Adrian Clingher, Andreas Malmendier
We determine normal forms for the Kummer surfaces associated with abelian surfaces of polarization of type $(1,1)$, $(1,2)$, $(2,2)$, $(2,4)$, and $(1,4)$. Explicit formulas for coordinates and moduli parameters in terms of Theta functions of genus two are also given. The normal forms in question are closely connected to the generalized Riemann identities for Theta functions of Mumford's.
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We study isogeny relations between K3 surfaces and Kummer surfaces. Specifically, we prove a Torelli-type theorem for the existence of rational maps from K3 surfaces to Kummer surfaces, and a Kummer sandwich theorem for K3 surfaces with Shioda-Inose structure.
We give explicit equations of smooth Jacobian Kummer surfaces of degree 8 by theta functions. As byproducts, we can write down Rosenhain's 80 hyperpanes and 32 lines on these Kummer surfaces explicitly.
A Nikulin configuration is the data of $16$ disjoint smooth rational curves on a K3 surface. According to results of Nikulin, the existence of a Nikulin configuration means that the K3 surface is a Kummer surface, moreover the abelian surface from the Kummer structure is determined by the $16$ curves. A classical question of Shioda is about the existence of non isomorphic Kummer structures on the same Kummer K3 surface. The question was studied by several authors, and it was shown that the number of non-isomorphic Kummer structures is finite, but no explicit geometric construction of such stru…
K3 surfaces of Kummer type in characteristic two
本文研究了特征二域上包含Kummer构型的K3曲面，刻画了这类曲面的几何结构。
The automorphism groups of Kummer surfaces in characteristic two and their complex analogues
本文研究了特征二下Kummer表面的自同构群及其复类似，探讨了K3表面的Picard晶格结构。
In 1899, Hutchinson presented a way to obtain a three-parameter family of Hessians of cubic surfaces as blowups of Kummer surfaces. We show that this family consists of those Hessians containing an extra class of conic curves. Based on this, we find the invariant of a cubic surface C in pentahedral form that vanishes if its Hessian is in Hutchinson's family, and we give an explicit map between cubic surfaces in pentahedral form and blowups of Kummer surfaces.
The aim of this paper is to describe the geometry of the generic Kummer surface associated to a $(1,2)$-polarized abelian surface. We show that it is the double cover of a weak del Pezzo surface and that it inherits from the del Pezzo surface an elliptic fibration with twelve singular fibers of type $I_2.$
This paper classifies Enriques surfaces whose K3-cover is a fixed Picard-general Jacobian Kummer surface. There are exactly 31 such surfaces. We describe the free involutions which give these Enriques surfaces explicitly. As a biproduct, we show that Aut(X) is generated by elements of order 2, which is an improvement of the theorem of S. Kondo.
We find two natural spherical functors associated to the Kummer surface and analyse how their induced twists fit with Bridgeland's conjecture on the derived autoequivalence group of a complex algebraic K3 surface.
We show that every supersingular K3 surface in characteristic 2 with Artin invariant less than or equal to 2 is obtained by the Kummer type construction of Schroeer.
We apply our earlier results on Fourier-Mukai partners to answer definitively a question about Kummer surface structures, posed by T. Shioda 25 years ago.
In the first part of this paper we give a survey of classical results on Kummer surfaces with Picard number 17 from the point of view of lattice theory. We prove ampleness properties for certain divisors on Kummer surfaces and we use them to describe projective models of Kummer surfaces of (1,d)-polarized Abelian surfaces for d=1,2,3. As a consequence we prove that in these cases the Néron--Severi group can be generated by lines. In the second part of the paper we use Kummer surfaces to obtain results on K3 surfaces with a symplectic action of the group (\Z/2\Z)^4. In particular we describe th…
We study generalized Kummer surfaces Km$_{3}(A)$, by which we mean the K3 surfaces obtained by desingularization of the quotient of an abelian surface $A$ by an order $3$ symplectic automorphism group. Such a surface carries $9$ disjoint configurations of two smooth rational curves $C,C'$ with $CC'=1$. This $9{\bf A}_{2}$-configuration plays a role similar to the Nikulin configuration of $16$ disjoint smooth rational curves on (classical) Kummer surfaces. We study the (generalized) question of T. Shioda: suppose that Km$_{3}(A)$ is isomorphic to Km$_{3}(B)$, does that imply that $A$ and $B$ ar…
Tropical Kummer quartic surfaces
本文研究了热带库默四次曲面的性质及其与非阿基米德赋值域上库默四次曲面的关联，引入了有理多面体轨道群的概念并展示了其忠实嵌入。
This is a brief history of discovery and later study of Kummer surfaces. The article is based on the author's Oliver Club talk at Cornell University on October 10, 2019 delivered exactly 101 years since the first talk at the club given by John Hutchinson.
