Vik. S. Kulikov
In this note we give examples of Zariski's pairs $B_{1,m}, B_{2,m}$ ($m \in N$ and $m \geq 5$) of plane cuspidal curves such that (i) $B_{i,m}$ is the discriminant curve of a generic morphism $f_{i,m}:S_i \to P^2$, $i=1, 2$, (ii) $S_1$ and $S_2$ are homeomorphic surfaces of general type, (iii) $f_{i,m}$ is given by linear three-dimensional subsystem of the mth canonical class of $S_i$.
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A series of Zariski pairs and four Zariski triplets were found by using lattice theory of K3 surfaces. There is a Zariski triplet of which one member is a deformation of another.
In this paper we give an elementary proof of the Zariski-Lipman conjecture for log canonical spaces.
On the unirationality of conic bundles with discriminant of degree eight
本文研究了判别次数为八的锥面的不可理化性，通过构造有理多截面和变形理论分析，揭示了其在不同域上的不可理化性质。
Mordell-Weil groups and Zariski triples
本文构造了三条12次不可约曲线，它们尖点数相同但Alexander多项式不同，从而给出Zariski三元组的新例子，并进一步研究了相关椭圆三维流形的Mordell-Weil群生成元及一般基变换下等奇异变形空间的维数。
We will describe the topological type of the discriminant curve of the morphism $(\ell, f)$, where $\ell$ is a smooth curve and $f$ is an irreducible curve (branch) of multiplicity less than five or a branch that the difference between its Milnor number and Tjurina number is less than 3. We prove that for a branch of these families, the topological type of the discriminant curve is determined by the semigroup, the Zariski invariant and at most two other analytical invariants of the branch.
In this paper we prove the Zariski-Lipman conjecture for log canonical spaces.
We prove that termination of lower dimensional flips for generalized klt pairs implies termination of flips for log canonical generalized pairs with a weak Zariski decomposition. Moreover, we prove that the existence of weak Zariski decompositions for pseudo-effective generalized klt pairs implies the existence of minimal models for such pairs.
Let $X$ be a normal variety. Assume that for some reduced divisor $D \subset X$, logarithmic 1-forms defined on the snc locus of $(X, D)$ extend to a log resolution $\tilde X \to X$ as logarithmic differential forms. We prove that then the Lipman-Zariski conjecture holds for $X$. This result applies in particular if $X$ has log canonical singularities. Furthermore, we give an example of a 2-form defined on the smooth locus of a three-dimensional log canonical pair $(X, \emptyset)$ which acquires a logarithmic pole along an exceptional divisor of discrepancy zero, thereby improving on a similar…
We study graded rings associated to big divisors on LC pairs whose difference with the log-canonical divisor is nef. For divisors that are positive enough at the LC centers of the pair, we prove the finite generation of such rings if the pair is DLT or the dimension is low, given that a Zariski decomposition exists.
We continue our study of the relation between log minimal models and various types of Zariski decompositions. Let $(X,B)$ be a projective log canonical pair. We will show that $(X,B)$ has a log minimal model if either $K_X+B$ birationally has a Nakayama-Zariski decomposition with nef positive part, or that $K_X+B$ is big and birationally it has a Fujita or CKM Zariski decomposition. Along the way we introduce polarized pairs $(X,B+P)$ where $(X,B)$ is a usual projective pair and $P$ is nef, and study the birational geometry of such pairs.
We consider a version of the Lipman-Zariski conjecture for logarithmic vector fields and logarithmic $1$-forms on pairs. Let $(X,D)$ be a pair consisting of a normal complex variety $X$ and an effective Weil divisor $D$ such that the sheaf of logarithmic vector fields (or dually the sheaf of reflexive logarithmic $1$-forms) is locally free. We prove that in this case the following holds: If $(X,D)$ is dlt, then $X$ is necessarily smooth and $\lfloor D\rfloor $ is snc. If $(X,D)$ is lc or the logarithmic $1$-forms are locally generated by closed forms, then $(X,\lfloor D\rfloor)$ is toroidal.
On the termination of flips for log canonical generalized pairs
该研究在代数几何领域证明了4维伪有效NQC对数典范广义配对的翻转终止性，通过验证3维对应情形的结论，并建立低维翻转终止性到高维容许弱扎里斯基分解情形的推导关系。
We show that the Zariski canonical stratification of complex hypersurfaces is locally bi-Lipschitz trivial along the strata of codimension two. More precisely, we study Zariski equisingular families of surface, not necessarily isolated, singularities in $\mathbb{C}^3$. We show that a natural stratification of such a family given by the singular set and the generic family of polar curves provides a Lipschitz stratification in the sense of Mostowski. In particular, such families are bi-Lipschitz trivial by trivializations obtained by integrating Lipschitz vector fields.
