Q-Gorenstein deformations of non-normal surfaces

准戈尔登环研究II：准戈尔登性质的变形

A Study of quasi-Gorenstein rings II: Deformation of quasi-Gorenstein property

本文研究了准戈尔登环的变形问题，探讨了在不同假设下准戈尔登性质的保持情况，并提供了反例。

Gorenstein and normal properties of the subregular variety of a variety of special lattices over Witt vectors

We recall the projective variety parametrizing a family of special lattices over Witt vectors. It is normal and Gorenstein. In this article, we prove that there exists a particular set of subvarieties in it that are also normal and Gorenstein. The set contains the subregular variety.

A differential graded Lie algebra controlling the Poisson deformations of an affine Poisson variety

We construct a differential graded Lie algebra $\fg$ controlling the Poisson deformations of an affine Poisson variety. We analyse $\fg$ in the case of affine Gorenstein toric Poisson varieties. Moreover, explicit description of the second and third Hochschild cohomology groups is given for three-dimensional affine Gorenstein toric varieties.

The versal Deformation of an isolated toric Gorenstein Singularity

Given a lattice polytope Q in R^n, we define an affine scheme M(Q) that reflects the possibilities of splitting Q into a Minkowski sum. On the other hand, Q induces a toric Gorenstein singularity Y, and we construct a flat family over M(Q) with Y as special fiber. In case Y has an isolated singularity only, this family is versal. (This revised version contains the proof now.)

Gorenstein代数连通和的变形

Deformations of the connected sum of Gorenstein algebras

研究Gorenstein代数连通和的变形，证明Hilbert方案在n≥12时非约化，并构造极化代数的非约化点，揭示分形结构。

Halfcanonical Gorenstein curves of codimension four

Recent work of Schenck, Stillman and Yuan arXiv:2011.10871 outlines all possible Betti tables for Artin Gorenstein algebras $A$ with regularity($A$) = 4 = codim($A$). We populate the second half of this list with examples of stable curves, and ask if there are further possible constructions. The problem of deformation between curves with the same Hilbert series but different Betti tables is ongoing work, but our work solves one case: a deformation (due to Jan Stevens) between a reducible curve corresponding to Betti table type 2.7 in arXiv:2011.10871 and the curve obtained as the intersection …

On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties

Firstly, we see that the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension $\leq 2$ which are the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities. Secondly, we show that the deformation spaces of isolated Gorenstein toric $3$-fold singularities appear, in a weak sense, as singularities of the K-moduli stack of K-semistable Fano varieties of every dimension $\geq 3$. As a consequence, we prove that the number of local …

On the Normal Sheaf of Gorenstein Curves

We show that any tetragonal Gorenstein integral curve is a complete intersection in its respective $3$-fold rational normal scroll S, implying that the normal sheaf on $C$ embedded in S, and in $\mathbb{P}^{g-1}$ as well, is unstable for $g\geq 5$, provided that $S$ is smooth. We also compute the degree of the normal sheaf of any singular reduced curve in terms of the Tjurina and Deligne numbers, providing a semicontinuity of the degree of the normal sheaf over suitable deformations, revisiting classical results of the local theory of analytic germs.

Smoothings of schemes with non-isolated singularities

In this paper we study the deformation and Q-Gorenstein deformation theory of schemes with non-isolated singularities. We obtain obstruction spaces for the existence of deformations and also for local deformations to exist globally. Finally we obtain explicit criteria in order for a pure and reduced scheme of finite type over a field $k$ to have smoothings and Q-Gorenstein smoothings.

$\mathbb{Q}$-Gorenstein smoothings of surfaces and degenerations of curves

In this paper we mainly describe $\mathbb{Q}$-Gorenstein smoothings of projective surfaces with only Wahl singularities which have birational fibers. For instance, these degenerations appear in normal degenerations of the projective plane, and in boundary divisors of the KSBA compactification of the moduli space of surfaces of general type [KSB88]. We give an explicit description of them as smooth deformations plus 3-fold birational operations, through the flips and divisorial contractions in [HTU13]. We interpret the continuous part (smooth deformations) as degenerations of certain curves in …

Poincaré series and deformations of Gorenstein local algebras with low socle degree

Let $K$ be an algebraically closed field of characteristic $0$, and let $A$ be an Artinian Gorenstein local commutative and Noetherian $K$--algebra, with maximal ideal $M$. In the present paper we prove a structure theorem describing such kind of $K$--algebras satisfying $M^4=0$. We use this result in order to prove that such a $K$--algebra $A$ has rational Poincaré series and it is always smoothable in any embedding dimension, if $\dim_K M^2/M^3 \le 4$. We also prove that the generic Artinian Gorenstein local $K$--algebra with socle degree three has rational Poincaré series, in spite of the f…

One-parameter families containing three-dimensional toric Gorenstein singularities

For affine toric varieties, the vector space T1 (containing the infinitesimal deformations) will be interpreted via Minkowski summands of cross cuts of the defining polyhedral cone. This result will be applied to study the deformation theory of (in particular non-isolated) three-dimensional Gorenstein singularities.

