D. A. Stepanov
The dual complex associated to a resolution of singularities generalizes the notion of a resolution graph of a surface singularity to any dimension. We show that homotopy type of the dual complex is an invariant of an isolated singularity.
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We discuss the relation between transposition mirror symmetry of Berlund and Hübsch for bimodal singularities and polar duality of Batyrev for associated toric K3 hypersurfaces. We also show that homological mirror symmetry for singularities implies the geometric construction of Coxeter-Dynkin diagrams of bimodal singularities by Ebeling and Ploog.
This is a note on toroidalization, formulated as the problem of resolution of singularities of morphisms in the logarithmic category. It is submitted to the proceedings of the Barrett conference held at the University of Tennessee at Knoxville in April 2002.
A recent paper of Bresciani and Vistoli studies fields of moduli using residual gerbes. One of their applications is to twisted forms of quotient singularities. We discuss a direct approach to automorphisms and twisted forms, which also works in many wildly ramified cases.
This is a short note on the relation between the graded stable derived categories of 14 exceptional unimodal singularities and the derived category of K3 surfaces obtained as compactifications of the Milnor fibers. As a corollary, we obtain a basis of the numerical Grothendieck group similar to the one given by Ebeling and Ploog (arXiv:0809.2738).
In this paper we shall prove that the singular locus of a symplectic singularity has no codimension 3 irreducible components. As a corollary, a symplectic singularity is terminal if and only if its singular locus has codimension $\geq 4$. It is hoped that a symplectic singularity has much stronger properties.
For a weighted quasihomogeneous two dimensional hypersurface singularity, we define a smoothing with unipotent monodromy and an isolated graded normal singularity. We study the natural weighted blow up of both the smoothing and the surface. In particular, we describe our construction for the quasihomogeneous singularities of type I, the 14 unimodal exceptional singularities and we relate it to their stable replacement.
For an isolated hypersurface singularity which is neither simple nor simple elliptic, it is shown that there exists a distinguished basis of vanishing cycles which contains two basis elements with an arbitrary intersection number. This implies that the set of Coxeter-Dynkin diagrams of such a singularity is infinite, whereas it is finite for the simple and simple elliptic singularities. For the simple elliptic singularities, it is shown that the set of distinguished bases of vanishing cycles is also infinite. We also show that some of the hyperbolic unimodal singularities have Coxeter-Dynkin d…
We study the singularities of the projective dual variety.
We investigate unibranched singularities of dual varieties of even-dimensional smooth projective varieties in characteristic 2.
We collect some classical results about holomorphic 1-forms of a reduced complex curve singularity, in particular of a complete intersection, and use them to compare the Milnor number, the Tjurina number and the dimension of the torsion part of the 1-forms.
There are strange dual pairs of bimodal singularities that are not assigned an invertible projectivisation in EbelingPloog. We study families of $K3$ surfaces associated to such pairs.
In this revised version, the mistake of the author confusing the weak transform and strict transform, pointed out by E. Bierstone, is corrected. It gives a self-contained proof of (embedded) resolution of singularities over a field of characteristic zero following the inductive algorithm of Encinas and Villamayor. A chapter explaining the background motivation for the invariants used (based upon a lecture by Villamayor) and another containing some examples demonstrating the subtle points are added.
We study del Pezzo fibrations of degree 1 with terminal singularities. A connection between singularities on del Pezzo surfaces of degree 1 and Kodaira's classification of elliptic singular fibers will be studied in this paper. By this investigation we obtain several results on birational maps of del Pezzo fibrations of degree 1 with terminal singularities.
For a convenient and Newton non-degenerate singularity, the Milnor number is computed from the complement of its Newton diagram in the first quadrant, so-called Kouchnirenko's formula. In this paper, we consider tropical curves dual to subdivisions of this complement for a plane curve singularity and show the existence of a tropical curve satisfying a certain formula, which looks like a well-known formula for a real morsification due to A'Campo and Gusein-Zade.