Geometry of weak contact conics to irreducible quartics with 2 nodes and 1 cusp via rational elliptic surfaces and Zariski pairs
研究弱接触圆锥与具有2个节点和1个尖点的不可约四次曲线的几何结构，通过理性椭圆面和扎里斯基对构造度数为7和8的扎里斯基对。
We show that minimal models of $\mathbb{Q}$-factorial NQC log canonical generalised pairs exist, assuming the existence of minimal models of smooth varieties. More generally, we prove that on a $\mathbb{Q}$-factorial NQC log canonical generalised pair $ (X,B+M) $ we can run an MMP with scaling of an ample divisor which terminates, assuming that it admits an NQC weak Zariski decomposition or that $K_X+B+M$ is not pseudoeffective. As a consequence, we establish several existence results for minimal models and Mori fibre spaces.
The notion of Zariski pairs for projective curves in $\mathbb P^2$ is known since the pioneer paper of Zariski \cite{Zariski} and for further development, we refer the reference in \cite{Bartolo}.In this paper, we introduce a notion of Zariski pair of links in the class of isolated hypersurface singularities. Such a pair is canonically produced from a Zariski (or a weak Zariski ) pair of curves $C=\{f(x,y,z)=0\}$ and $C'=\{g(x,y,z)=0\}$ of degree $d$ by simply adding a monomial $z^{d+m}$ to $f$ and $g$ so that the corresponding affine hypersurfaces have isolated singularities at the origin. Th…
We compute the canonical model of the cover of Shimura curves $X_0(2) \to X(1)$ for the cubic field of discriminant 13^2 described at the end of Elkies' paper "Shimura curves for level 3 subgroups of the (2,3,7) triangle group". Last, we list the coordinates of some rational CM points on X(1).
We construct the canonical structure of an irreducible projective variety on the set of connected curves of degree $d$ in $\Bbb P^n$ with rational components (some components can be multiple). The set of rational curves is open subset in the Zariski topology under this structure.
Let $Σ$ be a smooth projective surface, let $f' : S' \to Σ$ be a double cover of $Σ$ and let $μ: S \to S'$ be the canonical resolution. Put $f = f'\circμ$. An irreducible curve $C$ on $Σ$ is said to be a splitting curve with respect to $f$ if $f^*C$ is of the form $C^+ + C^- + E$, where $C^- = σ_f^*C^+$, $σ_f$ being the covering transformation of $f$ and all irreducible components of $E$ are contained in the exceptional set of $μ$. In this article, we show that a kind of "reciprocity" of splitting curves holds for a certain pair of curves on rational ruled surfaces. As an application, we consi…
There is a canonical isomorphism between the coarse moduli spaces of somooth hyperelliptic curves of genus g and binary forms of degree 2g+2 with nonzero discriminant. In this paper, we study the extension of this isomorphism to the compactification of these moduli spaces. The main result is that the canonical isomorphism above extends to a holomorphic map f_g from the compactication of the moduli space of hyperelliptic curves of genus g to the compactification of the moduli space of binary forms of degree 2g+2. Moreover, we work out how the boundary components are contracted by the map f_g.
Curve semistable Higgs bundles and smooth projective varieties whose canonical bundle is ample
本文研究典范丛丰富的光滑射影簇上的 Simpson 系统，证明其 Higgs 丛稳定性及 Guggenheimer–Yau 不等式，并在等号时得到判别类消失与曲线半稳定性，同时给出代数双曲性及正实常数下界。
We extend results of Cogolludo-Agustin and Libgober relating the Alexander polynomial of a plane curve $C$ with the Mordell--Weil rank of certain isotrivial families of jacobians over $\mathbf{P}^2$ of discriminant $C$. In the second part we introduce a height pairing on the $(2,3,6)$ quasi-toric decompositions of a plane curve. We use this pairing and the results in the first part of the paper to construct a pair of degree 12 curves with 30 cusps and Alexander polynomial $t^2-t+1$, but with distinct height pairing. We use the height pairing to show that these curves from a Zariski pair.
