Alessandra Sarti
In a previous paper, math.AG/0409419, we described six families of K3-surfaces with Picard-number 19, and we identified surfaces with Picard-number 20. In these notes we classify some of the surfaces by computing their transcendental lattices. Moreover we show that the surfaces with Picard-number 19 are birational to a Kummer surface which is the quotient of a non-product type abelian surface by an involution.
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We develop an algorithm computing the transcendental lattice and the Mordell--Weil group of an extremal elliptic surface. As an example, we compute the lattices of four exponentially large series of surfaces
A note on the primitive cohomology lattice of a projective surface
本文研究射影曲面的本原上同调格，证明在不定情形下它可由相交格和本原丰沛类的自交数确定，并给出 Godeaux 曲面、Kunev 曲面及 Horikawa 曲面等例子，同时讨论了一般型典范极化曲面的负定本原格结果。
The goal of the present paper is two-fold. First, we present a classification of algebraic K3 surfaces polarized by the lattice H+E_8+E_7. Key ingredients for this classification are: a normal form for these lattice polarized K3 surfaces, a coarse moduli space and an explicit description of the inverse period map in terms of Siegel modular forms. Second, we give explicit formulas for a Hodge correspondence that relates these K3 surfaces to principally polarized abelian surfaces. The Hodge correspondence in question underlies a geometric two-isogeny of K3 surfaces.
We treat non-symplectic automorphisms on $K3$ surfaces which act trivially on the Néron-Severi lattice. In this paper, we classify non-symplectic automorphisms of prime-power order, especially 2-power order on $K3$ surfaces, i.e., we describe their fixed locus.
We simplify the usual statement of the Torelli theorem for complex Enriques surfaces, by means of a lattice-theoretic trick. This allows easy proofs of several known results, which previously required intricate arithmetic arguments. The main new result is that the moduli space has contractible universal cover.
A (smooth) K3 surface X defined over a field k of characteristic 0 is called singular if the Néron-Severi lattice NS (X) of X over the algebraic closure of k is of rank 20. Let X be a singular K3 surface defined over a number field F. For each embedding σof F into the complex number field, we denote by T(X^σ) the transcendental lattice of the complex K3 surface X^σobtained from X by σ. For each prime ideal P of F at which X has a supersingular reduction X_P, we define L(X, P) to be the orthogonal complement of NS(X) in NS(X_P). We investigate the relation between these lattices T(X^σ) and L(X,…
We extend the notion of lattice polarization for K3 surfaces to families over a (not necessarily simply connected) base, in a way that gives control over the action of monodromy on the algebraic cycles, and discuss the uses of this new theory in the study of families of K3 surfaces admitting fibrewise symplectic automorphisms. We then give an application of these ideas to the study of Calabi-Yau threefolds admitting fibrations by lattice polarized K3 surfaces.
From the product of two elliptic curves, Shioda and Inose constructed an elliptic $K3$ surface having two $\mathrm{II}^*$ fibers. Its Mordell-Weil lattice structure depends on the morphisms between the two elliptic curves. In this paper, we give a method of writing down generators of the Mordell-Weil lattice of such elliptic surfaces when two elliptic curves are $3$-isogenous. In particular, we obtain a basis of the Mordell-Weil lattice for the singular $K3$ surfaces $X_{[3,3,3]}$, $X_{[3,2,3]}$ and $X_{[3,0,3]}$.
Using our results about Lorentzian Kac--Moody algebras and arithmetic mirror symmetry, we give six series of examples of lattice-polarized K3 surfaces with automorphic discriminant.
The Mordell-Weil lattice of an Inose surface arising from isogenous elliptic curves
该研究针对Inose曲面（含两条$II^{*}$纤维的椭圆K3曲面），提出寻找对应一般次数椭圆曲线同源的有理截面的方法，并给出5次、6次同源生成的Inose曲面的Mordell-Weil格基实例。
Using toric geometry, lattice theory, and elliptic surface techniques, we compute the Picard Lattice of certain K3 surfaces. In particular, we examine the generic member of each of M. Reid's list of 95 families of Gorenstein K3 surfaces which occur as hypersurfaces in weighted projective 3-spaces. The results appear in a multipage table near the end of the paper. As an application, we are able to determine whether the mirror family (in the sense of mirror symmetry for K3 surfaces) for each one is also on Reid's list.
Using results of our papers [19], [20] and [21] about classification of degenerations of Kahlerian K3 surfaces with finite symplectic automorphism groups, we classify Picard lattices of Kahlerian K3 surfaces. By classification we understand classification depending on their possible finite symplectic automorphism groups and their non-singular rational curves if a Picard lattice is negative definite.