This paper studies formality of the differential graded algebra $RHom(E,E)$, where $E$ is a semistable sheaf on a K3 surface. The main tool is Kaledin's theorem on formality in families. For a large class of sheaves $E$, this DG algebra is formal, therefore we have an explicit description of the singularity type of the moduli space of semistable sheaves at the point represented by $E$. This paper also explains why Kaledin's theorem fails to apply in the remaining case.
A note on simple $K3$ singularities and families of weighted $K3$ surfaces
本文探讨简单 $K3$ 奇点的 Seifert 形式与加权 $K3$ 曲面族的 Picard 格，并给出联系两者的数值公式。
Ebeling and Ploog \cite{EbelingPloog} studied a duality of bimodular singularities which is part of the Berglund--H$\ddot{\textnormal{u}}$bsch mirror symmetry. Mase and Ueda \cite{MU} showed that this duality leads to a polytope mirror symmetry of families of $K3$ surfaces. We discuss in this article how this symmetry extends to a symmetry between lattices.
We study the sheaf of locally square integrable holomorphic section of vector bundle with semi-positive curved singular Hermitian metric. We confirm the coherence when its induced determinant metric has analytic singularities.
In this article, we show that if $X$ is an excellent surface with rational singularities, the constant sheaf $\mathbb{Q}_{\ell}$ is a dualizing complex. In coefficient $\mathbb{Z}_{\ell}$, we also prove that the obstruction for $\mathbb{Z}_{\ell}$ to become a dualizing complex lying on the divisor class groups at singular points. As applications, we study the perverse sheaves and the weights of $\ell$-adic cohomology groups on such surfaces.
The dual complex of a singularity is defined, up-to homotopy, using resolutions of singularities. In many cases, for instance for isolated singularities, we identify and study a "minimal" representative of the homotopy class that is well defined up-to piecewise linear homeomorphism. This is derived from a more global result concerning dual complexes of dlt pairs. As an application, we also show that the dual complex of a log terminal singularity as well as the one of a simple normal crossing degeneration of a family of rationally connected manifolds are contractible.
Wild quotient surface singularities whose dual graphs are not star-shaped
本研究利用Kato对数结构与对数正则性理论解答Lorenzini的二维野商奇点相关问题，证明奇点消解例外曲线对偶图至少含一个节点，并构造出对偶图至少含两个节点的野商奇点实例。
This article is motivated by the original Casson invariant regarded as an integral lifting of the Rochlin invariant. We aim to defining an integral lifting of the Adams e-invariant of stably framed 3-manifolds, perhaps endowed with some additional structure. We succeed in doing so for manifolds which are links of normal complex Gorenstein smoothable singularities. These manifolds are naturally equipped with a canonical $\SU$-frame. To start we notice that the set of homotopy classes of $\SU$-frames on the stable tangent bundle of every closed oriented 3-manifold is canonically a $\mathbb Z$-to…
The dual complex can be associated to any resolution of singularities whose exceptional set is a divisor with simple normal crossings. It generalizes to higher dimensions the notion of the dual graph of a resolution of surface singularity. The homotopy type of the dual complex does not depend on the choice of a resolution and thus can be considered as an invariant of singularity. In this preprint we show that the dual complex is homotopy trivial for resolutions of 3-dimensional terminal singularities and for resolutions of Brieskorn singularities. We also review our earlier results on resoluti…
A note on complex plane curve singularities up to diffeomorphism and their rigidity
本文证明了在实光滑微分同胚下等价的平面曲线奇点必然复同构或共轭复同构，给出了基于泰勒级数的构造性证明，并将结论推广至非不可约超曲面情形。
A short note on Du Bois singularities
研究Du Bois奇点在基变换和纤维积下的行为，证明了在特征零的可嵌入簇中Du Bois奇点从任何域扩张下降，以及具有有理奇点的簇与具有Du Bois奇点的簇的乘积仍具有Du Bois奇点。
We prove homological mirror symmetry for Lefschetz fibrations obtained as disconnected sums of polynomials of types A or D. The proof is based on the behavior of the Fukaya category under the addition of a polynomial of type D.
