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3
votes
0answers
145 views

Some examples where the plurigenera are nonconstant, when the fibres have worse singularities than canonical

Let start with a definition Invariance of plurigenera: Choose $m$ large enough so that $mK_F$ has a non-zero global section for some fibre $F$. For any fibre $F$, we have $K_F = K_{X/D}~_{|F}$. So ...
1
vote
1answer
150 views

General Reference for surface singularities

Is there any "standard" reference for (rational) singularities on algebraic surfaces? I'm aware of Artin's papers and the one of Brieskorn (Rationale Singularitäten komplexer Flächen), but they seem ...
1
vote
0answers
79 views

Why has $L$ finitely many singularities? [closed]

Bézout's theorem Bézout's theorem for curves states that, in general, two algebraic curves of degrees $m$ and $n$ intersect in m·n points and cannot meet in more than $m·n$ points unless they have a ...
4
votes
1answer
92 views

The volume around a singular isolated root when equalities are loosened

Suppose I have a system of polynomial equations in $n$ real variables $f_i(x_1,\ldots,x_n)=0$, $i=1,\ldots,m$, such that $0$ is an isolated solution. Now I replace each of the equations with a ...
2
votes
1answer
144 views

Kahler Ricci flow in Fano fibration

Let $f:X\to Y$ be a Fano fibration of Kahler manifolds $X, Y$. Then why the Kahler Ricci flow $$\frac{\partial \omega}{\partial t}=-Ric(\omega(t))$$ starting of $[\omega_0]=f^*(\omega_Y)+c_1(X)$ ...
11
votes
0answers
158 views

Canonical scheme structure on the singular locus of a variety

I first asked this question on Math StackExchange but no answers were given. Let $X$ be subvariety of affine space $\mathbb{A}_{k}^n$, where $k$ is a field, and suppose $X$ is given by equations ...
6
votes
1answer
543 views

Valuation of an ideal in a two-dimensional regular local ring

Let $f,g$ be two coprime elements in the ring $K[[x,y]]$, with $K$ a field. What is the smallest integer $n$ such that the inclusion of ideals $$(x^n)\subset (f,g)$$ holds in $K[[x,y]]$? Can we ...
2
votes
0answers
74 views

Singularities of algebraic curves, and torsion of the pull-back of the differential module by the normalisation

The problem in the following : given an algebraic curve $C$, it's well-known that a smooth projective model of $C$ can be construct as the set of discrete valuations $v$ on it's function field ...
2
votes
0answers
154 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 ...
0
votes
0answers
129 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 ...
4
votes
1answer
229 views

Complexifying a real-analytic singularity

This is probably a well-known issue, but I could not find a clear discussion in the literature, and I think others could find it useful. Consider a real-analytic function germ $f:(\mathbb R^2,0) ...
2
votes
1answer
241 views

Controlling singularities on log mmp

Suppose all my varieties are complex threefolds $X\rightarrow Y$ over some smooth base curve germ $Y$. We can assume the fibres are Del Pezzo surfaces with generic smooth fibre. If I do (relative) ...
6
votes
1answer
222 views

Are codimension one foliations of $\mathbb{R}^{n}-\{0\}$ with compact leaves, stable at origin?

Assume that we have a codimension one foliation of $\mathbb{R}^{n}-\{0\}$ with compact leaves. Is it true to say that the foliation is stable at origin:That is: for every neighborhood $V$ of ...
2
votes
1answer
344 views

Reference request: English translation of Brieskorn 1970 paper

Is there any english (or french) translation of the following paper by Brieskorn (1970)? Brieskorn, E., "Die Monodromie der Isolierten Singularitäten von Hyperflächen", Manuscripta Mathematica 2 ...
3
votes
1answer
178 views

Rationality of higher dimensional du Val singularities

I am interested in the isolated singularity defined over $\mathbb{C}$ by $$ x_1^2+\cdots + x_n^2+x_{n+1}^k=0, $$ where $n>2$ and $k>2$. I would like to know whether this singularity is ...
3
votes
0answers
82 views

Is there a correspondence between counting curves in P^2 blown up at a point and counting curves in P^2?