On lattice-polarized K3 surfaces
本文对格极化和拟极化K3表面的定义进行了修改，引入了小锥概念并描述了具有ADE奇点的模空间。
Using results (especially see Remark 1.14.7) of our paper "Integral symmetric bilinear forms and some of their applications", 1979, we clarify relation between Kahlerian K3 surfaces and Niemeier lattices. We want to emphasise that all twenty four Niemeier lattices are important for K3 surfaces, not only the one which is related to the Mathieu group.
We present a method of Zariski-van Kampen type for the calculation of the transcendental lattice of a complex projective surface. As an application, we calculate the transcendental lattices of complex singular K3 surfaces associated with an arithmetic Zariski pair of maximizing sextics of type $A_{10}+A_{9}$ that are defined over a real quadratic field and are conjugate to each other over the field of rational numbers.
In this article, we focus on a new perspective of automorphisms of complex 2-tori, reviewing previous works from a lattice-theoretic point of view. In particular, we give a classification of families of symplectic and non-symplectic automorphisms of prime order.
We introduce the BCOV formula for the lattice polarized K3 surfaces. We find that it yields cusp forms expressed by certain eta products for many families of rank 19 lattice polarized K3 surfaces over $\mathbb{P}^{1}$. Moreover, for Clingher-Doran's family of $U\oplus E_{8}(-1)\oplus E_{7}(-1)$-polarized K3 surfaces, we obtain the Igusa cusp forms $χ_{10}$ and $χ_{12}$ from the formula. Inspired by the arithmetic properties of mirror maps studied by Lian-Yau, we also derive the K3 differential operators for all the genus zero groups of type $Γ_{0}(n)_{+}$.
We describe a notion of lattice polarization for rational elliptic surfaces and weak del Pezzo surfaces, and describe the complex moduli of the former and the Kähler cone of the latter. We then propose a version of mirror symmetry relating these two objects, which should be thought of as a form of Fano-LG correspondence. Finally, we relate this notion to other forms of mirror symmetry, including Dolgachev-Nikulin-Pinkham mirror symmetry for lattice polarized K3 surfaces and the Gross-Siebert program.
We prove that the integral polarized Hodge structure on the transcendental lattice of a sextic Fermat surface is decomposable. This disproves a conjecture of Kulikov related to a Hodge theoretic approach to proving the irrationality of the very general cubic fourfold.
In this expository note, we review the standard formulation of mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, and compare this construction to a description of mirror symmetry for K3 surfaces which relies on a sublattice of the Picard lattice. We then show how to combine information about the Picard group of a toric ambient space with data about automorphisms of the toric variety to identify families of K3 surfaces with high Picard rank.
Using the theory of holes of the Leech lattice and Borcherds method for the computation of the automorphism group of a K3 surface, we give an effective bound for the set of isomorphism classes of projective models of fixed degree for certain K3 surfaces.
We construct the equivariant analytic lattice cohomology associated with the analytic type of a complex normal surface singularity whenever the link is a rational homology sphere. It is the categorification of the equivariant geometric genus of the germ. This is the analytic analogue of the topological lattice cohomology, associated with the link of the germ, and indexed by the spin$^c$--structures of the link (which is a categorification of the Seiberg--Witten invariant and conjecturally it is isomorphic with the Heegaard Floer cohomology).
In this paper we investigate two stratifications of the moduli space of elliptically fibred K3 surfaces. The first comes from Shimada's classification of connected components of elliptically fibred K3 surfaces and is closely related to the root lattice of the fibration. The second is the monodromy stratification defined by Bogomolov, Petrov and Tschinkel. The main result of the paper is a classification of all positive-dimensional ambi-typical strata, that is strata which are both Shimada root strata and monodromy strata. We further discuss the connection with moduli spaces of lattice-polarise…
We construct a K3 surface over an algebraically closed field of characteristic 2 which contains two sets of 21 disjoint smooth rational curves such that each curve from one set intersects exactly 5 curves from the other set. This configuration is isomorphic to the configuration of points and lines on the projective plane over the finite field of 4 elements. The surface admits a finite automorphism group which contains the group of automorphisms of the plane which acts on the configuration of each set of 21 smooth rational curves, and the additional element of order 2 which interchanges the two…
On Néron-Severi lattices of Jacobian elliptic K3 surfaces
本文分类了有限自同构群 K3 曲面及无限自同构群且 Néron-Severi 格为 2-初等情形下具有有限 Mordell-Weil 群的 Jacobian 椭圆纤维化，并计算了相应格论重数。
The aim of this article is to show that the transpose-dual pairs in the sense of Ebeling-Ploog of singularities $(Z_{1,0},\, Z_{1,0})$, $(U_{1,0},\, U_{1,0})$, $(Q_{17},\, Z_{2,0})$, $(W_{1,0},\, W_{1,0})$ that are concluded to be polytope-dual by the author are actually lattice-dual.