Reflexive modules on normal Gorenstein Stein surfaces, their deformations and moduli

In this paper we generalize Artin-Verdier, Esnault and Wunram construction of McKay correspondence to arbitrary Gorenstein surface singularities. The key idea is the definition and a systematic use of a degeneracy module, which is an enhancement of the first Chern class construction via a degeneracy locus. We study also deformation and moduli questions. Among our main result we quote: a full classification of special reflexive MCM modules on normal Gorenstein surface singularities in terms of divisorial valuations centered at the singularity, a first Chern class determination at an adequate re…

Toric Q-Gorenstein Singularities

For an affine, toric Q-Gorenstein variety Y (given by a lattice polytope Q) the vector space T^1 of infinitesimal deformations is related to the complexified vector spaces of rational Minkowski summands of faces of Q. Moreover, assuming Y to be an isolated, at least 3-dimensional singularity, Y will be rigid unless it is even Gorenstein and dim Y=3 (dim Q=2). For this particular case, so-called toric deformations of Y correspond to Minkowski decompositions of Q into a sum of lattice polygons. Their Kodaira-Spencer-map can be interpreted in a very natural way. We regard the projective variety P…

Toric non-Gorenstein loci

We describe the non-Gorenstein loci of normal toric varieties. In the case of Hibi rings a combinatorial description is provided in terms of the underlying partially ordered set. As a non-toric application we compute the dimensions of the non-Gorenstein loci of the first secant variety of Segre varieties.

椭圆理想正常切锥的戈林斯坦性

Gorensteinness for normal tangent cones of elliptic ideals

本文研究了二维 excellent 正则戈林斯坦局部域中椭圆理想正常切锥的戈林斯坦性，并在特征零情况下对相关理想进行了分类。

Arithmetic and geometric deformations of $F$-pure and $F$-regular singularities

Given a normal $\mathbb{Q}$-Gorenstein complex variety $X$, we prove that if one spreads it out to a normal $\mathbb{Q}$-Gorenstein scheme $\mathcal{X}$ of mixed characteristic whose reduction $\mathcal{X}_p$ modulo $p$ has normal $F$-pure singularities for a single prime $p$, then $X$ has log canonical singularities. In addition, we show its analog for log terminal singularities, without assuming that $\mathcal{X}$ is $\mathbb{Q}$-Gorenstein, which is a generalization of a result of Ma-Schwede. We also prove that two-dimensional strongly $F$-regular singularities are stable under equal charac…

关于$\mathbb{Q}$-Gorenstein曲面的注记

A note on $\mathbb{Q}$-Gorenstein surfaces

本文在代数闭域上构造了一个典范环非有限生成的正规射影$\mathbb{Q}$-Gorenstein曲面，并给出了正特征下$\mathbb{Q}$-Gorenstein曲面极小模型程序的一个反例。

Existence and deformations of Kahler-Einstein metrics on smoothable Q-Fano varieties

We prove the existence of Kahler-Einstein metrics on Q-Gorenstein smoothable, K-polystable Q-Fano varieties, and we show how these metrics behave, in the Gromov-Hausdorff sense, under Q-Gorenstein smoothings.

On Gorenstein Surfaces Dominated by P^2

In this paper we prove that a normal Gorenstein surface dominated by the projective plane P^2 is isomorphic to a quotient P^2/G, where G is a finite group of automorphisms of P^2 (except possibly for one surface V_8'). We can completely classify all such quotients. Some natural conjectures when the surface is not Gorenstein are also stated.

Gorenstein-duality for one-dimensional almost complete intersections-with an application to non-isolated real singularities

We give a generalization of the duality of a zero-dimensional complete intersection to the case of one-dimensional almost complete intersections, which results in a {\em Gorenstein module} $M=I/J$. In the real case the resulting pairing has a signature, which we show to be constant under flat deformations. In the special case of a non-isolated real hypersurface singularity $f$ with a one-dimensional critical locus, we relate the signature on the jacobian module $I/J_f$ to the Euler characteristic of the positive and negative Milnor fibre, generalising the result for isolated critical points. A…

Gorenstein stable surfaces with $K_X^2 = 1$ and $p_g>0$

In this paper we consider Gorenstein stable surfaces with $K^2_X=1$ and positive geometric genus. Extending classical results, we show that such surfaces admit a simple description as weighted complete intersection. We exhibit a wealth of surfaces of all possible Kodaira dimensions that occur as normalisations of Gorenstein stable surfaces with $K_X^2=1$; for $p_g=2$ this leads to a rough stratification of the moduli space. Explicit non-Gorenstein examples show that we need further techniques to understand all possible degenerations.

