K3 Surfaces Of Genus Research Articles

On some moduli spaces of bundles on $K3$ surfaces, II

We give many examples in which there exist infinitely many divisorial conditions on the moduli space of polarized K3 surfaces $(S,H)$ of degree $H^2=2g-2$, $g \geq 3$, and Picard number $rk N(S)=\rho(S)=2$ such that for a general K3 surface $S$ satisfying these conditions the moduli space of sheaves $M_S(r,H,s)$ is birationally equivalent to the Hilbert scheme $S[g-rs]$ of zero-dimensional subschemes of $S$ of lenght equal to $g-rs$. This result generalizes the main result of \cite{Nik1} when $g=rs+1$ and of \cite{Monat} when $r=s=2$, $g \geq 5$.

PICARD GROUP OF HYPERSURFACES IN TORIC 3-FOLDS

We show that the usual sufficient criterion for a very general hypersurface in a smooth projective manifold to have the same Picard number as the ambient variety can be generalized to quasi-smooth hypersurfaces in complete simplicial toric varieties. This sufficient condition always holds for very general K3 surfaces embedded in Fano toric 3-folds.

Remark on polarized K3 surfaces of genus 36

Smooth primitively polarized K3 surfaces of genus 36 are studied. It is proved that all such surfaces S, for which there exists an embedding \({{\rm R} \hookrightarrow {\rm Pic}(S)}\) of some special lattice R of rank 2, are parameterized up to an isomorphism by some 18-dimensional unirational algebraic variety. More precisely, it is shown that a general S is an anticanonical section of a (unique) Fano threefold with canonical Gorenstein singularities.

On the gonality sequence of an algebraic curve

For any smooth irreducible projective curve $X$, the gonality sequence $\{d_r \;| \; r \in \mathbb N \}$ is a strictly increasing sequence of positive integer invariants of $X$. In most known cases $d_{r+1}$ is not much bigger than $d_r$. In our terminology this means the numbers $d_r$ satisfy the slope inequality. It is the aim of this paper to study cases when this is not true. We give examples for this of extremal curves in $\PP^r$, for curves on a general K3-surface in $\PP^r$ and for complete intersections in $\PP^3$.

Transcendental obstructions to weak approximation on general K3 surfaces

We construct an explicit K3 surface over the field of rational numbers that has geometric Picard rank one, and for which there is a transcendental Brauer–Manin obstruction to weak approximation. To do so, we exploit the relationship between polarized K3 surfaces endowed with particular kinds of Brauer classes and cubic fourfolds.

Modular properties of nodal curves on K3 surfaces

In this article we are going to address the following issues: (1) the first is a rigidity property for pairs (S, C) consisting of a general projective K3 surface S, and a curve C obtained as the normalization of a nodal, hyperplane section \({\hat C\hookrightarrow S}\). We prove that a non-trivial deformation of a pair (S, C) induces a non-trivial deformation of C; (2) the second question concerns the Wahl map of curves C obtained as above. We prove that the Wahl map of the normalization of a nodal curve contained in a general projective K3 surface is non-surjective. In both cases, we impose upper bounds on the number of nodes of \({\hat C}\).

K3 surfaces with Picard rank 20

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 apply our methods to general singular K3 surfaces, that is, those with Neron–Severi group of rank 20, but not necessarily generated by divisors over ℚ.

A Note on Seshadri constants on general K3 surfaces

We prove a lower bound on the Seshadri constant ε ( L ) on a K3 surface S with Pic S ≃ Z [ L ] . In particular, we obtain that ε ( L ) = α if L 2 = α 2 for an integer α. To cite this article: A.L. Knutsen, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Explicit correspondences of a K3 surface with itself

Let be a K3-surface with a polarization of degree , . We consider the moduli space of sheaves over with a primitive isotropic Mukai vector . This space is again a K3-surface. In earlier papers, we gave necessary and sufficient conditions (in terms of the Picard lattice ) for and to be isomorphic. Here we show that these conditions imply the existence of an isomorphism between and which is a composite of certain universal geometric isomorphisms between moduli of sheaves over and Tyurin's geometric isomorphism between moduli of sheaves over and itself. It follows that a general K3-surface with is isomorphic to if and only if there is an isomorphism which is a composite of universal isomorphisms and Tyurin's isomorphism.

Coherent Sheaves on General K3 Surfaces and Tori

Let M be a K3 surface or an even-dimensional compact torus. We show that the category of coherent sheaves on M is independent from the choice of the complex structure, if this complex structure is generic.

