L. Barbieri-Viale, A. Rosenschon, V. Srinivas
We prove a Lefschetz (1,1)-Theorem for proper seminormal varieties over the complex numbers. The proof is a non-trivial geometric argument applied to the isogeny class of the Lefschetz 1-motive associated to the mixed Hodge structure on H^2.
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For a binary quartic form $ϕ$ without multiple factors, we classify the quartic K3 surfaces $ϕ(x,y)=ϕ(z,t)$ whose Neron-Severi group is (rationally) generated by lines. For generic binary forms $ϕ$, $ψ$ of prime degree without multiple factors, we prove that the Neron-Severi group of the surface $ϕ(x,y)=ψ(z,t)$ is rationally generated by lines.
Let $Z$ be a closed subscheme of a smooth complex projective complete intersection variety $Y\subseteq \Ps^N$, with $dim Y=2r+1\geq 3$. We describe the Néron-Severi group $NS_r(X)$ of a general smooth hypersurface $X\subset Y$ of sufficiently large degree containing $Z$.
In this paper we extend the well known theorem of Angelo Lopez concerning the Picard group of the general space projective surface containing a given smooth projective curve, to the intermediate Néron-Severi group of a general hypersurface in any smooth projective variety.
P. Stiller computed the Picard numbers of several families of elliptic surfaces, the rank of the Néron-Severi groups of these surfaces. However he did not give the generators of these groups. In this paper we give $\mathbb Q$-bases of these groups explicitly. If these surfaces are rational, then we also show that they are $\mathbb Z$-bases.
Let $X \hookrightarrow \mathbb{P}^r$ be a smooth projective variety defined by homogeneous polynomials of degree $\leq d$. We give explicit upper bounds on the order of the torsion subgroup $(\mathrm{NS} \, X)_{\mathrm{tor}}$ of the Néron-Severi group of $X$. The bounds are derived from an explicit upper bound on the number of irreducible components of either the Hilbert scheme $\mathbf{Hilb}_Q X$ or the scheme $\mathbf{CDiv}_n X $parametrizing the effective Cartier divisors of degree $n$ on $X$. We also give an upper bound on the number of generators of $(\mathrm{NS} \, X)[\ell^\infty]$ unifo…
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。
Let $X \hookrightarrow \mathbb{P}^r$ be a smooth projective variety defined by homogeneous polynomials of degree $\leq d$ over an algebraically closed field. Let $\mathbf{Pic}\, X$ be the Picard scheme of $X$. Let $\mathbf{Pic}^0 X$ be the identity component of $\mathbf{Pic}\, X$. The Néron--Severi group scheme of $X$ is defined by $\mathbf{NS}\, X = (\mathbf{Pic}\, X)/(\mathbf{Pic}^0 X)_{\mathrm{red}}$. We give an explicit upper bound on the order of the finite group scheme $(\mathbf{NS}\, X)_{\mathrm{tor}}$ in terms of $d$ and $r$. As a corollary, we give an upper bound on the order of the f…
We work out properties of smooth projective varieties over a (not necessarily algebraically closed) field that admit collections of objects in the bounded derived category of coherent sheaves that are either full exceptional, or numerically exceptional of maximal length. Our main result gives a necessary and sufficient condition on the Neron-Severi lattice for a smooth projective surface with holomorphic Euler characteristic 1 to admit a numerically exceptional collection of maximal length, consisting of line-bundles. As a consequence we determine exactly which complex surfaces with vanishing …
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…
Assuming the Tate conjecture and the computability of étale cohomology with finite coefficients, we give an algorithm that computes the Néron-Severi group of any smooth projective geometrically integral variety, and also the rank of the group of numerical equivalence classes of codimension p cycles for any p.
Let X be a smooth complex projective variety. The group of autoequivalences of the derived category of X acts naturally on its singular cohomology H(X, Q) and we denote by Geq(X) the Zariski closure of its image in Gl(H(X, Q)). We study the relation of the Lie algebra LieGeq(X) and the Neron-Severi Lie algebra gNS(X) in case X has trivial canonical line bundle. At the same time for mirror symmetric families of (weakly) Calabi-Yau varieties we consider a conjecture of Kontsevich on the relation between the monodromy of one family and the group Geq(X) for a very general member X of the other fam…
We give a natural parameterization of the Néron-Severi group of a product $A = E\times E'$ of two elliptic curves in terms of quadratic forms. As an application, we determine (in the non-CM case) whether $A$ contains a smooth curve of any fixed genus. We also determine whether $A$ admits a very ample line bundle of any fixed degree. In particular, we determine which of these abelian surfaces embed in $\mathbb{P}^4$, i.e. which come from the Horrocks-Mumford bundle.