Divisorial models of normal varieties

We prove that the canonical ring of a canonical variety in the sense of de Fernex and Hacon is finitely generated. We prove that canonical varieties are klt if and only if R(-K_X) is finitely generated. We introduce a notion of nefness for non-Q-Gorenstein varieties and study some of its properties. We then focus on these properties for non-Q-Gorenstein toric varieties.

Demi-Normal Surface Singularities

We prove semi-rationalification and semi-log-canonicalization for Gorenstein demi-normal surfaces. That is, given a Gorenstein demi-normal surface X with semi-rational (respectively, semi-log canonical) singularities in an open set U with complement a finite set of points, there is a proper birational morphism f : Y --> X such that f is an isomorphism over U and Y has only semi-rational (respectively, semi-log canonical) singularities. We proceed by passing to the normalization and then gluing along the conductor in an appropriate rationalification or log-canonicalization of the normalization …

A(n)奇点的导出McKay对应的非交换形变

Non-Commutative Deformations of Derived McKay Correspondence for A(n) singularities

本文证明导出McKay对应可扩展到A(n)环面曲面奇点的交换与非交换crepant解消的半泛非交换形变之间，从而推广了经典的导出等价结果。

Gorenstein models of del Pezzo surfaces of degree 1 over Dedekind schemes

Let R be a Dedekind scheme, $ν$ its generic point, X and V del Pezzo surfaces of degree 1 over R that are Gorenstein Mori fiber spaces (as 3-folds germs over the ground field). We study birational maps $ϕ:X\dasharrow V$ over R which are isomorphisms over the generic point of R. We put down normal forms of such transformations (in suitable coordinates) and give some properties of X and V. In particular, we prove the uniqueness of a smooth model.

Deformations of zero-dimensional schemes and applications

In this thesis we consider the geometry of the Hilbert scheme of points in P^n, concentrating on the locus of points corresponding to the Gorenstein subschemes of P^n. New results are given, most importantly we provide tools for constructing flat families and analysis of finite Gorenstein algebras and expose their efficiency by proving smoothability of certain families of algebras. Much of the existing theory and folklore is reviewed, providing a micro-encyclopaedic reference.

椭圆m重点曲线奇点的通用形变

The versal deformation of elliptic m-fold point curve singularities

本文给出了n维仿射空间中过原点的n+1条直线处于一般位置的奇点L_{n+1}^n的通用形变的显式高度对称方程，并证明了当n>4时其基空间同构于L_n^{n-1}的全空间，进而通过归纳得出基空间不可约且为Gorenstein环，同时讨论了与模空间紧化的联系。

非光滑的来自自反多胞形的Calabi-Yau三维流形

Non-smoothable Calabi-Yau threefolds from reflexive polytopes

本文构造了具有孤立Gorenstein典范奇点且不可光滑化的Calabi-Yau三维流形，利用Altmann形变理论与Corti-Filip-Petracci准则，并在Kreuzer-Skarke分类中识别出约8.27%的多胞形存在此阻碍。

Singularities of non-$\mathbb{Q}$-Gorenstein varieties admitting a polarized endomorphism

In this paper, we discuss a generalization of log canonical singularities in the non-$\mathbb{Q}$-Gorenstein setting. We prove that if a normal complex projective variety has a non-invertible polarized endomorphism, then it has log canonical singularities in our sense. As a corollary, we give an affirmative answer to a conjecture of Broustet and Höring.

Numerically Gorenstein surface singularities are homeomorphic to Gorenstein ones

Consider a normal complex analytic surface singularity. It is called Gorenstein if the canonical line bundle is holomorphically trivial in some punctured neighborhood of the singular point and is called numerically Gorenstein if this line bundle is topologically trivial. The second notion depends only on the topological type of the singularity. Laufer proved in 1977 that, given a numerically Gorenstein topological type of singularity, every analytical realization of it is Gorenstein if and only if one has either a Kleinian or a minimally elliptic topological type. The question to know if any n…

Simply connected surfaces of general type in positive characteristic via deformation theory

Algebraically simply connected surfaces of general type with p_g=q=0 and 1\le K^2\le 4 in positive characteristic (with one exception in K^2=4) are presented by using a Q-Gorenstein smoothing of two-dimensional toric singularities, a generalization of Lee-Park's construction in the field of complex numbers to the positive characteristic case, and Grothendieck's specialization theorem for the fundamental group.

