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 $dim NS_m(X)=1$ if and only if $dim NS_m(F)=1$. When $X$ is a threefold (i.e. $m=2$) we deduce a new characterization for the factoriality of $X$, i.e. that $X$ is factorial if and only if $dim NS_2(F)=1$. This allows us to give examples of factorial threefolds, in some case with many singularities. During the proof of the announced results, we show that the quotient of the middle cohomology of $F$ by the cycle classes coming from $X$ is irreducible under the monodromy action induced by the hypersurfaces of degree $n$ containing $X$. As consequences we deduce a Noether-Lefschetz Theorem for a projective complete intersection with isolated singularities, and, also using a recent result on codimension two Hodge conjecture, in the case $X\subset \Ps^{5}$ is a threefold as before, we deduce that the general hypersurface $F$ of degree $n$ containing $X$ verifies Hodge conjecture.
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.
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.
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…
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.
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 …
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…
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 prove that the Klein cubic threefold $F$ is the only smooth cubic threefold which has an automorphism of order 11. We compute the period lattice of the intermediate Jacobian of $F$ and study its Fano surface $S$. We compute also the set of fibrations of $S$ onto a curve of positive genus and the intersection between the fibres of these fibrations. These fibres generate an index 2 sub-group of the Néron-Severi group and we obtain a set of generators of this group. The Néron-Severi group of $S$ has rank $25=h^{1,1}$ and discriminant $11^{10}$.
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.
Let $G$ be a finite abelian group which acts symplectically on a K3 surface. The Néron-Severi lattice of the projective K3 surfaces admitting $G$ symplectic action and with minimal Picard number is computed by Garbagnati and Sarti. We consider a $4$-dimensional family of projective K3 surfaces with $\mathbb{Z}_2^2$ symplectic action which do not fall in the above cases. If $X$ is one of these K3 surfaces, then it arises as the minimal resolution of a specific $\mathbb{Z}_2^3$-cover of $\mathbb{P}^2$ branched along six general lines. We show that the Néron-Severi lattice of $X$ with minimal Pic…
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.
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.
Let $X$ be a general hypersurface of degree $md$ in the weighted projective space with weights $1,1,1,m$ for some for $d\geq 2$ and $m\geq 3$. We prove that the Seshadri constant of the ample generator of the Néron-Severi space at a general point $x\in X$ lies in the interval $\left[\sqrt{d}- \frac d m, \sqrt{d}\right]$ and thus approaches the possibly irrational number $\sqrt d$ as $m$ grows.
For every supersingular $K3$ surface $X$ in characteristic 2, there exists a homogeneous polynomial $G$ of degree 6 such that $X$ is birational to the purely inseparable double cover of a projective plane defined by $w^2=G$. We present an algorithm to calculate from $G$ a set of generators of the numerical Néron-Severi lattice of $X$. As an application, we investigate the stratification defined by the Artin invariant on a moduli space of supersingular $K3$ surfaces of degree 2 in characteristic 2.
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 study a family of surfaces of general type that arises from the intersections of two translates of the theta divisor on a principally polarized complex abelian fourfold. In particular we determine the Néron-Severi lattices of these surfaces and show that for a general abelian fourfold the monodromy group of the associated variation of Hodge structures is a subgroup of index at most two in the Weyl group W(E_6), whereas for a Jacobian variety it degenerates to a subgroup of the symmetric group S_6.
We prove that the degree of a nonconstant morphism from a smooth projective 3-fold $X$ with Néron-Severi group ${\bf Z}$ to a smooth 3-dimensional quadric is bounded in terms of numerical invariants of $X$. In the special case where $X$ is a 3-dimensional cubic we show that there are no such morphisms. The main tool in the proof is Miyaoka's bound on the number of double points of a surface.
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 del Pezzo surface of degree 1, and let G be the simple Lie group of type E_8. We construct a locally closed embedding of a universal torsor over X into the G-orbit of the highest weight vector of the adjoint representation. This embedding is equivariant with respect to the action of the Neron-Severi torus T of X identified with a maximal torus of G extended by the group of scalars. The T-invariant hyperplane sections of the torsor defined by the roots of G are inverse images of the 240 exceptional curves on X. This extends the main result of our previous work to del Pezzo surfaces o…
A K3-surface is a (smooth) surface which is simply connected and has trivial canonical bundle. In these notes we investigate three particular pencils of K3-surfaces with maximal Picard number. More precisely the general member in each pencil has Picard number 19 and each pencil contains four surfaces with Picard number 20. These surfaces are obtained as the minimal resolution of quotients X/G, where $G\subset SO(4,\R)$ is some finite subgroup and $X\subset ¶_3(\C)$ denotes a G-invariant surface. The singularities of X/G come from fix points of G on X or from singularities of X. In any case the…
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…
Maschke's Calabi-Yau threefold is the double cover of projective three space branched along Maschke's octic surface. This surface is defined by the lowest degree invariant of a certain finite group acting on a four dimensional vector space. Using this group, we show that the middle Betti cohomology group of the threefold decomposes into the direct sum of 150 two-dimensional Hodge substructures. We exhibit one dimensional families of rational curves on the threefold and verify that the associated Abel-Jacobi map is non-trivial. By counting the number of points over finite fields, we determine t…
Let $C\subset \mathbb P^r$ be an integral projective curve. We define the speciality index $e(C)$ of $C$ as the maximal integer $t$ such that $h^0(C,ω_C(-t))>0$, where $ω_C$ denotes the dualizing sheaf of $C$. In the present paper we consider $C\subset \mathbb P^5$ an integral degree $d$ curve and we denote by $s$ the minimal degree for which there exists a hypersurface of degree $s$ containing $C$. We assume that $C$ is contained in two smooth hypersurfaces $F$ and $G$, with $deg(F)=n>k=deg (G)$. We assume additionally that $F$ is Noether-Lefschetz general, i.e. that the $2$-th Néron-Severi g…
Let $X$ be a projective variety admitting a polarized (or more generally, int-amplified) endomorphism. We show: there are only finitely many contractible extremal rays; and when $X$ is $\mathbb{Q}$-factorial normal, every minimal model program is equivariant relative to the monoid $SEnd(X)$ of all surjective endomorphisms, up to finite index. Further, when $X$ is rationally connected and smooth, we show: there is a finite-index submonoid $G$ of $SEnd(X)$ such that $G$ acts via pullback as diagonal (and hence commutative) matrices on the Neron-Severi group; the full automorphisms group $Aut(X)$…
For a K3 surface over an algebraically closed field of odd characteristic, the representation of the automorphism group on the global two forms is finite. If the K3 surface is supersingular, it is isomorphic to the representation on the discriminant group of the Neron-Severi group. If the K3 surface is of finite height, the representation on the (etale or crystalline) transcendental cycles is also finite and there is a canonical projection from the representation on the transcendental cycles to the representation on the two forms. We prove that, if the base field is an algebraic closure of a f…
Let $X$ be a projective variety over an algebraically closed field $k$ of arbitrary characteristic $p \ge 0$. A surjective endomorphism $f$ of $X$ is $q$-polarized if $f^\ast H \sim qH$ for some ample Cartier divisor $H$ and integer $q > 1$. Suppose $f$ is separable and $X$ is $\mathbb{Q}$-Gorenstein and normal. We show that the anti-canonical divisor $-K_X$ is numerically equivalent to an effective $\mathbb{Q}$-Cartier divisor, strengthening slightly the conclusion of Boucksom, de Fernex and Favre (Theorem C) and also covering singular varieties over an algebraically closed field of arbitrary…
