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### 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 ...
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### How would you call a variety that is locally a complete intersection up to defect c?

Let $X$ be an equidimensional variety of dimension $n$ over a field that can be covered by open subvarieties of certain intersections of $N-n$ hypersurfaces in $P^N$ (for a large enough $N$; we ...
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### Number of singular fibers in families of hypersurfaces

Consider the projection map $$\pi: X = V(t_0 f + t_1 gh) \to \mathbf P^1,$$ where $[t_0: t_1]$ are the homogeneous coordinates on $\mathbf P^1$, $f=f(x_0, \dots, x_n)$ is a homogeneous polynomial of ...
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### When can one find holomorphic sections vanishing at a point to a certain order?

Let $X$ be a compact complex manifold (say of dimension $2$) and $L \rightarrow X$ a holomorphic line bundle. Consider the following statements: Statement $A_0$: Given any point $p\in X$, there ...
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### A condition on isolated singularity

Suppose $F: {\mathbb C}^N \to {\mathbb C}$ defines a singularity at the origin (for simplicity one can assume that $F$ is a quasi-homogeneous polynomial). Suppose it is nondegenerate, i.e., $dF(z) = 0$...
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### Real structure in the mixed Hodge structure associated to an isolated singularity

We know that a mixed Hodge structure on a complex vector space $H$ with an integral lattice $H_{\mathbb Z}$ consists of the weight filtration and the Hodge filtration. For an isolated hypersurface ...
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### Obtaining non-normal varieties by pushout

In his answer to this MO question, Karl Schwede claimed that every non-normal variety can be obtained by an appropriate pushout diagram, as sketched in that answer. This would give substance to the ...
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### How can one determine if a singularity is simple?

Let $f(z_1,z_2,\dots ,z_n)$ be an analytic function in $\mathbb{C}[[z_1,z_2,\dots ,z_n]]$ whose leading term defines an isolated singularity at the origin. If we have the following types of ...
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### Discriminant of a polynomial in two variables

I want to compute the discriminant of the following polynomial $$F(X,Y)=X^mY^n+\sum_{i=0}^{m-1}\sum_{j=0}^{n-1}c_{ij}X^iY^j.$$ Here the discriminate means the equation $D(c_{i,j})$ in the variables ...
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### Singularities induced by the toric ambient spaces