Let $X$ be $\mathbb{P}^2$ blownup at one point and $\beta := d L -2E \in H_2(X, \mathbb{Z})$, where $L$ and $E$ denote the class of a line and the exceptional divisor respectively. Let ...
10
votes
2answers
524 views

Analytical formula for topological degree

At the first page of the following article http://arxiv.org/pdf/1004.1018v1.pdf [edit: the formula on the arXiv differs from the formula in the published paper, and the formula displayed below is the ...
5
votes
0answers
108 views

Gysin exact sequence for a singular subvariety

Let $k$ be an algebraically closed field (I'm interested in a characteristic $p>0$ specific example) and let $X$ be a (smooth if needed) algebraic variety. Let $Y \subset X$ be a (possibly) ...
1
vote
0answers
133 views

reference for weighted blow-up

Let $(0\in X)$ be a germ of a normal 3-fold with a singular point $0$ (over $\mathbb{C}$). We think of $X$ as a small neighborhood of $0$ (for studying singularity). If we can think $X$ as a ...
3
votes
1answer
288 views

Stratification of complex algebraic varieties

Let $V$ be a complex quasi-projective variety, we know from H. Whitney's and B Teissier works on stratifications of algebraic varieties that $V$ has an intrinsic stratification $$X_0\subset ...
2
votes
0answers
86 views

Is it obvious that the defining conditions to obtain a particular singularity are well-defined on the quotient space?

Let $~f:\mathbb{C}^2 \rightarrow \mathbb{C}$ be a holomorphic function vanishing at the origin, with the following properties: $$ f_{00}, ~f_{10}, ~f_{01}, ~f_{20}, ~f_{11} =0,~~f_{20} \neq 0 ...
1
vote
0answers
69 views

Is a variety a local complete intersection if it is locally a complement of to a smooth $N$-dimensional affine of $N-m$ affine subvarieties?

If an equidimensional variety $V$ of dimension $m$ is locally a set-theoretic complete intersection (i.e., it can be covered by open subvarieties of certain intersections of $N-n$ hypersurfaces in ...
2
votes
0answers
66 views

quasi-ordinary singularities on a versal deformation?

Let $V$ be a variety over $\mathbb{C}$ and suppose $O$ is a singular point of $V$. Are there conditions on $(V,O)$ such that a versal deformation $W$ of $(V,O)$ has only quasi-ordinary singularities. ...
9
votes
2answers
427 views

Which weighted projective spaces (and their finite quotients) are local complete intersections?

Let $G$ be a finite subgroup of $\textrm{Gl}_{n+1}(k)$ (where $k$ is an algebraically closed field). My question is: do there exist examples of $G$ such that the corresponding quotient $P$ of ...
2
votes
0answers
393 views

Weil Petersson metric on moduli space of Calabi Yau manifolds

Let $f:(X,D)\to Y$ be a holomorphic fibre space where $D$ is divisor with conic singularities and let fibres $(X_s,D_s)$ are log Calabi-Yau pair .i.e $K_X+D$ is nummerically trivial, then we have ...
3
votes
1answer
171 views

Jaffe's exact sequence

Let $X$ be a normal projective rational surface over $\mathbb{C}$ with finitely generated divisor class group $\text{Cl}(X)$. Consider the exact sequence $$0 \rightarrow \text{Pic}(X) \rightarrow ...
6
votes
0answers
89 views

Toric Degenerations and Nearby Cycles

Suppose that $f: X \to \mathbb{A}^1$ is a toric degeneration in the sense of Nishinou-Siebert. In other words let X be a (possibly singular) toric variety equipped with a (not necessarily proper) ...
1
vote
0answers
77 views

If there exists an immersion, then does a neighbourhood of a singular rational curve contain a genuine cuspidal point?

Let $X$ be a compact complex surface and $u_1, u_2: \mathbb{P}^1 \longrightarrow X$ be rational curves that are not multiply covered that represents a class $\beta \in H_2(X, \mathbb{Z})$. Suppose ...
0
votes
0answers
66 views

Characterization of Singular locus

Let A be a complete regular local ring over a field k and B be a complete normal local ring over a field k. We assume that (Krull-dimension of A) > 1. We consider the ring homomorphism f: A ---> B, ...
6
votes
1answer
215 views

Infinitesimal deformations of a singular projective surface

Let $X$ be a normal projective surface with just two singular points $x_1,x_2\in X$, where $X$ has rational quotient singularities. Assume that both the singularities in $x_1$ and in $x_2$ admit a ...
5
votes
0answers
113 views

Framed singular knots

I've recently run across what one might (and I suspect people probably do) call framed singular knots, or maybe singular ribbon knots. Regardless of the name, what I mean is the following: Let $D$ be ...
0
votes
0answers
43 views

Singularities of the product of a $(\mathbb{C}^*)$-surface with $\mathbb{C}$

Recall that any normal $\mathbb{C}^*$-surface is Cohen-Macaulay and there exists normal $\mathbb{C}^*$-surfaces whose singularities are not rational. Does anyone know an example of a normal ...
1
vote
0answers
90 views

Is there any explicit result on the triangulated category of singularities of a curve?