We construct the analytic lattice cohomology associated with the analytic type of any complex normal surface singularity. It is the categorification of the geometric genus of the germ, whenever the link is a rational homology sphere. It is the analytic analogue of the topological lattice cohomology, associated with the link of the germ whenever it is a rational homology sphere. This topological lattice cohomology is the categorification of the Seiberg--Witten invariant, and conjecturally it is isomorphic with the Heegaard Floer cohomology. We compare the two lattice cohomologies: in some simpl…
For a smooth canonically embedded curve $C$ of genus $9$ together with a pencil $|L|$ of degree $6$, we study the relative canonical resolution of $C\subset X\subset \mathbb{P}^8$, where $X$ is the scroll swept out by the pencil $|L|$. We show that the second syzygy bundle in this resolution of $C\subset X$ is unbalanced. The proof reveals a new geometric connection between the universal Brill--Noether variety $\mathcal{W}^1_{9,6}$ and a moduli space $\mathcal{F}^\mathfrak{h}$ of lattice polarized $K3$ surfaces (for a certain rank $3$ lattice $\mathfrak{h}$). As a by-product we prove the unira…
Let X and Y be supersingular K3 surfaces defined over an algebraically closed field. Suppose that the sum of their Artin invariants is 11. Then there exists a certain duality between their Néron-Severi lattices. We investigate geometric consequences of this duality. As an application, we classify genus one fibrations on supersingular K3 surfaces with Artin invariant 10 in characteristic 2 and 3, and give a set of generators of the automorphism group of the nef cone of these supersingular K3 surfaces. The difference between the automorphism group of a supersingular K3 surface X and the automorp…
Degenerations and Fibrations of K3 Surfaces: Lattice Polarisations and Mirror Symmetry
本文研究K3曲面Tyurin退化及椭圆纤维化的格极化理论，建立兼容现有定义的格极化概念，并探讨两种设置间的镜像对称对应。
We study threefolds fibred by K3 surfaces admitting a lattice polarization by a certain class of rank 19 lattices. We begin by showing that any family of such K3 surfaces is completely determined by a map from the base of the family to the appropriate K3 moduli space, which we call the generalized functional invariant. Then we show that if the threefold total space is a smooth Calabi-Yau, there are only finitely many possibilities for the polarizing lattice and the form of the generalized functional invariant. Finally, we construct explicit examples of Calabi-Yau threefolds realizing each case…
We compute the transcendental lattices of the singular K3 surfaces belonging to three pencils of K3 surfaces, namely the Apéry-Fermi pencil with transcendental lattice $U\oplus \langle 12 \rangle$, the Verrill's pencil with transcendental lattice $U \oplus \langle 6 \rangle$ and another pencil linked to Verrill's pencil with transcendental lattice $U \oplus \langle 24 \rangle$. Many corollaries are deduced. For example, some singular K3 surfaces belong to different pencils or are Kummer surfaces of K3 surfaces of another pencil.
First we solve the problem of finding minimal degree families on toric surfaces by reducing it to lattice geometry. Then we describe how to find minimal degree families on, more generally, rational complex projective surfaces.
Let $L$ be an even, hyperbolic lattice with infinitely many simple $(-2)$-roots. We call $L$ a Borcherds lattice if it admits an isotropic vector with bounded inner product with all the simple $(-2)$-roots. We show that this is the case if and only if $L$ has zero entropy, or equivalently if and only if all symmetries of $L$ preserve some isotropic vector. We obtain a complete classification of Borcherds lattices, consisting of $194$ lattices. In turn this provides a classification of hyperbolic lattices of rank $\ge 5$ with virtually solvable symmetry group. Finally, we apply these general re…
The main result of this paper describes the normalizer of a finite parabolic subgroup of a (possibly infinite) Coxeter group. We use this to compute the automorphism groups of some Lorentzian lattices and K3 surfaces.
This article extends Bayer-Fluckiger's theorem on characteristic polynomials of isometries on an even unimodular lattice to the case where the isometries have determinant $-1$. As an application, we show that the logarithm of every Salem number of degree $20$ is realized as the topological entropy of an automorphism of a nonprojective K3 surface.