Noether-Horikawa曲面的正规稳定退化

Normal stable degenerations of Noether-Horikawa surfaces

本文研究Noether-Horikawa曲面的正规稳定退化，分类了仅含$\mathbb{Q}$-Gorenstein可光滑化对数典范奇点的曲面，并给出整体可光滑化判据及模空间边界层描述。

Canonical rings of surfaces whose canonical system has base points

In this paper we propose an approach to investigate the canonical rings of surfaces of general type whose canonical system has isolated base points and yields a birational map onto its image. We apply then the method in the concrete case of surfaces with $K^2=7$ and $p_g=4$, where we give three different descriptions of the canonical ring (under the presence of base points). The main result in the paper consists in showing an explicit deformation of the above canonical rings to the canonical rings of surfaces whose canonical system has no base points (the latter were already classified by the …

Discrepancies of non-$\Q$-Gorenstein varieties

We give an example of a non $\Q$-Gorenstein variety which is canonical but not klt, and whose canonical divisor has an irrational valuation. We also give an example of an irrational jumping number and we prove that there are no accumulation points for the jumping numbers of normal non-$\Q$-Gorenstein varieties with isolated singularities.

"Weakly" Elliptic Gorenstein Singularities of Surfaces

The main message of the paper is that for Gorenstein singularities, whose (real) link is rational homology sphere, the Artin--Laufer program can be continued. Here we give the complete answer in the case of elliptic singularities. The main result of the paper says that in the case of an elliptic Gorenstein singularity whose link is rational homology sphere, the geometric genus is a topological invariant. Actually, it is exactly the length of the elliptic sequence in the minimal resolution (or, equivalently, in S. S.-T. Yau's terminology: these singularities are maximally elliptic). In the pape…

Singularities on normal varieties

In this paper we generalize the definitions of singularities of pairs and multiplier ideal sheaves to pairs on arbitrary normal varieties, without any assumption on the variety being Q-Gorenstein or the pair being log Q-Gorenstein. The main features of the theory extend to this setting in a natural way.

On Gonality, Scrolls, and Canonical Models of Non-Gorenstein Curves

Let $C$ be an integral and projective curve; and let $C'$ be its canonical model. We study the relation between the gonality of $C$ and the dimension of a rational normal scroll $S$ where $C'$ can lie on. We are mainly interested in the case where $C$ is singular, or even non-Gorenstein, in which case $C'\not\cong C$. We first analyze some properties of an inclusion $C'\subset S$ when it is induced by a pencil on $C$. Afterwards, in an opposite direction, we assume $C'$ lies on a certain scroll, and check some properties $C$ may satisfy, such as gonality and the kind of its singularities. At t…

Singularities with respect to Mather-Jacobian discrepancies

As is well known, the "usual discrepancy" is defined for a normal Q-Gorenstein variety. By using this discrepancy we can define a canonical singularity and a log canonical singularity. In the same way, by using a new notion, Mather-Jacobian discrepancy introduced in recent papers we can define a "canonical singularity" and a "log canonical singularity" for not necessarily normal or Q-Gorenstein varieties. In this paper, we show basic properties of these singularities, behavior of these singularities under deformations and determine all these singularities of dimension up to 2.

一些局部Calabi-Yau流形的形变

Deformations of some local Calabi-Yau manifolds

本文研究孤立有理Gorenstein奇点的某些互反消解的形变，特别关注三维情形下好的（对数）消解与小消解的例子，并得到关于允许好的互反消解的正则三维流形奇点分类的部分结果。

Test ideals in non-Q-Gorenstein rings

Suppose that $X = \Spec R$ is an $F$-finite normal variety in characteristic $p > 0$. In this paper we show that the big test ideal $τ_b(R) = \tld τ(R)$ is equal to $\sum_Δ τ(R; Δ)$ where the sum is over $Δ$ such that $K_X + Δ$ is $\bQ$-Cartier. This affirmatively answers a question asked by various people, including Blickle, Lazarsfeld, K. Lee and K. Smith. Furthermore, we have a version of this result in the case that $R$ is not even necessarily normal.

Homological flips and homological flops

We introduce a notion of homological flips and homological flops. The former includes the class of all flips between Gorenstein normal varieties; while the latter includes the class of all flops between Cohen-Macaulay normal varieties whose contracted variety is quasi-Gorenstein. Our main theorem shows that certain local cohomology complexes are dual to each other under homological flips/flops. We give some preliminary applications of this duality to relate the derived categories under flip/flop. Further applications are in [Yeu20b].