We construct and study curves with low H-constants on abelian and K3 surfaces. Using the Kummer $(16_{6})$-configurations on Jacobian surfaces and some $(16_{10})$-configurations of curves on $(1,3)$-polarized Abelian surfaces, we obtain examples of configurations of curves of genus $>1$ on a generic Jacobian K3 surface with H-constants $<-4$.
We find formulas for the birational maps from a Kummer surface K and its dual K^* to their common minimal desingularization S. We show how the nodes of K blow up. Then we give a description of the group of linear automorphisms of S.
Pseudo-automorphisms of rational threefolds and Kummer surfaces
本文将45个经典Kummer曲面自同构提升为$\mathbb{P}^3$沿6点15线爆破所得三维簇的伪自同构，并描述了由这些已知伪自同构生成的群的基本域。
Generalized Kummer surfaces over finite fields
本文改进了库萨拉定理，研究有限群作用于阿贝尔面的商为K3表面的情况，并计算了有限域上超奇异广义库默表面的弗罗贝尼乌斯迹。
The aim of this paper is to show that using some natural curve arrangements in algebraic surfaces and Hirzebruch-Kummer covers one cannot construct new examples of ball-quotients, i.e., minimal smooth complex projective surfaces of general type satisfying equality in the Bogomolov-Miyaoka-Yau inequality.
We pose the problem to determine explicit defining equations of various elliptic fibrations on a given $K3$ surface, and study the case of the Kummer surfaces of the product of two elliptic curves.
By showing a structural result on how the Rozansky-Witten invariants of Hilbert schemes of points on a K3 surface and the Generalised Kummer varieties depend on the dimension, we give a method how to compute a large class of Rozansky-Witten invariants in terms of Chern numbers.
Nikulin proved that the isometries induced on the second cohomology group of a K3 surface $X$ by a finite abelian group $G$ of symplectic automorphisms are essentially unique. Moreover he computed the discriminant of the sublattice of $H^2(X, \Z)$ which is fixed by the isometries induced by $G$. However for certain groups these discriminants are not the same of those found for explicit examples. Here we describe Kummer surfaces for which this phenomena happens and we explain the difference.
A generalized Kummer surface $X=Km(T,G)$ is the resolution of a quotient of a torus $T$ by a finite group of symplectic automorphisms $G$. We complete the classification of generalized Kummer surfaces by studying the two last groups which have not been yet studied. For these surfaces, we compute the associated Kummer lattice $K_{G}$, which is the minimal primitive sub-lattice containing the exceptional curves of the resolution $X\to T/G$. We then prove that a K3 surface is a generalised Kummer surface of type $Km(T,G)$ if and only if its Néron-Severi group contains $K_{G}$. For smooth-orbifold…
We prove that a Kummer surface defined over a complete strictly Henselian discretely valued field $K$ of residue characteristic different from 2 admits a strict Kulikov model after finite base change. The Kulikov models we construct will be schemes, so our results imply that the semistable reduction conjecture is true for Kummer surfaces in this setup, even in the category of schemes. Our construction of Kulikov models is closely related to an earlier construction of Künnemann, which produces semistable models of Abelian varieties. It is well-known that the special fibre of a strict Kulikov mo…
We investigate Calabi--Yau three folds which are small resolutions of fiber products of elliptic surfaces with section admitting reduced fibers. We start by the classification of all fibers that can appear on such varieties. Then, we find formulas to compute the Hodge numbers of obtained three folds in terms of the types of singular fibers of the elliptic surfaces. Next we study Kummer fibrations associated to these fiber products.
