By Theorem 1.1 in this paper: ...
1
vote
2answers
203 views

Small birational maps and singularities of the pair

Let $f:X\dashrightarrow Y$ be a small birational map, where $X,Y$ are normal $\mathbb{Q}$-factorial varieties. Let $\Delta_X\subset X$ be an effective $\mathbb{Q}$-divisor such that the pair ...
2
votes
1answer
177 views

Some Kind of Resolution of Singularites

Let $X \subseteq \mathbb{P}^n$ be a projective variety. I would like to have a morphism $f: \tilde{X}\to X \subseteq \mathbb{P}^n$ where $f$ is finite and birational, $f^* ...
2
votes
2answers
331 views

Finite Quotients and Resolutions of Singularities

So, I feel like I'm missing something obvious, but I have the following situation: Let $X\to Y$ be a finite group quotient of schemes (in fact, varieties) by the finite group $G$. Let $\tilde{Y}\to ...
0
votes
1answer
432 views

Simple normal crossing divisors

I found the following definition. A Weil divisor $D = \sum_{i}D_i \subset X$ on a smooth variety $X$ is simple normal crossing if for every point $p \in X$ a local equation of $D$ is ...
0
votes
1answer
152 views

Singular irreducible quadrics

Let $Q\subset\mathbb{P}^n$ be the quadric hypersurface defined by $$x_0^2+x_1^2+...+x_k^2 =0.$$ If $2\leq k\leq n-1$ then $Q$ is irreducible and $Sing(Q)$ is a linear space of dimension $n-k-1$. If ...
1
vote
1answer
86 views

Pushforward of $K_X+D$ on the non-snc locus

Let $f:Y\rightarrow X$ be a birational morphism of smooth projective varieties, $F$ an effective divisor on $X$, $D=f^{-1}F_{\mathrm{red}}+\mathrm{Ex}(f)$, $B$ a smooth subvariety of $Y$ contained in ...
4
votes
1answer
183 views

Simultaneous resolution of singularities in special cases of flat families of projective varieties

Let $\pi:\mathcal{X} \to B$ be a flat family of projective varieties. Assume that $B$ is irreducible. Suppose that $\mathcal{X}$ is smooth except for a closed subscheme, say $Y$ which is isomorphic to ...
4
votes
2answers
281 views

Varieties with big anti-canonical divisor

I recently heard about the following problem: Let $X$ be a projective variety with klt singularities and such that $-K_X$ is big. Is $X$ a Mori Dream Space ? Now, $-K_X$ big if and only if $-K_X ...
7
votes
1answer
215 views

How can one show that orbit closures in representations of a linear quiver don't have small resolutions?

Let $1\to \cdots\to n$ be a linear quiver of length $n$. Let $\mathbf{d}=(d_1,\dots,d_n)$ be a dimension vector. It's well known (for example, by Gabriel's theorem, but also by basic linear algebra) ...
1
vote
1answer
120 views

Small resolutions are automatically crepant?

Page 17 of the following survey: http://arxiv.org/abs/1103.5380 makes the claim that small resolutions, meaning resolutions such that the exceptional set is in codimension at least two, are ...
9
votes
2answers
487 views

Resolution of unpleasant singularity

I've been working on some varieties defined by taking some quotients of group actions, and the resolutions have been straightforward... until now. E.g., consider $\mathbb{C}^2$ with the action ...
2
votes
1answer
217 views

Surfaces singular along a curve

Let $C\subset\mathbb{P}^3$ be a smooth curve a degree $d$ and genus $g$. Let $\mathcal{S}$ be the system of surfaces of degree $k$ in $\mathbb{P}^3$ containing $C$ with multiplicity $\beta$. What is ...
6
votes
1answer
527 views

Bertini's Theorem

Let $p_1,...,p_n\in\mathbb{P}^{N}$ be general points. Consider the linear system $|L|$ of hypersurfaces of degree $d$ in $\mathbb{P}^{N}$ with prescribed multiplicities $m_1,...,m_n$ at $p_1,...,p_n$. ...
0
votes
1answer
214 views

Kawamata-Log-Terminal pairs

Let $p_1,...,p_n\in\mathbb{P}^3$ be general points, and let $\Delta\subset\mathbb{P}^3$ be a general surface of degree $d$ with points of multiplicity $m_i$ at $p_i$ for $i = 1,...,n$. Consider the ...
1
vote
1answer
69 views

Determining the desingularization from the complete local ring

Suppose I have a curve $C$ over a field $k$ and that $p$ is a singular point of $C$. Let $f : X \to C$ be the desingularization of $C$ at $p$. Then for each $s \in f^{-1}(p)$ we have a map of local ...
4
votes
1answer
176 views

Blowing up rational singularities

Let $X$ be a projective surface embedded into $\mathbb{P}^n_{\mathbb{C}}$ having at most rational singularities. Let $\tilde{X} \to X$ be the minimal resolution of $X$. Is it possible to embed ...
3
votes
0answers
133 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
3answers
188 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
101 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 ...
3
votes
0answers
190 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
464 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
221 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
294 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
121 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
286 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
235 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. ...
9
votes
2answers
594 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
163 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 ...
3
votes
0answers
264 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
100 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
92 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 ...
2
votes
1answer
190 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 $\pi_1(\widetilde{X/G})$ ...