Néron--Severi groups of proper schemes over finite fields
研究有限域上恰当概型的奈龙 - 塞维里群，通过证明第一陈类映射，将\(\text{NS}(\overline{X})\otimes\mathbb{Z}_{\ell}\)与特定扎里斯基局部平凡类群等同，得到有限域类似定理，无需半正规性和不可约性。
André and Maulik--Poonen proved that for any smooth proper family $X\to B$ of varieties over an algebraically closed field of characteristic $0$, there is a closed fiber whose Néron-Severi group has the same rank as that of the Néron-Severi group of the geometric generic fiber. We prove the analogous statement over algebraically closed fields of characteristic $p>0$ which are not isomorphic to $\overline{\mathbb{F}}_p$. Furthermore, we prove that for any algebraically closed field $k$ of characteristic $p>0$ and smooth proper family $X\to B$ of $k$-varieties, there exists a dense open subvarie…
For a K3 surface of finite height over a field of odd characteristic, there exists a smooth lifting to the ring of Witt vectors such that the reduction map from the Picard group of the generic fiber to the Picard group of the special fiber is isomorphic. In this paper, using this result, we give a criterion for a K3 surface of finite height over a field of odd characteristic to be an Enriques K3 surface or to be a K3 surface in terms of the Neron-Severi lattice. Then we show every Kummer surface has an Enriques involution. We also give a classification of K3 surfaces of Picard rank of 20 over …
Brauer and Neron-Severi groups of surfaces over finite fields
本文给出了不假设Tate猜想的有限域上曲面的Artin-Tate公式版本，建立了Brauer群与Néron-Severi群相关项的等式，并据此给出了依赖于p秩的阿贝尔曲面Brauer群大小的精确估计。
We apply Tate's conjecture on algebraic cycles to study the Néron-Severi groups of varieties fibered over a curve. This is inspired by the work of Rosen and Silverman, who carry out such an analysis to derive a formula for the rank of the group of sections of an elliptic surface. For a semistable fibered surface, under Tate's conjecture we derive a formula for the rank of the group of sections of the associated Jacobian fibration. For fiber powers of a semistable elliptic fibration $E --> C$, under Tate's conjecture we give a recursive formula for the rank of the Néron-Severi groups of these f…
It is known that K3 surfaces S whose Picard number rho (= rank of the Neron-Severi group of S) is at least 19 are parametrized by modular curves X, and these modular curves X include various Shimura modular curves associated with congruence subgroups of quaternion algebras over Q. In a family of such K3 surfaces, a surface has rho=20 if and only if it corresponds to a CM point on X. We use this to compute equations for Shimura curves, natural maps between them, and CM coordinates well beyond what could be done by working with the curves directly as we did in ``Shimura Curve Computations'' (199…
For a projective variety $Z$ and for any integer $p$, define the $p$-th Néron-Severi group $NS_p(Z)$ of $Z$ as the image of the cycle map $A_{p}(Z)\to H_{2p}(Z; \mathbb{C})$. Now let $X\subset \Ps^{2m+1}$ ($m\geq 1$) be a projective variety of dimension $2m-1$, with isolated singularities, complete intersection of a smooth hypersurface of degree $k$, with a hypersurface of degree $n>max\{k, 2m+1\}$, and let $F$ be a general hypersurface of degree $n$ containing $X$. We prove that the natural map $NS_m(X)\to NS_m(F)$ is surjective, and that if $dim NS_m(F)=1$ then $dim NS_m(X)=1$. In particular…
In this paper we define certain analogues of the volume of a divisor - called asymptotic cohomological functions - and investigate their behaviour on the Neron--Severi space. We establish that asymptotic cohomological functions are invariant with respect to the numerical equivalence of divisors, and that they give rise to continuous functions on the real Neron--Severi space. To illustrate the theory, we work out these invariants for abelian varieties, smooth surfaces, and certain homogeneous spaces.
We introduce a new vector space associated to projective variety, the Weil Neron-Severi space, which we show is finitely generated and contains the usual Neron-Severi space as a subspace. We define the Nef cone of Weil divisor and the cone of Weil curves. We study these cones, and prove a new Cone theorem. We use this theorem to propose a Minimal Model Program without flips.