Fourier--Mukai partners and generalized Kummer structures on generalized Kummer surfaces of order $3$
本文研究3阶广义Kummer曲面与原Abel曲面的相互决定关系，按除子平方模9的情形刻画超越格、Hodge结构及Fourier–Mukai伙伴何时推出曲面同构。
Kummer surfaces and quadratic line complexes in characteristic two
本文探讨了在特征二条件下二次线复形与库默表面之间的关系及其经典理论的类比。
We prove that the complex cobordism class of any hyper-Kähler manifold of dimension $2n$ is a unique combination with rational coefficients of classes of products of punctual Hilbert schemes of $K3$ surfaces. We also prove a similar result using the generalized Kummer varieties instead of punctual Hilbert schemes. As a key step, we establish a closed formula for the top Chern character of their tangent bundles.
A generalized Kummer surface $X$ obtained as the quotient of an abelian surface by a symplectic automorphism of order 3 contains a $9\mathbf{A}_{2}$-configuration of $(-2)$-curves. Such a configuration plays the role of the $16\mathbf{A}_{1}$-configurations for usual Kummer surfaces. In this paper we construct $9$ other such $9\mathbf{A}_{2}$-configurations on the generalized Kummer surface associated to the double cover of the plane branched over the sextic dual curve of a cubic curve. The new $9\mathbf{A}_{2}$-configurations are obtained by taking the pullback of a certain configuration of $…
Inseparable Kummer surfaces
本文定义特征 2 中由 16 条光滑有理曲线及可除性条件刻画的超奇异 K3 曲面，证明其可被数值不变量类似阿贝尔曲面、光滑轨迹具阿贝尔群结构的非正规曲面 A 不可分二重覆盖。
Kummer surfaces for the selfproduct of the cuspidal rational curve
针对特征2下经典Kummer构造对超奇异阿贝尔曲面失效的问题，本文用有理尖点曲线自乘积和无穷小群概型作用给出新的Kummer型构造，得到Artin不变量为1、2的超奇异K3曲面，它们属于纯不可分基变换后同步消解得到的1维族。
Kummer surfaces associated with group schemes
本研究在特征2情形下引入由群概形mu_2作用于有理尖点曲线自乘积得到的Kummer曲面，证明其恰为Artin不变量σ≤3的超奇异K3曲面，给出曲线构型刻画并说明其可作为单连通Enriques曲面的正规K3型覆盖。
Brill-Noether loci on an Enriques surface covered by a Jacobian Kummer surface
针对被 Jacobian Kummer 曲面覆盖的 Enriques 曲面，本文证明其特定秩 2 稳定向量丛模空间中存在真第一 Brill-Noether 轨迹。
We define Enriques varieties as a higher dimensional generalization of Enriques surfaces and construct examples by using fixed point free automorphisms on generalized Kummer varieties. We also classify all automorphisms of generalized Kummer varieties that come from an automorphism of the underlying abelian surface.
Perverse filtration for generalized Kummer varieties of fibered surfaces
本文研究2维群概形纤维化的广义Kummer簇导出的真态射，证明其关联的反常层滤具有乘性。
The chiral de Rham complex is a sheaf of vertex algebras Ω^ch_M on any nonsingular algebraic variety or complex manifold M, which contains the ordinary de Rham complex as the weight zero subspace. We show that when M is a Kummer surface, the algebra of global sections is isomorphic to the N = 4 superconformal vertex algebra with central charge 6. Previously, CP^n was the only manifold where a complete description of the global section algebra was known.
