Alex Degtyarev
We partially prove and partially disprove Oka's conjecture on the fundamental group/Alexander polynomial of an irreducible plane sextic. Among other results, we enumerate all irreducible sextics with simple singularities admitting dihedral coverings and find examples of Alexander equivalent Zariski pairs of irreducible sextics.
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We complete the proof of Oka's conjecture on the Alexander polynomial of an irreducible plane sextic. We also calculate the fundamental groups of irreducible sextics with a singular point adjacent to $J_{10}$.
We complete the equisingular deformation classification of irreducible singular plane sextic curves. As a by-product, we also compute the fundamental groups of the complement of all but a few maximizing sextics.
We compute the fundamental group of the complement of each irreducible sextic of weight eight or nine (in a sense, the largest groups for irreducible sextics), as well as of 169 of their derivatives (both of and not of torus type). We also give a detailed geometric description of sextics of weight eight and nine and of their moduli spaces and compute their Alexander modules; the latter are shown to be free over an appropriate ring.
We calculate the fundamental groups $π=π_1(P^2\setminus B)$ for all irreducible plane sextics $B\subset¶^2$ with simple singularities for which $π$ is known to admit a dihedral quotient $D_{10}$. All groups found are shown to be finite, two of them being of large order: 960 and 21600.
The existence of Alexander-equivalent Zariski pairs dealing with irreducible curves of degree 6 was proved by A. Degtyarev. However, up to now, no explicit example of such a pair was available (only the existence was known). In this paper, we construct the first concrete example.
We construct explicit geometric models for and compute the fundamental groups of all plane sextics with simple singularities only and with at least one type $\bold E_8$ singular point. In particular, we discover four new sextics with nonabelian fundamental groups; two of them are irreducible. The groups of the two irreducible sextics found are finite. The principal tool used is the reduction to trigonal curves and Grothendieck's {\it dessins d'enfants}
We classify projective symmetries of irreducible plane sextics with simple singularities which are stable under equivariant deformations. We also outline a connection between order~2 stable symmetries and maximal trigonal curves.
A nonspecial divisor in the moduli space of cubic fourfolds via 10-nodal plane sextics
本文通过研究10节点平面六次曲线，发现立方四维流形模空间中存在一个非诺特-勒夫施茨类型的不可约除子。
Let S be the variety of irreducible sextics with six cusps as singularities. Let W be one of irreducible components of W. Denoting by M_4 the space of moduli of smooth curves of genus 4, the moduli map of W is the rational map from W to M_4 sending the general point of W, corresponding to a plane curve D, to the point of M_4 parametrizing the normalization curve of D. The number of moduli of W is, by definition the dimension of the image of W with respect to the moduli map. We know that this number is at most equal to seven. In this paper we prove that both irreducible components of S have num…
A conjecture of Beauville and Voisin states that for an irreducible symplectic variety X, any polynomial relation between classes of divisors and the Chern classes of X which holds in cohomology already holds in the Chow groups. We verify the conjecture for a very general double EPW sextic.
We analyze irreducible plane sextics whose fundamental group factors to $D_{14}$. We produce explicit equations for all curves and show that, in the simplest case of the set of singularities $3A_6$, the group is $D_{14}\times Z_3$.
We develop a geometric approach to the study of plane sextics with a triple singular point. As an application, we give an explicit geometric description of all irreducible maximal sextics with a type $\bold E_7$ singular point and compute their fundamental groups. All groups found are finite; one of them is nonabelian.
Conics in sextic $K3$-surfaces in $\mathbb{P}^4$
本文确定$\mathbb{P}^4$中光滑六次$K3$曲面所含圆锥曲线至多285条、实情形下实圆锥曲线至多261条，且两种极值构型中的圆锥曲线均不可约。
On fake ES-irreducibile components of certain strata of smooth plane sextics
本文构造了首批伪ES-不可约分支的例子，并分类了仅具有阶数不超过3的自同构的光滑平面六次曲线的自同构群。
Eisenbud Popescu and Walter have constructed certain special 4-dimensional sextic hypersurfaces as Lagrangian degeneracy loci. We prove that the natural double cover of a generic EPW-sextic is a deformation of the Hilbert square of a K3-surface and that the family of such varieties is locally complete for deformations that keep the hyperplane class of type (1,1) - thus we get an example similar to that (discovered by Beauville and Donagi) of the Fano variety of lines on a cubic 4-fold. Conversely suppose that X is an irreducible symplectic 4-fold numerically equivalent to the Hilbert square of…
Let $S \subset \mathbb P^3$ be a very general sextic surface over complex numbers. Let $\mathcal{M}(H, c_2)$ be the moduli space of rank $2$ stable bundles on $S$ with fixed first Chern class $H$ and second Chern class $c_2$. In this article we study the configuration of points of certain reduced zero dimensional subschemes on $S$ satisfying Cayley-Bacharach property, which leads to the existence of non-trivial sections of a general memeber of the moduli space for small $c_2$. Using this study we will make an attempt to prove Mestrano-Simpson conjecture on the number of irreducible components …
Parametric equations of plane sextic curves with a maximal set of double points
给出所有至多有双点且总米尔诺数最大（等于19）的不可约平面射影六次曲线的显式参数方程，找到最小可能次数数域上的参数化并选坐标使系数尽可能小。
We study the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of \bigwedge^3{\mathbb C}^6 by the natural action of SL_6, call it M. This is a compactification of the moduli space of smooth double EPW-sextics. We determine the stable points, the irreducible components of the GIT boundary and their dimensions. There are strong analogies with the moduli space of cubic 4-folds: we prove a result which is analogous to a theorem of Laza stating that cubic 4-folds with simple singularities are stable. Our final goal (not achieved in the present paper) is to understand th…
Irreducibility of moduli of vector bundles over a very general sextic Surface
本文证明了在$\mathbb{P}^3$中非常一般的六次光滑超曲面上，具有特定陈类条件的$μ$-稳定秩2无挠层模空间是不可约的。
We compute the fundamental groups of all irreducible plane sextics constituting classical Zariski pairs
Periods of Oka K3 surfaces are dense
该研究给出首个Oka K3曲面例子，证明K3曲面局部Kuranishi族中Oka纤维点集及零Néron–Severi群的Oka纤维子轨迹均为稠密G_δ集，还得到Oka Kummer曲面并证其周期域的稠密性。
Varieties of Nodal surfaces, coding theory and Discriminants of cubic hypersurfaces. Part 1: Generalities and nodal K3 surfaces. Part 2: Cubic Hypersurfaces, associated discriminants. Part 3: Nodal quintics. Part 4: Nodal sextics
本文通过为射影节点曲面附加二元码，系统研究了P^3中节点曲面的Severi簇的不可约分量及其关联层级，完整分类了节点K3曲面，并揭示了Togliatti五次曲面及最大节点数六次曲面与三次超曲面判别式之间的深刻联系。
We derive explicit defining equations for a number of irreducible maximizing plane sextics with double singular points only. For most real curves, we also compute the fundamental group of the complement; all groups found are abelian. As a by-product, examples of Zariski pairs in the strongest possible sense are constructed.
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