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 classify K3 surfaces with non-symplectic automorphism of order 16 in full generality. We show that the fixed locus contains only rational curves and points and we completely classify the seven possible configurations. If the Néron-Severi group has rank 6, there are two possibilities and if its rank is 14, there are five possibilities. In particular if the action of the automorphism is trivial on the Néron-Severi group, then we show that its rank is six.
André used Hodge-theoretic methods to show that in a smooth proper family X to B of varieties over an algebraically closed field k of characteristic 0, there exists a closed fiber having the same Picard number as the geometric generic fiber, even if k is countable. We give a completely different approach to André's theorem, which also proves the following refinement: in a family of varieties with good reduction at p, the locus on the base where the Picard number jumps is nowhere p-adically dense. Our proof uses the ``p-adic Lefschetz (1,1) theorem'' of Berthelot and Ogus, combined with an anal…
We study tetrahedral quartics in projective space. We address their projective geometry, Neron-Severi lattice and automorphism group.
We give a brief systematic overview of a few results concerning the Néron--Severi lattices of Fermat varieties and Delsarte surfaces.
Chai's conjecture for semiabelian Jacobians
本文证明Chai关于半阿贝尔簇基变换导子可加性的猜想在正常曲线雅可比簇情形成立，推广了Raynaud构造，并证明了完美剩余域下Néron模型存在性的两个猜想。
The space of Hodge cycles of the general Prym variety is proved to be generated by its Neron-Severi group.
Given a principally polarized abelian variety $(A,Θ)$, we give a characterization of all elliptic curves that lie on $A$ in terms of intersection numbers of divisor classes in its Néron-Severi group.
Galois-invariant Néron--Severi ranks of Fermat surfaces over number fields: a Galois module, closed forms, a threshold, and exact tables
研究数域上费马曲面的伽罗瓦不变奈龙 - 塞维里秩，通过显式特征平均计算，给出\(\gcd(d,6)=1\)时封闭形式及\(4\leq d\leq30\)的有理秩，证明相关公式，建立阈值和轨道规则，给出精确表格及平均阶，结果无条件且有辅助文件支持。
Specialization of Néron-Severi groups in positive characteristic
本文研究正特征下Néron-Severi群的特化问题，利用晶体上同调的变分Tate猜想等工具证明了几何Picard秩在大量闭点处与generic纤维一致，推广了特征零的相关结果，具有多方面应用。
We determine all complex K3 surfaces with Picard rank 20 over Q. Here the Neron-Severi group has rank 20 and is generated by divisors which are defined over Q. Our proof uses modularity, the Artin-Tate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that Mordell-Weil rank 18 over Q is impossible for an elliptic K3 surface. We then apply our methods to general singular K3 surfaces, i.e. with Neron-Severi group of rank 20, but not necessarily generated by divisors over Q.
We show that, as in de Rham cohomology over the complex numbers, the value of the entropy of an automorphism of the surface over a finite field $\F_q $ is taken on the span of the Néron-Severi group inside of $\ell$-adic cohomology. v2: (some) typos removed, exposition (partly) improved.
For a smooth complex projective variety, the rank of the Néron-Severi group is bounded by the Hodge number h^{1,1}. Varieties with rk NS = h^{1,1} have interesting properties, but are rather sparse, particularly in dimension 2. We discuss in this note a number of examples, in particular those constructed from curves with special Jacobians.
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。
Let $Z$ be a closed subscheme of a smooth complex projective variety $Y\subseteq \Ps^N$, with $\dim\,Y=2r+1\geq 3$. We describe the intermediate Néron-Severi group (i.e. the image of the cycle map $A_r(X)\to H_{2r}(X;\mathbb{Z})$) of a general smooth hypersurface $X\subset Y$ of sufficiently large degree containing $Z$.
We derive a characterization of the complex projective K3 surfaces which have automorphisms of positive entropy in term of their Néron-Severi lattices. Along the way, we classify the projective K3 surfaces of zero entropy with infinite automorphism groups and we determine the projective K3 surfaces of Picard number at least five with almost abelian automorphism groups, which gives an answer to a long standing question of Nikulin.