A Nikulin configuration is the data of $16$ disjoint smooth rational curves on a K3 surface. According to a well known result of Nikulin, if a K3 surface contains a Nikulin configuration $\mathcal{C}$, then $X$ is a Kummer surface $X=Km(B)$ where $B$ is an Abelian surface determined by $\mathcal{C}$. Let $B$ be a generic Abelian surface having a polarization $M$ with $M^{2}=k(k+1)$ (for $k>0$ an integer) and let $X=Km(B)$ be the associated Kummer surface. To the natural Nikulin configuration $\mathcal{C}$ on $X=Km(B)$, we associate another Nikulin configuration $\mathcal{C}'$; we denote by $B'…
We describe all the elliptic fibrations with section on the Kummer surface X of the Jacobian of a very general curve C of genus 2 over an algebraically closed field of characteristic 0, modulo the automorphism group of X and the symmetric group on the Weierstrass points of C. In particular, we compute elliptic parameters and Weierstrass equations for the 25 different fibrations and analyze the reducible fibers and Mordell-Weil lattices. This answers completely a question posed by Kuwata and Shioda in 2008.
K3 surfaces associated with varieties of generalized Kummer type
本文通过Hodge理论将广义库默尔型流形与K3表面关联，并利用模空间证明了相关K3表面和Weil型四维阿贝尔簇的Hodge猜想。
We compute explicitly the Chow motive of any generalized Kummer variety associated to any abelian surface. In fact, it lies in the rigid tensor subcategory of the category of Chow motives generated by the Chow motive of the underlying abelian surface. One application of this calculation is to show that the Hodge conjecture holds for arbitrary products of generalized Kummer varieties. As another application, all numerically trivial 1-cycles on arbitrary products of generalized Kummer varieties are smash-nipotent.
Higher Chow cycles on a family of Kummer surfaces
本文在二维Kummer曲面族上构造了类型为(2,1)的高阶Chow环，并证明其不可分解部分生成秩至少18的子群，方法结合有限群作用与Picard-Fuchs算子。
Kummer Surfaces, Isogenies and Theta Functions
本文基于西塔函数框架讨论主极化阿贝尔曲面的(n,n)同源及库默尔曲面的几何与计算问题，关注分裂同源情况并探索将里歇洛特同源扩展到高阶以开发高效算法。
Grothendieck's period conjecture for Kummer surfaces of self-product CM type
研究自积CM型椭圆曲线的库默表面，证明格罗滕迪克周期猜想成立，周期同构在动机周期torsor中稠密，且表面幂上的动机类为代数类。
We prove the equisingular rigidity of the singular Hirzebruch-Kummer coverings $X(n, \mathcal{L})$ of the projective plane branched on line configurations $\mathcal{L}$, satisfying some technical condition. In the case, $\mathcal{L}$ = the complete quadrangle, we give explicit equations of the Hirzebruch-Kummer covering $S_n$ (=the minimal desingularisation of $X(n, \mathcal{L})$) in a product of four Fermat curves of degree n. Since $S_n$ is the $(\mathbb{Z}/n)^5$ covering of the Del Pezzo surface $Y_5$ of degree 5 branched on the 10 lines, these equations are derived from explicit equations …
We study threefolds fibred by Kummer surfaces associated to products of elliptic curves, that arise as resolved quotients of threefolds fibred by certain lattice polarized K3 surfaces under a fibrewise Nikulin involution. We present a general construction for such surfaces, before specializing our results to study Calabi-Yau threefolds arising as resolved quotients of threefolds fibred by mirror quartic K3 surfaces. Finally, we give some geometric properties of the Calabi-Yau threefolds that we have constructed, including expressions for Hodge numbers.
We develop a power structure over the Grothendieck ring of varieties relative to an abelian monoid, which allows us to compute the motivic class of the generalized Kummer scheme. We obtain a generalized version of Cheah's formula for the Hilbert scheme of points, which specializes to Gulbrandsen's conjecture for Euler characteristics. Moreover, in the surface case we prove a conjecture of Göttsche for geometrically ruled surfaces, and we obtain an explicit formula for the virtual motive of the generalized Kummer scheme in dimension three.
The hyper-Kummer construction
本文介绍超库默尔构造，它是经典库默尔构造的高维类似物。通过该构造关联超凯勒流形，证明了一些平行于经典结果的结论，还提出构造方法，比较了相关流形，证明了多个猜想，产生了丰富的\(K3\)曲面配置并证明了周动机阿贝尔性等。
We relate the Brauer group of a Kummer surface to the Brauer group of the corresponding abelian surface. For many pairs of elliptic curves over the rational numbers we prove that the Kummer surface attached to their product has trivial Brauer group.