It is well known that the Gauss map for a complex plane curve is birational, whereas the Gauss map in positive characteristic is not always birational. Let $q$ be a power of a prime integer. We study a certain plane curve of degree $q^2+q+1$ for which the Gauss map is inseparable with inseparable degree $q$. As a special case, we show a relation between the dual curve of the Fermat curve of degree $q^2+q+1$ and the Ballico-Hefez curve.
In this paper we collect some results on the obstruction spaces for rational surface singularities and minimally elliptic surface singularities. Based on the known results we calculate higher obstruction spaces for such surface singularities. The results imply that in general the higher obstruction spaces of deforming semi-log-canonical surfaces do not vanish. We apply the calculation result to show that there is no Li-Tian and Behrend-Fantechi style virtual fundamental class on such moduli space of semi-log-canonical surfaces.
It is well known that the exceptional set in a resolution of a rational surface singularity is a tree of rational curves. We generalize the combinatoric part of this statement to higher dimensions and show that the highest cohomologies of the dual complex associated to a resolution of an isolated rational singularity vanish. We also prove that the dual complex associated to a resolution of an isolated hypersurface singularity is simply connected. As a consequence, we show that the dual complex associated to a resolution of a 3-dimensional Gorenstein terminal singularity has the homotopy type o…
Given an algebroid plane curve $f=0$ over an algebraically closed field of characteristic $p\geq 0$ we consider the Milnor number $μ(f)$, the delta invariant $δ(f)$ and the number $r(f)$ of its irreducible components. Put $\bar μ(f)=2δ(f)-r(f)+1$. If $p=0$ then $\bar μ(f)=μ(f)$ (the Milnor formula). If $p>0$ then $μ(f)$ is not an invariant and $\bar μ(f)$ plays the role of $μ(f)$. Let $\mathcal N_f$ be the Newton polygon of $f$. We define the numbers $μ(\mathcal N_{f})$ and $r(\mathcal N_{f})$ which can be computed by explicit formulas. The aim of this note is to give a simple proof of the ine…
We study lattice polarizations of five exceptional pairs of families of K3 surfaces obtained via compactifications of strange dual pairs of bimodal singularities. We show that the polarizations induced by embedding these paired families in paired toric varieties obtained from polar dual reflexive polytopes cannot be mirror lattices and identify mirror sublattices.
We consider minimal resolutions of the singularities for weighted projective planes of type $\mathbb{P}(e^2, f^2, g^2)$, where $e, f, g$ satisfy the Markoff equation $ e^2 + f^2 + g^2 = 3efg$. We give a complete classification of such resolutions in terms of continued fractions similar to classical work of Frobenius. In particular, we investigate the behaviour of resolutions under mutations and describe a Cantor set emerging as limits of continued fractions.
Positivity, singularities, and boundedness
本文梳理双有理几何中正性、奇点与有界性之间的新近联系。
Teissier singularities
本文介绍了 Teissier 奇点，并指出其在正特征下的奇点消解中可发挥类似于特征零中拟常奇点的作用。
A total mass is the weighted count of continuous homomorphisms from the absolute Galois group of a local field to a finite group. In the preceding paper, the authors observed that in a particular example, two total masses coming from two different weightings are dual to each other. We discuss the problem how general such a duality holds and relate it to the existence of simultaneous resolution of singularities, using the wild McKay correspondence and the Poincaré duality for stringy invariants. We also exhibit several examples.
In this paper, we extract natural invariants of a singularity by using the Deligne weight filtration on the cohomology of an exceptional fibre of a resolution, and also on the intersection cohomology of the link. Our primary goal is to study and give natural bounds on the weights in terms of direct images of differential forms. These bounds can be made explicit for various standard classes such as rational, isolated normal Cohen-Macaulay and toroidal singularities, and lead to strong restrictions on the topology of these singularities. A secondary goal of this paper is to make the weight filtr…
P-resolutions of a cyclic quotient singularity are known to be in one-to-one correspondence with the components of the base space of its semi-universal deformation. Stevens and Christophersen have shown that P-resolutions are parametrized by so-called chains representing zero or, equivalently, certain subdivisions of polygons. I give here a purely combinatorial proof of the correspondence between subdivisions of polygons and P-resolutions.