The Abstract for this revised version is the following: Let k be a number field, let C be a smooth, projective and geometrically integral k-curve and let p:X --> C be a Severi-Brauer k-fibration of squarefree index. Various authors have studied the cokernel of the natural map CH_{0}(X/C)-->\bigoplus_{v}CH_{0}(X_{v}/C_{v}), where CH_{0}(X/C) is the kernel of p_{*}:CH_{0}(X)-->CH_{0}(C). In this paper I obtain an exact sequence which relates the Tate-Shafarevich group of the kernel of the above map to the Tate-Shafarevich group of the Neron-Severi torus of X. I then obtain conditions under which…
We prove Bertini type theorems for the inverse image, under a proper morphism, of any Schubert variety in an homogeneous space. Using generalisations of Deligne's trick, we deduce connectedness results for the inverse image of the diagonal in $X^2$ where $X$ is any isotropic grassmannian. We also deduce simple connectedness properties for subvarieties of $X$. Finally we prove transplanting theorems {à} la Barth-Larsen for the Picard group of any isotropic grassmannian of lines and for the Neron-Severi group of some adjoint and coadjoint homogeneous spaces.
Remark on a theorem of Oguiso
本文证明Oguiso本原性判据的一个变体，并将其用于验证Oguiso原始判据无法处理的Calabi-Yau三维流形上某类双有理自同构的本原性。
Let X be a complex algebraic K3 surface or a supersingular K3 surface in odd characteristic. We present an algorithm by which, under certain assumptions on X, we can calculate a finite set of generators of the image of the natural homomorphism from the automorphism group of X to the orthogonal group of the Néron-Severi lattice of X. We then apply this algorithm to certain complex K3 surfaces, among which are singular K3 surfaces whose transcendental lattices are of small discriminants.
Densities of Picard Rank Jumps for K3 Surfaces Over Global Fields
本文给出全局域上K3曲面约化处Picard秩跳跃的精确密度公式，并证明存在无穷多个约化具有固定平方类的Néron-Severi元素，且一般模点对应的密度为零。
Let X be as smooth complex projective variety with Neron-Severi group isomorphic to Z, and D an irreducible divisor with normal crossing singularities. Assume r is equal to 2 or 3. We prove that if the fundamental group of X doesn't have irreducible PU(r) representations, then the fundamental group of X-D doesn't have irreducible U(r) representations. The proof uses the non-existence of certain stable parabolic bundles. We also obtain a similar result for GL(2) when D is smooth and X is a complex surface.
To a smooth projective variety $X$ whose Chow group of $0$-cycles is $\mathbf Q$-universally trivial one can associate its torsion index $\mathrm{Tor}(X)$, the smallest multiple of the diagonal appearing in a cycle-theoretic decomposition à la Bloch-Srinivas. We show that $\mathrm{Tor}(X)$ is the exponent of the torsion in the Néron-Severi-group of $X$ when $X$ is a surface over an algebraically closed field $k$, up to a power of the exponential characteristic of $k$.
We prove some finiteness theorems for the Picard functor of an algebraic stack, in the spirit of SGA 6, exp. XII and XIII. In particular, we give a stacky version of Raynaud's relative representability theorem, we give sufficient conditions for the existence of the torsion component of the Picard functor, and for the finite generation of the Neron-Severi groups or of the Picard group itself. We give some examples and applications. In an appendix, we prove the semicontinuity theorem for a (non necessarily tame) algebraic stack.
Let $Y$ be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of $\mathbb{Q}$. In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes $\text{Br}\, Y/ \text{Br}_1\, Y$ is finite. We study this quotient for the family of surfaces that are geometrically isomorphic to a product of isogenous non-CM elliptic curves, as well as the related family of geometrically Kummer surfaces; both families can be characterized by their geometric Néron-Severi lattices. Over a field of characteristic $0$, we prove that the existe…
We propose an explicit conjectural lift of the Neron-Severi Lie algebra of a hyperkähler variety $X$ of $K3^{[2]}$-type to the Chow ring of correspondences ${\rm CH}^\ast(X \times X)$ in terms of a canonical lift of the Beauville-Bogomolov class obtained by Markman. We give evidence for this conjecture in the case of the Hilbert scheme of two points of a $K3$ surface and in the case of the Fano variety of lines of a very general cubic fourfold. Moreover, we show that the Fourier decomposition of the Chow ring of $X$ of Shen and Vial agrees with the eigenspace decomposition of a canonical lift …
We prove that the Neron-Severi groups of several complex Fermat surfaces are generated by lines. Specifically, we obtain these new results for all degrees up to 100 that are relatively prime to 6. The proof uses reduction modulo a supersingular prime. The techniques are developed in detail. They can be applied to other surfaces and varieties as well.