This question is related to this MO question. Let $X$ be a projective curve over a field $\mathbb{C}$. We have the bounded derived category of coherent sheaves $D^b_{coh}(X)$ and the derived category ...
7
votes
0answers
170 views

Singularities of an analytic function over a non-archimedean field

What do we know about the types of singularities that a convergent power series over a non-archimedean field can have? More specifically: i) What types of essential singularities can occur? ii) Are ...
1
vote
2answers
204 views

Cohen-Macaulayness of the direct image of the canonical sheaf

Let $Y$ be a normal projective variety and let $f:X\to Y$ be a desingularization. Define $\mathcal K_X=f_*\omega_X$, the Grauert--Riemenschneider canonical sheaf of $X$. It is independent of the ...
4
votes
1answer
746 views

the blowing up of a plane curve playing me tricks.

Sorry for the easy question but this is driving me crazy. Consider the blowing up of the curve $(y^2-x^3)^2+y^5$ at the origin. On the first blowing up, on the chart that intersects the exceptional ...
6
votes
4answers
1k views

minimal resolution of singularities

What is the minimal resolution of singularities of the surface $S^2(X^3+Y^3+Z^3)-3(S^2+T^2)XYZ=0$ which is a subset of $\mathbb{P}^1\times\mathbb{P}^2$ Please note that in this equation ...
4
votes
0answers
120 views

$\mathbb{Q}$-factoriality of singularities

I would like to understand if a certain variety is $\mathbb{Q}$-factorial (i.e., if every Weil divisor $D$ has a multiple $mD$ that is Cartier). This property can be deduced by a local picture around ...
3
votes
1answer
197 views

Analytically but not algebraically smoothable singularity

Are there examples of algebraic singularities which may be smoothed analytically but not algebraically? It certainly seems possible, but if not, why? Are there conditions under which this becomes ...
0
votes
2answers
138 views

Hochster-Roberts Theorem reciprocal

Given a Cohen-Macaulay ring $R$ over a field of characteristic zero and $G$ a reductive algebraic group acting on $R$, then the ring of ivanriants $R^G$ is also Cohen-Macaulay. This is known as ...
8
votes
2answers
338 views

Implicit Function Theorem on Singular Varieties

Let $X$ and $Y$ be two complex reduced affine algebraic or analytic varieties, possibly singular. Take a regular proper function $$f\colon X \to Y $$ and assume that it is bijective at the level of ...
8
votes
2answers
1k views

“Arithmetic genus” of a plane curve singularity.

I believe that the following questions are very basic, but I don't know how to get a reference. Consider a curve in the plane $C\in \mathbb C^2$ with a singularity at $0$ and suppose it is ...
2
votes
1answer
198 views

Is there a formula for the number of rational cuspidal curves in surfaces other than P^2?

Let $M$ be a two dimensional compact complex manifold and $A \in H_2(M, \mathbb{Z})$ a fixed homology class. Define a rational curve in $M$ to be $\textit{1-cuspidal}$ if the singularities of the ...
3
votes
0answers
150 views

Implicit function theorem for singularities

I am looking for an implicit function theorem which holds also on singular spaces, at least if the singularities are "mild". For example, let $0 = z^2 - x y + z w + w^2 + \epsilon w$ define a ...
3
votes
0answers
227 views

Hypersurface with singularities

I heard once about one open problem. That was about existing a hypersurface of a small degree (5? or 6?) passing through some number (5? 6?) of 3-fold points and 2-fold lines (3 lines?). It was said ...
1
vote
0answers
78 views

Global topological equivalence of Morse functions

Two Morse functions $f$ and $g$ are called topologicaly equivalent if there are diffeomorphism $h$ of the source and orientational-preserving diffeomorphism $k$ of the target such that $f=k\circ ...
1
vote
1answer
118 views

About 3-fold log canonical singularity

As far as I know, log canonical surface singularities were classified. How about higher dimensional case? I especially want to know whether given 3-fold singularity is log canonical or not. Let $f$ ...
0
votes
0answers
173 views

projective map from $\overline{\mathcal{M}}_{0,n}$

Suppose I have a morphism $f:\overline{\mathcal{M}}_{0,n} \to \mathbb{P}^N$ birational onto its image, and I know exactly what $F$-curves are contracted (or "dually", what divisors are contracted). ...
6
votes
0answers
100 views

When is a smooth function locally equivalent to a truncation of its Taylor series?

Let $U \subset \mathbb{R}^n$ be an open set and let $f:U \rightarrow \mathbb{R}$ be a smooth proper function. For $p \in \mathbb{R}^n$ let $$T_p(x_1,\ldots,x_n) = \sum_{i_1,\ldots,i_n=0}^{\infty} ...
3
votes
3answers
749 views

Poll about your proof of resolution of singularities and a request for advice

The questions first: What is the proof of resolution of singularities that you know? Why am I asking?: There are a number of proofs of resolution of singularities of varieties over a field of ...