This note explains a construction of a Poisson manifold whose symplectic foliation describes a deformation of a moduli space of meromorphic connections with unramified irregular singularities. In particular, this deformation of the moduli space corresponds to the unfolding of irregular singularities of the meromorphic connections. This is an announcement of some results in the forthcoming paper.
This note provides a connection between the Hodge spectrum and the Hodge ideals recently developed by Mustaţǎ--Popa in the case of weighted homogeneous isolated singularities.
In this note we introduce Cox rings of singularities and explicitly compute them in the case of du Val singularities $\mathbb{D}_n,\mathbb{E}_6,\mathbb{E}_7$ and $\mathbb{E}_8$.
These are the notes for my lecture ``Resolution of Sigularities in Charcteristic 0" given at the AMS Summer Institute at Seattle. It gives a self contained proof of the strong Hironaka resolution theorem.
The purpose of this note is to show that in characteristic two there exists a smooth Fano variety for which the square of the anti-canonical bundle violates Kodaira vanishing. This is used in turn to construct non-Cohen-Macaulay canonical singularities in characteristic two.
Cohomological Chow Groups of codimension one of varieties with isolated singularities
本文研究了具有孤立奇点的代数簇的上同调Chow群，通过不同条件下的计算揭示了其结构特性。
Previously, we introduced a duality transformation for Euler $G$--Frobenius algebras. Using this transformation, we prove that the simple $A,D,E$ singularities and Pham singularities of coprime powers are mirror self--dual where the mirror duality is implemented by orbifolding with respect to the symmetry group generated by the grading operator and dualizing. We furthermore calculate orbifolds and duals to other $G$--Frobenius algebras which relate different $G$--Frobenius algebras for singularities. In particular, using orbifolding and the duality transformation we provide a mirror pairs for …
This note shows that the orbifold Jacobian algebra associated to each invertible polynomial defining an exceptional unimodal singularity is isomorphic to the (usual) Jacobian algebra of the Berglund-Hübsch transform of an invertible polynomial defining the strange dual singularity in the sense of Arnold.
Notes on rational chain connectedness
本文通过最小模型程序扩展了有理链连通性定理，证明了复分析kawamata log终端奇点的解析性纤维具有有理链连通性。
The Bogomolov-Gieseker-Koseki inequality on surfaces with canonical singularities in arbitrary characteristic
本文将Bogomolov-Gieseker不等式从特征零光滑射影曲面推广到任意特征、具典范奇点的射影曲面，并推广其若干经典应用。
Defect of irreducible plane curves with simple singularities
本研究围绕Dimca提出的平面曲线亏量展开，推导了含结点、普通尖点与普通三重点的不可约平面曲线的亏量下界，证明高Arnold指数的约化简单奇异平面曲线不可能是自由曲线。
A singularity is said to be weakly-exceptional if it has a unique purely log terminal blow up. This is a natural generalization of the surface singularities of types $D_{n}$, $E_{6}$, $E_{7}$ and $E_{8}$. Since this idea was introduced, quotient singularities of this type have been classified in dimensions up to most $4$. This note extends that classification to dimension $5$.
This note gives two examples of surfaces with normal crossing singularities. In the first example the canonical ring is not finitely generated. In the second, the canonical line bundle is not ample but its pull back to the normalization is ample. The latter answers in the negative a problem left unresolved in [EGA,III.2.6.2] and raised again by Viehweg.
We introduce a lifting property for local cohomology, which leads to a unified treatment of the dualizing complex for flat morphisms with semi-log-canonical, Du Bois or F-pure fibers. As a consequence we obtain that, in all 3 cases, the cohomology sheaves of the relative dualizing complex are flat and commute with base change. We also derive several consequences for deformations of semi-log-canonical, Du Bois and F-pure singularities.