Elliptic constant cycle curves on Kummer surfaces
本文通过超越中间 Jacobian 计算 Kummer 曲面上椭圆常循环曲线的阶，并证明任意正整数都可实现为某 K3 曲面上常循环曲线的阶。
In this work we study the Humbert-Edge's curves of type 5, defined as a complete intersection of four diagonal quadrics in $\mathbb{P}^5$. We characterize them using Kummer surfaces and using the geometry of these surfaces we construct some vanishing thetanulls on such curves. In addition, we describe an argument to give an isomorphism between the moduli space of Humbert-Edge's curves of type 5 and the moduli space of hyperelliptic curves of genus 2, and we let see how this argument can be generalized to state an isomorphism between the moduli space of hyperelliptic curves of genus $g=\frac{n-…
The good reduction of generalized Kummer surfaces in the non-supersingular case
本文针对非超奇异情形，建立了阿贝尔曲面带循环群作用时广义Kummer曲面好约化的判别准则，推广了前人关于Kummer曲面的相关结果。
We construct a topological embedding of the maximal connected component of Bridgeland stability conditions of a (twisted) Abelian surface into the distinguished connected component of the stability manifold of the associated (twisted) Kummer surface. We use methods developed for orbifold conformal field theories.
For an Abelian surface $A$ with a symplectic action by a finite group $G$, one can define the partition function for $G$-invariant Hilbert schemes \[Z_{A, G}(q) = \sum_{d=0}^{\infty} e(\text{Hilb}^{d}(A)^{G})q^{d}.\] We prove the reciprocal $Z_{A,G}^{-1}$ is a modular form of weight $\frac{1}{2}e(A/G)$ for the congruence subgroup $Γ_{0}(|G|)$, and give explicit expressions in terms of eta products. Refined formulas for the $χ_{y}$-genera of $\text{Hilb}(A)^{G}$ are also given. For the group generated by the standard involution $τ: A \to A$, our formulas arise from the enumerative geometry of t…
We show that a Hilbert scheme of conics on a Fano fourfold double cover of $\mathbb{P}^2\times\mathbb{P}^2$ ramified along a divisor of bidegree $(2,2)$ admits a $\mathbb{P}^1$-fibration with base being a hyper-Kähler fourfold. We investigate the geometry of such fourfolds relating them with degenerated EPW cubes, with elements in the Brauer groups of $K3$ surfaces of degree $2$, and with Verra threefolds studied in [Ver04]. These hyper-Kähler fourfolds admit natural involutions and complete the classification of geometric realizations of anti-symplectic involutions on hyper-Kähler $4$-folds o…
Given an abelian surface $A$, defined over a discrete valuation field and having good reduction, does the attached Kummer surface $\mathrm{Km}(A)$ also have good reduction? In this paper we give an affirmative answer in the extreme case, that is, when the abelian surface has supersingular reduction in characteristic $2$.
We study the geometry of Nieto's quintic threefold (Barth & Nieto, J. Alg. Geom. 3, 1994) and the Kummer and abelian surfaces that correspond to special loci.
We discuss several geometric features of a Kummer surface associated with a (1,2)-polarized abelian surface defined over the field of complex numbers. In particular, we show that any such Kummer surface can be modeled as the double cover of the projective plane branched along six lines, three of which meet a common point. The proof uses certain explicit pencils of plane quartic bielliptic genus-three curves whose associated Prym varieties are naturally (1,2)-polarized abelian surfaces.
We study effective bounds for Brauer groups of Kummer surfaces associated to the Jacobians of curves of genus $2$ defined over number fields.
For an abelian surface $A$, we explicitly construct two new families of stable vector bundles on the generalized Kummer variety $K_n(A)$ for $n\geqslant 2$. The first is the family of tautological bundles associated to stable bundles on $A$, and the second is the family of the "wrong-way" fibers of a universal family of stable bundles on the dual abelian surface $\widehat{A}$ parametrized by $K_n(A)$. Each family exhibits a smooth connected component in the moduli space of stable bundles on $K_n(A)$, which is holomorphic symplectic but not simply connected, contrary to the case of K3 surfaces.
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