Let $\Delta \subset \mathbb{R}^4$ be a (reflexive) polytope and $X$ be the hypersurfacedefined by a generic section of the any-canonical bundle of the toric variety $\mathbb{P}_{\Delta}$. Are there ...
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Let X a normal integral scheme over a base field scheme, assumedd to be singular and an integer $n$ Let $\mathcal{O}=k[[t]]$, we consider the arc space $X(\mathcal{O})$ which is a $k$- pro-scheme and $... 0answers 190 views ###$n$-Fold Framed Functions Suppose that$M$is a manifold. One can consider a suitably constructed space of generalized framed Morse functions on$M$, let's call it$\mathrm{Fun}^\mathrm{fr}(M)$. This space is known to be ... 1answer 490 views ### Higher Cerf Theory Morse functions on a manifold$M$are defined as smooth maps$f:M \rightarrow \mathbb{R}$, such that at the critical points we can find local coordinates so that $$f(x_1,\dots,x_n)=-x_1^2-x_2^2-\dots-... 3answers 410 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 ... 0answers 356 views ### Singularities arising from the Minimal Model Program (an algebraic point of view) I will start the story by the end: Is there some characterization of (some of) the singularities arising from the Minimal Model Program (canonical, terminal, log-...) in terms of commutative algebra ?... 2answers 560 views ### Singular points of algebraic varieties and parametrization by Puiseux series Let V\subset \mathbb{R}^n (or \mathbb{C}^n if that makes anything easier) be an algebraic variety and p\in V a possibly singular point. Let U\subset V be a sufficiently small neighborhood of ... 0answers 74 views ### on reductive monoids which are gorenstein Let M a reductive monoid, i.e. a integral normal affine scheme, which is a monoid whose group of units is a connected reductive group. By Rittatore http://www.cmat.edu.uy/cmat/docentes/alvaro/... 1answer 224 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 0... 0answers 73 views ### Complexity of mappings (forms) in R. Thom's “Structural stability and morphogenesis” In his "Structural stability and morphogenesis", R. Thom (especially in the chapter about dynamics of forms) among other things speculates about a notion of complexity of a "form" (mapping between ... 0answers 251 views ### generalization Abhyankar's lemma This question is related to a question I already asked on MO (smooth quotient out of a singular variety?), but I realized later that the hypotheses where not precise enough in my former question. Let ... 1answer 352 views ### smooth quotient out of a singular variety? If X is a smooth quasi-projective variety over \mathbb{C} and G is a finite group acting faithfully on X, then the Shepard-Todd theorem gives us some criterion for X/G to be smooth. My ... 1answer 297 views ### Picard group generated by effective divisors: counterexample? Let X be an integral variety defined over an algebraically closed field k of characteristic 0 with finitely generated Picard group Pic(X) and such that k[X]^\times=k^\times (i.e. the only ... 1answer 323 views ### A geometric characterization of smooth points of a complex algebraic variety Let X^m\subset \mathbb{C}^n be an irreducible m-dimensional complex algebraic subvariety. Let \mathbb{C}^n be equipped with the standard Hermitian metric. Fix an arbitrary point p\in X. Let ... 1answer 169 views ### Castelnuovo's rationality criterion on singular surfaces? Let S be a projective surface over an algebraically closed field. Suppose that q(S)=h^1(\mathcal O_S)=0 and P_2(S)=h^0(\mathcal O_S(2K_S))=0. If S is smooth, Castelnuovo's rationality ... 0answers 263 views ### A strong form of implicit function theorem (what happens when the derivative is degenerate?) (this can be considered as some ad) Consider the system of equations F(x,y)=0. (Here x, y are multi-variables. The equations are over a local ring. e.g. polynomial/analytic/formal/C^\infty ... 2answers 408 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 -\... 1answer 101 views ### Complement of bifurcation variety I am reading a seminal paper of Arnold "Normal forms of functions near degenerate critical points, the Weyl group of A_k, D_k, E_k and lagrangian singularities". Let f\colon \mathbb{C}^n\to \... 0answers 128 views ### Birational contraction to a \mathbb{Q}-Gorenstein Variety Given a birational contraction morphism X\rightarrow Y of complex normal algebraic varieties. If Y is a smooth variety, what kind of singularities can appear on X? I would be grateful of any ... 1answer 157 views ### Intrinsically proving a singularity is rational In general, how to prove a variety has rational singularities intrinsically? i.e., don't use the Artin's criterion concerning the exceptional locus. And what kinds of varieties have only rational ... 1answer 403 views ### The singularity of the algebraic stack and the singularity of the coarse moduli space It is possible that an algebraic stack is smooth while the coarse moduli space is not smooth. I want to know what is relationship between the singularity of the algebraic stack and that of its coarse ... 1answer 275 views ### Simple example of isolated critical point with non-semisimple monodromy Consider a polynomial map f :\mathbb{C}^{n+1} \rightarrow \mathbb{C} with f(0)=0 (no constant term) and with isolated critical point at 0 \in \mathbb{C}^{n+1}. We can choose a disc D of some ... 1answer 510 views ### Topological degree and polynomial degree Let F:\mathbb{C}^n\to \mathbb{C}^n be a homeomorphism homogeneous of degree 1 (i.e., F(tx)=tF(x), t>0) and g:\mathbb{C}^n\to \mathbb{C} a homogeneous polynomial of degree k. Let L (0\... 1answer 206 views ### Analytic vector fields on surfaces which have infinite number of singularities Let X be an analytic vector field on a compact oriantable surface S with volume form \omega. We denote the set of its singularities by Z(X). A local question Is there an analytic vector ... 0answers 349 views ### Presence of singular points in the trajectory of a double pendulum Watching the trajectory of a double pendulum, I caught myself wondering if it would be possible to prove that the path the second pendulum makes contains "cusps" or singular points. Upon investigating ... 0answers 70 views ### Three- dimensional astroid and catastrophe maps Is a three-dimensional astroid curve (2\cos^3u,2\sin^3u,3\cos2u) a part of a bifurcation set of some catastrophe map? 1answer 562 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. ... 1answer 112 views ### Properties of singularities that are preserved by categorical quotients Let G be a reductive group acting on an affine singular variety X, and let X/G be the categorical quotient. I know that if X has rational singularities, then so does X/G (http://link.... 0answers 392 views ### Blowdown and contraction I am sorry, my question is very naive. 2nd Edit: Let us suppose that V is a smooth complex projective variety, and Y\subset V is a smooth divisor and has an ample conormal line bundle. We would ... 1answer 250 views ### fearful of defining equivalent germs for non isolated singularities Two power series G(x_1, \ldots, x_n) and F(x_1, \ldots, x_n) are equivalent over \mathbb{C} if there is an automorphism of the ring \mathbb{C}[[x_1, \ldots, x_n]] given by x_1 \to \phi(x_1, \... 1answer 73 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 ... 0answers 157 views ### geometric irregularities in pde's The following question is intended for a person more acquainted with the works of Yves Laurent. see: http://archive.numdam.org/article/ASENS_1987_4_20_3_391_0.pdf (French) http://link.springer.com/... 2answers 386 views ### Cohomology of the tangent sheaf of \mathbb{P}(1,2,3) Using the exact sequence$$0\mapsto\mathcal{O}_{\mathbb{P}^{2}}\rightarrow\mathcal{O}_{\mathbb{P}^{2}}(1)^{\oplus 3}\rightarrow T_{\mathbb{P}^{2}}\mapsto 0$$it is easy to compute H^{1}(\mathbb{P}^... 1answer 302 views ### For what varieties do we have results on the category of singularities? Let X be a singular variety. Define the (triangulated) category of singularities (as in Orlov's paper) as the Verdier quotient of the derived category of coherent sheaves on X modulo the full ... 1answer 227 views ### Deformations of quotient singularities Let Y be an affine scheme over a field of characteristic zero. Suppose we have a group G acting on Y and that the subset of Y of points with non-trivial stabilizer is in codimension greater or ... 1answer 456 views ### Relation between Milnor ring and middle dimensional homology of hypersurface I have suspected that the following is well-known: If P is a homogeneous polynomial of degree d in n variables (for example, Fermat quintic x_1^5 + \cdots + x_5^5). The Milnor ring is {\... 0answers 87 views ### Lagrangean equations for the generating function of quadrangulations Let M(z) be the generating function of edge-rooted connected quadrangulations, with z marking the number of edges. I derived the following Lagrangean equations for M(z):$$M(z) = \psi(L(z)),~\... 1answer 440 views ### Why can you deform singularities in two dimensions but not in higher dimensions? I've been trying to read this paper to understand deformations of surface quotient singularities. I'm particularly interested in when one can deform certain cyclic quotient singularities into other ... 2answers 343 views ### Whitney stratification and affine grassmanian Let$G$a simply connected group over$\mathbb{C}$and$Gr:=G(\mathbb{C}((t)))/G(\mathbb{C}[[t]])$the affine grassmannian. By Cartan decomposition we have a partition of stratas indexed by$\lambda\...
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. ...
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 ...