We study Serre duality in the singularity category of an isolated Gorenstein singularity and find an explicit formula for the duality pairing in terms of generalised fractions and residues. For hypersurfaces we recover the residue formula of the string theorists Kapustin and Li. These results are obtained from an explicit construction of complete injective resolutions of maximal Cohen-Macaulay modules.
For analytic map germs $f: (\mathbb{R}^n, 0)\to (\mathbb{R}, 0)$ having an isolated critical value in the origin with $\dim V(f)>0$ and satisfying the transversality property of D.B. Massey we show that for $c>0$ a large enough constant, and $k$ a large enough natural number the local Milnor-Lê fibers of $f$ and of the isolated singularity $g=f-c(\sum_{i=1}^n x_i^2)^k$ satisfies the following. There exists a homotopy equivalence between the negative Milnor-Lê fiber of $f$, to which a cobordism between its boundary and the link of $f$ has been adjoined, and the negative Milnor-Lê fiber of $g$.
For a given irreducible projective variety $X$, the closure of the set of all hyperplanes containing tangents to $X$ is the projectively dual variety $X^{\vee}$. We study the singular locus of projectively dual varieties of certain Segre-Plücker embeddings in series of papers. In this work we give a classification of the irreducible components of the singular locus of the dual Grassmannian. Basically, it admits two components: cusp type and node type which are degeneracies of a certain Hessian matrix, and the closure of the set of tangent planes having more than one critical point, respectivel…
These lecture notes provide a unified overview of most known canonical desingularization methods in characteristic zero. It starts with discussing the classical method, and then proceeds with the recently discovered ones: logarithmic desingularization of logarithmic schemes and morphisms, weighted methods: logarithmic and non-logarithmic ones, desingularization in wider contexts: quasi-excellent schemes and formal schemes, complex and non-archimedean spaces.
Let $(R,m,k)$ be an excellent local ring of equal characteristic. Let $j$ be a positive integer such that $H_m^i(R)$ has finite length for every $0\leq i <j$. We prove that if $R$ is $F$-injective in characteristic $p>0$ or Du Bois in characteristic $0$, then the truncated dualizing complex $τ_{>-j}ω_R^\bullet$ is quasi-isomorphic to a complex of $k$-vector spaces. As a consequence, $F$-injective or Du Bois singularities with isolated non-Cohen-Macaulay locus are Buchsbaum. Moreover, when $R$ has $F$-rational or rational singularities on the punctured spectrum, we obtain stronger results.
We first generalize classical Auslander-Reiten duality for isolated singularities to cover singularities with a one-dimensional singular locus. We then define the notion of CT modules for non-isolated singularities and we show that these are intimately related to noncommutative crepant resolutions (NCCRs). When R has isolated singularities, CT modules recover the classical notion of cluster tilting modules but in general the two concepts differ. Then, wanting to generalize the notion of NCCRs to cover partial resolutions of Spec R, in the main body of this paper we introduce a theory of modify…
Let $X$ be a smooth irreducible nondegenerate projective variety and let $X^*$ denote its dual variety. It is well known that $σ_2(X)^*$, the dual of the 2-secant variety of $X$, is a component of the singular locus of $X^*$. The locus of bitangent hyperplanes, i.e. hyperplanes tangent to at least two points of $X$, is a component of the sigular locus of $X^*$. In this paper we provide a sufficient condition for this component to be of maximal dimension and show how it can be used to determine which dual varieties of Grassmannians are normal. That last part may be compared to what has been don…
This short note is a supplement to the previous article with the same title. Here we treat a conical symplectic variety obtained as a finite covering of a (not necessarily normal) nilpotent orbit closure of a complex semisimple Lie algebra.
On holographic duals of certain isolated weighted Gorenstein cDV singularities
本文基于镜像对称性，证明了某些加权齐次戈林斯坦复合杜瓦奇点无crepant解，这对超对称电荷理论的膜背景有影响。
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