Let $X$ be a smooth compact projective variety over $\mathbb C$. Let $H^2(π_1(X),\mathbb R)^{1,1}$ be the intersection of $H^{1,1}(X,{\mathbb R})$ with the image of the map $H^2(π_1(X),{\mathbb R})\to H^2(X)$ induced by the classifying map $X\to Bπ_1(X)$. Let $NS(X)$ be the Néron-Severi group of $X$. Let $[ω]\in H^2(π_1(X),\mathbb R)^{1,1}+ NS(X)\otimes {\mathbb R}$. In this note, we prove that $[ω]$ is the cohomology class of a Kähler metric if and only if for every $d$-dimensional reduced closed algebraic subvariety $Z\subset X$, $[ω]^d.Z>0$.
We prove non-Archimedean analogs of results of Noguchi and Winkelmann showing algebraic degeneracy of rigid analytic maps to projective varieties omitting an effective divisor with sufficiently many irreducible components relative to the rank of the group they generate in the Neron-Severi group of the variety.
On K3 surfaces with hyperbolic automorphism groups
研究Picard数≥6的复射影K3表面的Néron-Severi晶格有限性，基于 genus 一纤维和Kikuta与Takatsu近期工作，证明其有限性。
We introduce the notion of a variety (or more generally a motive) of CM-type which generalises the well known notion of abelian variety of CM-type. Just as in that particular case it will turn out that the cohomology of the variety is determined by purely combinatorial data; the type of the variety. As applications we will show that the ł-adic representations are given by algebraic Hecke characters whose algebraic parts are determined by the type and give a method for computing the discriminant of the Néron-Severi group of super-singular Fermat surfaces.
We consider a Cartier divisor L on a d-dimensional complex projective variety X. It is well-known that the dimensions of the cohomomology groups H^i(X,O_X(mL)) grow at most like m^d, and it is natural to ask when one of these actually has maximal growth. For i = 0, this happens by definition exactly when L is big. Here we focus on the question of when one or more of the higher cohomology groups grows maximally. Our main result is that if one considers also small perturbations of the divisor in question, then the maximal growth of higher cohomology characterizes non-ample divisors. This criteri…
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。
In this paper, we prove, as the complex case, a supersingular K3 surface over a field of odd characteristic has an Enriques involution if and only if there exists a primitive embedding of the twice of the Enriques lattice into the Neron-Severi group such that the orthogonal complement of the embedding has no vector of self-intersection -2 using the Crystalline Torelli theorem. By this criterion and some lattice calculation, we prove that when the characteristic of the base field is p=19 or p>23, a superingular K3 surface is an Enriques K3 surface if and only if the Artin invariant is less than…
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,…
Let X be a smooth double cover of a geometrically ruled surface defined over a separably closed field of characteristic different from 2. The main result of this paper is a finite presentation of the 2-torsion in the Brauer group of X with generators given by central simple algebras over the function field of X and relations coming from the Néron-Severi group of X. In particular, the result gives a central simple algebra representative for the unique nontrivial Brauer class on any Enriques surface. An example demonstrating the applications to the study of rational points is given.
Let X be a K3 surface with the Neron-Severi lattice S_X and transcendental lattice T_X. Nukulin considered the kernel H_X of the natural representation Aut(X) ---> O(S_X) and proved that H_{X} is a finite cyclic group with phi(h(X))) | t(X) and acts faithfully on the space H^{2,0}(X) = C omega_{X}, where h(X) = ord(H_X), t(X) = rank T_X and phi(.) is the Euler function. Consider the extremal case where phi(h(X)) = t(X). In the situation where T_{X} is unimodular, Kondo has determined the list of t(X), as well as the actual realizations, and showed that t(X) alone uniquely determines the isomor…
Using the Shioda-Tate theorem and an adaptation of Silverman's specialization theorem, we reduce the specialization of Mordell-Weil ranks for abelian varieties over fields finitely generated over infinite finitely generated fields $k$ to the the specialization theorem for Néron-Severi ranks recently proved by Ambrosi in positive characteristic. More precisely, we prove that after a blow-up of the base surface $S$, for all vertical curves $S_x$ of a fibration $S \to U \subseteq \mathbf{P}^1_k$ with $x$ from the complement of a sparse subset of $|U|$, the Mordell-Weil rank of an abelian scheme o…
Let $X$ be a smooth projective variety over the complex numbers. One knows by the Cone Theorem that the closed cone of curves of $X$ is rational polyhedral whenever $c_1(X)$ is ample. For varieties $X$ such that $c_1(X)$ is not ample, however, it is in general difficult to determine the structure of $\bar NE(X)$. The purpose of this paper is to study the cone of curves of abelian varieties. Specifically, the abelian varieties $X$ are determined such that the closed cone $\bar NE(X)$ is rational polyhedral. The result can also be formulated in terms of the nef cone of $X$ or in terms of the sem…
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