Prime Fano threefolds and integrable systems

For a general K3 surface S of genus g, with 2 ≤ g ≤ 10, we prove that the intermediate Jacobians of the family of prime Fano threefolds of genus g containing S as a hyperplane section, form generically an algebraic completely integrable Hamiltonian system.

Normal generation and Clifford index on algebraic curves

For a smooth curve C it is known that a very ample line bundle \({\mathcal{L}}\) on C is normally generated if Cliff(\({\mathcal{L}}\)) < Cliff(C) and there exist extremal line bundles \({\mathcal{L}}\) (:non-normally generated very ample line bundle with Cliff(\({\mathcal{L}}\)) = Cliff(C)) with \({h^{1}(\mathcal{L}) \le 1}\) . However it has been unknown whether there exists an extremal line bundle \({\mathcal{L}}\) with \({h^{1}(\mathcal{L}) \ge 2}\) . In this paper, we prove that for any positive integers (g, c) with g = 2c + 5 and \({c \equiv 0}\) (mod 2) there exists a smooth curve of genus g and Clifford index c carrying an extremal line bundle \({\mathcal{L}}\) with \({h^{1}(\mathcal{L}) = 2}\) . In fact, a smooth quadric hypersurface section C of a general projective K3 surface always has an extremal line bundle \({\mathcal{L}}\) with \({h^{1}(\mathcal{L}) = 2}\) . More generally, if C has a line bundle \({\mathcal{M}}\) computing the Clifford index c of C with \({(3c/2) + 3 < {\deg} \mathcal{M} \leq g-1}\) , then C has such an extremal line bundle \({\mathcal{L}}\).

THE ABELIAN FIBRATION ON THE HILBERT CUBE OF A K3 SURFACE OF GENUS 9

In this paper we construct an abelian fibration over P3on the Hilbert cube of the primitive K3 surface of genus 9. After the abelian fibration constructed by Hassett and Tschinkel on the Hilbert square on the primitive K3 surface of genus 5, this is the second example where the abelian fibration is constructed directly on [Formula: see text]. The recent more general result of Sawon proves the existence of an abelian fibration on the Hilbert scheme [Formula: see text] of a primitive K3 surface S of degree 2g - 2 = m2(2n - 2). Our example provides an alternative proof in the case m = 2, n = 3. Furthermore we identify the general fiber with the Hilbert scheme of twisted cubic curves in a Fano 3-fold of genus 9, and interpret the addition law on this variety.

Nikulin involutions on K3 surfaces

We study the maps induced on cohomology by a Nikulin (i.e. a symplectic) involution on a K3 surface. We parametrize the 11-dimensional irreducible components of the moduli space of algebraic K3 surfaces with a Nikulin involution and we give examples of the general K3 surface in various components. We conclude with some remarks on Morrison–Nikulin involutions, these are Nikulin involutions which interchange two copies of E 8(−1) in the Néron Severi group.

The Hodge-𝒟-conjecture for 𝒦3 and Abelian surfaces

Let X X be a projective algebraic manifold, and CH k ( X , 1 ) \text {CH}^{k}(X,1) the higher Chow group, with corresponding real regulator r k , 1 ⊗ R : CH k ( X , 1 ) ⊗ R → H D 2 k − 1 ( X , R ( k ) ) \text {r}_{k,1}\otimes {{\mathbb R}}: \text {CH}^k(X, 1)\otimes {{\mathbb R}} \to H_{\mathcal D}^{2k-1}(X,{{\mathbb R}}(k)) . If X X is a general K3 surface or Abelian surface, and k = 2 k=2 , we prove the Hodge- D {\mathcal D} -conjecture, i.e. the surjectivity of r 2 , 1 ⊗ R \text {r}_{2,1}\otimes {{\mathbb R}} . Since the Hodge- D {\mathcal D} -conjecture is not true for general surfaces in P 3 \mathbb {P}^{3} of degree ≥ 5 \geq 5 , the results in this paper provide an effective bound for when this conjecture is true.

Some remarks on Brauer groups of K3 surfaces

We discuss the geometry of the genus one fibrations associated to an elliptic fibration on a K3 surface. We show that the two-torsion subgroup of the Brauer group of a general elliptic fibration is naturally isomorphic to the two torsion of the Jacobian of a curve associated to the fibration. We remark that this is related to Recillas’ trigonal construction. Finally we discuss the two-torsion in the Brauer group of a general K3 surface with a polarization of degree two.