曲线模空间中最大变化区域

Maximal Variation in the Moduli of Curves

本文研究了曲线模空间中最大变化区域的性质，揭示了非Gorenstein奇点对退化的影响，并建立了退化区域的分类方法。

Some properties and examples of log terminal+ singularities

In "Singularities on Normal Varieties", de Fernex and Hacon started the study of singularities on non-Q-Gorenstein varieties using pullbacks of Weil divisors. In "Log Terminal Singularities", the author of this paper and Urbinati introduce a new class of singularities, called log terminal+, or simply lt+, which they prove is rather well behaved. In this paper we will continue the study of lt+ singularities, and we will show that they satisfy a Bertini type result, inversion of adjunction and small deformation invariance, and that they are naturally related to rational singularities. Finally, w…

Kawachi's invariant for normal surface singularities

We study a useful numerical invariant of normal surface singularities, introduced recently by T. Kawachi. Using this invariant, we give a quick proof of the (well-known) fact that all log-canonical surface singularities are either elliptic Gorenstein or rational (without assuming a priori that they are Q-Gorenstein). The main result is a criterion (stated in terms of Kawachi's invariant) regarding global generation of adjoint linear systems on normal surfaces with boundary. Such results can be used in proving effective estimates for global generation on singular threefolds. The theorem of Ein-…

具有椭圆奇点的I曲面形变

Deformations of I-surfaces with elliptic singularities

本文研究具有简单椭圆或尖点奇点的I曲面（满足$K_S^2=1$，$p_g=2$的一般型代数曲面），在一般位置假设下证明其形变对奇点形变是versal的，并详细分析两个例外情况。

不可光滑化的曲线奇点

Non-smoothable curve singularities

本文研究曲线奇点的可光滑化问题，通过维数计数和Dedekind不同的半连续性两种方法，给出不可光滑化奇点的新例子，包括Gorenstein曲线奇点。

Finiteness of the polyhedral Q-codegree spectrum

In this paper we show that the spectrum of the Q-codegree of a d-dimensional lattice polytope is finite above any positive threshold in the class of lattice polytopes with α-canonical normal fan for any fixed α>0. For α=1/r this includes lattice polytopes with Q-Gorenstein normal fan of index r. In particular, this proves Fujita's Spectrum Conjecture for polarized varieties in the case of Q-Gorenstein toric varieties of index r.

Normal differential operators and deformation theory

This paper develops the theory of a sheaf of normal differential operators to a submanifold Y of a complex manifold X as a generalization of the normal bundle. We show that the global sections of this sheaf play an analogous role for formal deformations of Y to the role played by the normal bundle with respect to first-order deformations.

The jumping coefficients of non-Q-Gorenstein multiplier ideals

Let $\mathfrak a \subset \mathscr O_X$ be a coherent ideal sheaf on a normal complex variety $X$, and let $c \ge 0$ be a real number. De Fernex and Hacon associated a multiplier ideal sheaf to the pair $(X, \mathfrak a^c)$ which coincides with the usual notion whenever the canonical divisor $K_X$ is $\mathbb Q$-Cartier. We investigate the properties of the jumping numbers associated to these multiplier ideals. We show that the set of jumping numbers of a pair is unbounded, countable and satisfies a certain periodicity property. We then prove that the jumping numbers form a discrete set of real…

First order deformations of schemes with normal crossing singularities

We describe the sheaf T^1(X) of first order deformations of a reduced scheme with normal crossing singularities. In particular, we obtain a formula for T^1(X) in a suitable log resolution of X.

K-半稳定性和正规仿射锥的奇点

K-semistability and singularities of normal affine cones

本文在Collins-Székelyhidi意义下，通过K-稳定性给出了Q-Gorenstein正规仿射奇点的完整且最优刻画。

Severi inequality for varieties of maximal Albanese dimension

Let $X$ be a projective, normal, minimal and Gorenstein $n$-dimensional complex variety of general type. Suppose $X$ is of maximal Albanese dimension. We prove that $K^n_X \ge 2 n! χ(K_X)$

What is missing in canonical models for proper normal algebraic surfaces?

Smooth surfaces have finitely generated canonical rings and projective canonical models. For normal surfaces, however, the graded ring of multicanonical sections is possibly nonnoetherian, such that the corresponding homogeneous spectrum is noncompact. I construct a canonical compactification by adding finitely many non-Q-Gorenstein points at infinity, provided that each Weil divisor is numerically equivalent to a Q-Cartier divisor. Similar results hold for arbitrary Weil divisors instead of the canonical class.