We propose and study a generalized version of the Lipman-Zariski conjecture: let $(x \in X)$ be an $n$-dimensional singularity such that for some integer $1 \le p \le n - 1$, the sheaf $Ω_X^{[p]}$ of reflexive differential $p$-forms is free. Does this imply that $(x \in X)$ is smooth? We give an example showing that the answer is no even for $p = 2$ and $X$ a terminal threefold. However, we prove that if $p = n - 1$, then there are only finitely many log canonical counterexamples in each dimension, and all of these are isolated and terminal. As an application, we show that if $X$ is a projecti…
We conjecture that any connected component $Q$ of the moduli space of triples $(X,E=E_1+\dots+E_n,Θ)$ where $X$ is a smooth projective variety, $E$ is a normal crossing anti-canonical divisor with a 0-stratum, every $E_i$ is smooth, and $Θ$ is an ample divisor not containing any 0-stratum of $E$, is unirational. More precisely: note that $Q$ has a natural embedding into the Kollár-Shepherd-Barron-Alexeev moduli space of stable pairs, we conjecture that the induced compactification admits a finite cover by a complete toric variety. We construct the associated complete toric fan, generalizing th…
Quadratic points on double planes
本文研究高维代数簇上Zariski稠密二次点的几何刻画，证明在双覆盖情形下Vojta猜想可排除典范丛充分正时的其他稠密来源，并给出反例说明该条件不可省略。
Weakly Special Manifolds with no rational curves
本文在假设丰盛猜想和终端卡拉比-约当簇上存在稠密有理曲线的前提下，证明了无有理曲线的复投影弱特殊流形是阿贝尔簇的 étale 商。若流形包含稠密整个曲线，则在假设朗猜想下同样成立，表明非超广复投影流形包含阿贝尔簇的像。
In this manuscript we prove that if two cuspidal plane curves have equivalent braid monodromy factorizations, then they are smoothly isotopic in the plane. As a consequence of this and the Chisini conjecture, we obtain that if two discriminant curves (or branch curves in other terminology) of generic projections (to the plane) of surfaces of general type imbedded in a projective space by means of a multiple canonical class have equivalent braid monodromy factorizations, then the surfaces are diffeomorphic (if we consider them as real 4-folds).
Explicit Uniform Lower Bounds for the Canonical Height on Elliptic Curves over Abelian Extensions
本文通过显式方法为椭圆曲线在阿贝尔扩张上的诺尔-塔特高度提供了一个不依赖基域判别式的下界，改进了 Baker 的定理证明。
We explore a number of examples of special Lagrangian fibrations on non-compact Calabi-Yau manifolds invariant under torus actions. These include fibrations on crepant resolutions of canonical toric singularities (already found by Goldstein), proper versions of these fibrations, and fibrations on flat deformations of canonical toric singularities. We do this with an eye towards understanding the global structure and discriminant loci of such fibrations. The paper ends with some speculation, both about local mirror symmetry, and the connections of this work with the work of W.D. Ruan and D. Joy…
This is an extended, renovated and updated report on a joint work which the second named author presented at the Conference on Algebraic Geometry held at Saitama University, 15-17 of March, 1995. The main result is an inequality for the numerical type of singularities of a plane curve, which involves the degree of the curve, the multiplicities and the Milnor numbers of its singular points. It is a corollary of the logarithmic Bogomolov-Miyaoka-Yau's type inequality due to Miyaoka. It was first proven by F. Sakai at 1990 and rediscovered by the authors independently in the particular case of an…
Let $C \s \pr^2$ be an irreducible plane curve whose dual $C^* \s \pr^{2*}$ is an immersed curve which is neither a conic nor a nodal cubic. The main result states that the Poincaré group $π_1(\pr^2 \se C)$ contains a free group with two generators. If the geometric genus $g$ of $C$ is at least 2, then a subgroup of $G$ can be mapped epimorphically onto the fundamental group of the normalization of $C$, and the result follows. To handle the cases $g=0,1$, we construct universal families of immersed plane curves and their Picard bundles. This allows us to reduce the consideration to the case of…
Kummers, spinors, and heights
本文利用纯旋量理论显式描述了奇数超椭圆曲线雅可比簇上由 $|2Θ|$ 诱导的态射，并应用于有理点高度研究，证明了 Lang--Silverman 猜想的一个“密度 1”形式。
Given a parameterization $ϕ$ of a rational plane curve C, we study some invariants of C via $ϕ$. We first focus on the characterization of rational cuspidal curves, in particular we establish a relation between the discriminant of the pull-back of a line via $ϕ$, the dual curve of C and its singular points. Then, by analyzing the pull-backs of the global differential forms via $ϕ$, we prove that the (nearly) freeness of a rational curve can be tested by inspecting the Hilbert function of the kernel of a canonical map. As a by product, we also show that the global Tjurina number of a rational c…
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