On the structure of rigid semistable sheaves on algebraic surfaces

LetS be a smooth projective surface, letK be the canonical class ofS and letH be an ample divisor such thatH • K < 0. We prove that for any rigid sheafF (Ext¹ (F, F) = 0) that is Mumford-Takemoto semistable with respect toH there exists an exceptional set (E ₁ ,..., E _n ) of sheaves onS such thatF can be constructed from {E _i } by means of a finite sequence of extensions. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 692–700, November, 1998. The author wishes to express his gratitude to S. A. Kuleshov for useful discussions and to A. N. Rudakov and A. L. Gorodentsev for their attention to the present work. This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-01323 and by the INTAS Foundation.     

Moduli of sheaves and the Chow group of K3 surfaces

Let X be a projective complex K  3 surface. Beauville and Voisin singled out a 0-cycle _cX$c_X$ on X of degree 1 and Huybrechts proved that the second Chern class of a rigid simple vector-bundle on X   is a multiple of _cX$c_X$ if certain hypotheses hold. We believe that the following generalization of Huybrechts? result holds. Let M be a moduli space of stable pure sheaves on X with fixed cohomological Chern character: the set whose elements are second Chern classes of sheaves parametrized by the closure of M (in the corresponding moduli spaces of semistable sheaves) depends only on the dimension of M. We will prove that the above statement holds under some additional assumptions on the Chern character.     

Computing the canonical height on K3 surfaces

Let be a surface in given by the intersection of a (1,1)-form and a (2,2)-form. Then is a K3 surface with two noncommuting involutions and . In 1991 the second author constructed two height functions and which behave canonically with respect to and , and in 1993 together with the first author showed in general how to decompose such canonical heights into a sum of local heights . We discuss how the geometry of the surface is related to formulas for the local heights, and we give practical algorithms for computing the involutions , , the local heights , , and the canonical heights , .

     

Configurations in abelian categories. III. Stability conditions and identities

This is the third in a series on configurations in an abelian category A. Given a finite poset (I,?), an (I,?)-configuration(σ,ι,π) is a finite collection of objects σ(J) and morphisms ι(J,K) or in A satisfying some axioms, where J,K are subsets of I. Configurations describe how an object X in A decomposes into subobjects.The first paper defined configurations and studied moduli spaces of configurations in A, using the theory of Artin stacks. It showed well-behaved moduli stacks ObjA,MA(I,?) of objects and configurations in A exist when A is the abelian category coh(P) of coherent sheaves on a projective scheme P, or mod-KQ of representations of a quiver Q. The second studied algebras of constructible functions and stack functions on ObjA.This paper introduces (weak) stability conditions(τ,T,?) on A. We show the moduli spaces , , of τ-semistable, indecomposable τ-semistable and τ-stable objects in class α are constructible sets in ObjA, and some associated configuration moduli spaces constructible in MA(I,?), so their characteristic functions and are constructible.We prove many identities relating these constructible functions, and their stack function analogues, under pushforwards. We introduce interesting algebras of constructible and stack functions, and study their structure. In the fourth paper we show are independent of (τ,T,?), and construct invariants of A,(τ,T,?).     

Nodal Curves with General Moduli on K3 Surfaces

We investigate the modular properties of nodal curves on a low genus K3 surface. We prove that a general genus g curve C is the normalization of a δ-nodal curve X sitting on a primitively polarized K3 surface S of degree 2p ? 2, for 2 ≤ g = p ? δ < p ≤ 11. The proof is based on a local deformation-theoretic analysis of the map from the stack of pairs (S, X) to the moduli stack of curves ? _g that associates to X the isomorphism class [C] of its normalization.     

Singular rational curves on elliptic K3 surfaces

We show that on every elliptic K3 surface there are rational curves ${(R_i)}_{i\in \mathbb{N}}$ such that $R_i^2\to \infty$, that is, of unbounded arithmetic genus. Moreover, we show that the union of the lifts of these curves to $\mathbb{P}({\mathrm{\Omega }}_X)$ is dense in the Zariski topology. As an application, we give a simple proof of a theorem of Kobayashi in the elliptic case, that is, there are no globally defined symmetric differential forms.     

A finite group acting on the moduli space of K3 surfaces

We consider the natural action of a finite group on the moduli space of polarized K3 surfaces which induces a duality defined by Mukai for surfaces of this type. We show that the group permutes polarized Fourier-Mukai partners of polarized K3 surfaces and we study the divisors in the fixed loci of the elements of this finite group.

     

Order 9 automorphisms of K3 surfaces

Abstract

In this paper, we provide a complete classification of non-symplectic automorphisms of order 9 of complex K3 surfaces.     

On the Kodaira Dimension of the Moduli Space of K3 Surfaces II

We show that the Kodaira dimension of the moduli space of polarized K3 surfaces of degree 2n in non negative if n = 42, 43, 51, 53, 55, 57, 59, 61, 66, 67, 69, 74, 83, 85, 105, 119 or 133. We use an automorphic form associated with the fake monster Lie algebra constructed by Borcherds.     

K3 surfaces,Picard numbers and Siegel disks

If a K3 surface admits an automorphism with a Siegel disk, then its Picard number is an even integer between 0 and 18. Conversely, using the method of hypergeometric groups, we are able to construct K3 surface automorphisms with Siegel disks that realize all possible Picard numbers. The constructions involve extensive computer searches for appropriate Salem numbers and computations of algebraic numbers arising from holomorphic Lefschetz-type fixed point formulas and related Grothendieck residues.     

On Shioda-Inose structures of one-parameter families of K3 surfaces

We develop an algorithm to determine a one-parameter family of elliptic curves associated to a one-parameter family of K3 surfaces with generic Picard number 19 by a Shioda-Inose structure. The family of elliptic curves is determined up to an isomorphism and an isogeny. An application to a generalized congruence number problem is also discussed.     

Transcendental lattices of some K 3‐surfaces

In a previous paper, [12], we described six families of K 3‐surfaces (over ?) 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. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Orbits of Curves on Certain K3 Surfaces

In this paper, we study the family of algebraic K3 surfaces generated by the smooth intersection of a (1, 1) form and a (2, 2) form in defined over and with Picard number 3. We describe the group of automorphisms on V. For an ample divisor D and an arbitrary curve C ₀ on V, we investigate the asymptotic behavior of the quantity . We show that the limit

Classification of non‐symplectic automorphisms of order 3 on K3 surfaces

In this paper, we study non‐symplectic automorphisms of order 3 on algebraic K3 surfaces over ${\bb C}$ which act trivially on the Néron‐Severi lattice. In particular we shall characterize their fixed loci in terms of the invariants of 3‐elementary lattices. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Ordinary reduction of K3 surfaces

Let X be a K3 surface over a number field K. We prove that there exists a finite algebraic field extension E/K such that X has ordinary reduction at every non-archimedean place of E outside a density zero set of places.