Correspondence between K3 surfaces ( With an appendix by Igor Dolgachev)

In this paper we study the existence of correspondences between K3 surfaces X(k, m, n) with k, m, n ∈ N, Picard number 17, and transcendental lattices T(k, m, n) ∼= U(k)⊕U(m)⊕〈−2n〉. In a fundamental paper, Mukai [Mu] showed that correspondences between K3 surfaces exist if the transcendental lattices are Hodge isometric over Q. This construction holds if the Picard number of the surfaces is greater than or equal to 11. Nikulin [N2] later improved this result, obtaining the lower bound 5 for the Picard number. The aim of our work is to realize examples of K3 surfaces with transcendental lattice that are not Hodge isometric but such that a correspondence between them already exists. This in particular implies the existence of an algebraic cycle on the middle cohomology of the product of two surfaces arbitrarily chosen in the constructed family. In Sections 2 and 3 we recall some basic notions and results on lattices and correspondences. In Section 4 we consider a generic genus-2 curve and we show the existence of a correspondence between the Jacobian of the curve and a K3 surface with isomorphic transcendental lattice. Since this construction involves a second K3 surface whose transcendental lattice has quadratic form multiplied by 2, in Sections 5 and 6 we generalize this first example. First, we construct K3 surfaces “twisting” each direct summand of the transcendental lattice of the Jacobian by natural numbers. Then we find correspondences between them using both Mukai’s theorem and Shioda–Inose structures that translate the problem into a problem of looking for isogenies between abelian varieties. In this way we prove in Theorem 6.3 that all the K3 surfaces X(k, m, n) are in correspondence to each other. Finally, in Theorem 6.5 we show the existence of a correspondence between a general K3 surface of Picard rank 17 and a Kummer surface of the same rank having transcendental lattices that are Q-Hodge isomorphic. In the Appendix, Igor Dolgachev realizes a geometric correspondence between the K3 surfaces of Section 4.

Chow groups of surfaces with h2,0⩽1

We will investigate the geometry of rational equivalence classes of points on a surface S. We will show that if S is a general projective K3 surface then these equivalence classes are dense in the complex topology. We will also show that if S has the property that these equivalence classes are Zariski dense, then h 2,0( S)⩽1. To cite this article: C. Maclean, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

$K3$ surfaces of genus 8 and varieties of sums of powers of cubic fourfolds

The main result of this paper is that the variety of presentations of a general cubic form f in 6 variables as a sum of 10 cubes is isomorphic to the Fano variety of lines of a cubic 4-fold F', in general different from F = Z(f). A general K3 surface S of genus 8 determines uniquely a pair of cubic 4folds: the apolar cubic F(S) and the dual Pfaffian cubic F'(S) (or for simplicity F and F'). As Beauville and Donagi have shown, the Fano variety .TF' of lines on the cubic F' is isomorphic to the Hilbert scheme Hilb2 S of length two subschemes of S. The first main result of this paper is that Hilb2 S parametrizes the variety VSP(F, 10) of presentations of the cubic form f, with F = Z(f), as a sum of 10 cubes, which yields an isomorphism between NFF and VSP(F, 10). Furthermore, we show that VSP(F, 10) sets up a (6, 10) correspondence between F' and FF/ . The main result follows by a deformation argument. 1. PFAFFIAN AND APOLAR CUBIC 4-FOLDS ASSOCIATED TO K3 SURFACES OF GENUS 8 1.1. Let V be a 6-dimensional vector space over C. Fix a basis e,... ,e5 for V; then ei A ej for 0 i<j aijei Aej E A2V a skewsymmetric matrix M(g) = (aij), with aji = -aj. With Pliicker coordinates xij, the embedding of the Grassmannian G = G(2, V) in p14 = P(A2V) is then precisely the locus of rank 2 skewsymmetric 6 x 6 matrices / 0 X01 X02 X03 X04 X05 -x0i 0 X12 X13 X14 X15 M -X02 -X12 0 X23 X24 X25 -X03 -X13 -X23 0 X34 X35 -X04 -X14 -X24 -X34 0 X45 -X05 -X15 -X25 -X35 -X45 0 / Since the sum of two rank 2 matrices has rank at most 4, and any rank 4 skewsymmetric matrix is the sum of two rank 2 skewsymmetric matrices, the secant variety of G is the cubic hypersurface K defined by the 6 x 6 Pfaffian m of the matrix M. The dual variety of G in p14 = P(A2V*) is a cubic hypersurface K* = K (cf. [10]). K* is the secant variety of G* = G(V, 2), the Grassmannian of rank 2 quotient spaces of V, and of course G* C G. Received by the editors July 5, 1999. 2000 Mathematics Subject Classification. Primary 14J70; Secondary 14M15, 14N99. (2000 American Mathematical Society