Mirror symmetry for lattice-polarized abelian surfaces
本文研究了abelian surfaces的镜像对称性，构建了相应的模空间，并通过Néron-Severi lattices对self-mirror abelian surfaces进行了分类。
A semi-numerical algorithm for the homology lattice and periods of complex elliptic surfaces over the projective line
本文提出一种半数值算法，用于计算复射影直线上椭圆曲面的同调有效基并进行周期积分，从而启发式恢复Néron-Severi格、超越格及Mordell-Weil群与格等代数不变量，并提供SageMath实现。
We study the class of complex algebraic K3 surfaces admitting an embedding of H+E8+E8 inside the Neron-Severi lattice. These special K3 surfaces are classified by a pair of modular invariants, in the same manner that elliptic curves over the field of complex numbers are classified by the J-invariant. Via the canonical Shioda-Inose structure we construct a geometric correspondence relating K3 surfaces of the above type with abelian surfaces realized as cartesian products of two elliptic curves. We then use this correspondence to determine explicit formulas for the modular invariants.
This paper studies residual finiteness of lattices in the universal cover of $\mathrm{PU}(2,1)$ and applications to the existence of smooth projective varieties with fundamental group a cocompact lattice in $\mathrm{PU}(2,1)$ or a finite covering of it. First, we prove that certain lattices in the universal cover of $\mathrm{PU}(2,1)$ are residually finite. To our knowledge, these are the first such examples. We then use residually finite central extensions of torsion-free lattices in $\mathrm{PU}(2,1)$ to construct smooth projective surfaces that are not birationally equivalent to a smooth co…
On holomorphic conformal structures associated with lattice polarized K3 surfaces
本文探讨了格极化K3曲面族的Picard-Fuchs方程与周期域上可积全纯共形结构的联系，并计算了与一般Jacobian Kummer曲面相关的局部共形平坦全纯度量显式例子。
A note on Seshadri constants, Gromov widths of toric surfaces and lattice widths of polygons
本文利用矩多边形的格宽度研究辛环面 4 维流形的 Seshadri 常数与 Gromov 宽度，并建立两者宽度比的精确下界，从而回答前人提出的问题。
We explicitly construct modular forms on a $4$-dimensional bounded symmetric domain of type $IV$ based on the variation of the Hodge structures of $K3$ surfaces. We study the ring of our modular forms. Because of the Kneser conditions of the transcendental lattice of our family of $K3$ surfaces, our modular group has a good arithmetic property. Also, our results can be regarded as natural extensions of classical Siegel modular forms from the viewpoint of $K3$ surfaces.
A compact complex surface with positive definite intersection lattice is either the projective plane or a false projective plane. If the intersection lattice is negative definite, the surface is either a non-minimal secondary Kodaira surface, a non-minimal elliptic surface with $b_1=1$, or a class VII surface with $b_2>0$. In all cases the lattice is odd and diagonalizable over the integers.
Rational curves on lattice-polarised K3 surfaces
本文通过退化技术证明秩为2的K3格极化曲面上存在整节点有理曲线，并推广至更高秩格情形。
We use lattice theory to study the isogeny class of a K3 surface. Starting from isotropic Brauer classes, we construct isogenies via Kneser method of neighboring lattices. We also determine the fields of definition of isogenous K3 surfaces, and study Kneser construction over number fields. We then apply our results to relate conjectures about the finiteness of Brauer groups and Néron-Severi lattices of K3 surfaces.
The alternating group of degree 6 is located at the junction of three series of simple non-commutative groups : simple sporadic groups, alternating groups and simple groups of Lie type. It plays a very special role in the theory of finite groups. We shall study its new roles both in a finite geometry of certain pentagon in the Leech lattice and also in a complex algebraic geometry of $K3$ surfaces.
The aim of this paper is to give necessary and sufficient conditions for an integral polynomial to be the characteristic polynomial of a semi-simple isometry of some even unimodular lattice of given signature. This result has applications applications to automorphisms of K3 surfaces; in particular, we show that every Salem number of degree 4, 6, 8,1 2, 14 or 16 is the dynamical degree of an automorphism of a non-projective K3 surface.
Reflective hyperbolic 2-elementary lattices, K3 surfaces and hyperkahler varieties
本文计算大反射偶2-初等双曲格的Coxeter图及其极大抛物子图，并将其应用于K3曲面和超凯勒簇理论。
We establish a relationship between mirror symmetry for K3 surfaces and Arnold's strange duality for K3 surfaces. We compute various examples of mirror families. Among them the mirror moduli family for the moduli space of degree 2n polarized K3 surfaces. It turns out to be related to the moduli space of elliptic curves with level n.
A novel surface interrogation technique is proposed to compute the intersection of curves with spline surfaces in isogeometric analysis. The intersection points are determined in one-shot without resorting to a Newton-Raphson iteration or successive refinement. Surface-curve intersection is required in a wide range of applications, including contact, immersed boundary methods and lattice-skin structures, and requires usually the solution of a system of nonlinear equations. It is assumed that the surface is given in form of a spline, such as a NURBS, T-spline or Catmull-Clark subdivision surfac…
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