Kummer Surface Research Articles

On real quadric line complexes

We describe the topological types of the real parts of the Kummer surfaces associated with real three-dimensional quadric line complexes. The topological type of the real part of such a surface is shown to depend on the number of real singular points: it is determined by the number of such points if any exist, and otherwise the real part of the Kummer surface is either empty or consists of one or two tori.

K3 Surfaces Associated with Curves of Genus Two

It is known (work of Galluzzi, Lombardo, Dolgachev and Naruki) that there is a unique K3 surface X which corresponds to a genus 2 curve C such that X has a Shioda-Inose structure with quotient birational to the Kummer surface of the Jacobian of C. In this paper we give an explicit realization of X as an elliptic surface over P^1 with specified singular fibers of type II^* and III^*. We describe how the Weierstrass coefficients are related to the Igusa-Clebsch invariants of C.

Del Pezzo surfaces of degree 4 and their relation to Kummer surfaces

In this note, which has little pretence to originality, we clarify the relation between the geometry of del Pezzo surfaces of degree 4 and their realization as the zero set of two quadratic forms in five variables. We also review the classical description of the desingularized Kummer surface K constructed from the Jacobian J of a curve C of genus 2 as the zero set of three quadratic forms in six variables (Plucker, Kummer, Klein [7], [6], see [5] or [3] for a modern treatment). If C has a rational Weierstrass point, a partial diagonalization of this system gives rise to a natural projection onto a hyperplane, defining a finite morphism π : K → X of degree 2 onto a del Pezzo surface X of degree 4 (see [9, §6]). We show that X is the blow-up of Pk in the images of the five other Weierstrass points of C under the embedding of Pk as a conic in Pk. The morphism π sends the 16 lines on K to the 16 lines on X, and is equivariant with respect to the action of the subgroup of 2-division points J [2] ⊂ J . Thus π gives rise to a morphism from the twisted Kummer surface to the twisted del Pezzo surface. In our presentation it is obvious that all del Pezzo surfaces of degree 4 can be obtained in this way, an observation made by Victor Flynn in [4]. The fact that any 2-covering of J maps to a del Pezzo surface of degree 4 was first observed in [2], and used in [2], [1] and [9] to construct and visualize elements of order 2 in the Tate–Shafarevich group of J over Q using the theory of the Brauer– Manin obstruction on del Pezzo surfaces of degree 4. It was the author’s desire to understand the geometry behind these calculations that prompted him to write this note. I would like to thank Igor Dolgachev for useful discussions.

Explicit Kummer surface formulas for arbitrary characteristic

AbstractIfCis a curve of genus 2 defined over a fieldkandJis its Jacobian, then we can associate a hypersurfaceKin3toJ, called the Kummer surface ofJ. Flynn has made this construction explicit in the case when the characteristic ofkis not 2 andCis given by a simplified equation. He has also given explicit versions of several maps defined on the Kummer surface and shown how to perform arithmetic onJusing these maps. In this paper we generalize these results to the case of arbitrary characteristic.

Traces of the Group Law on the Kummer Surface of a Curve of Genus 2 in Characteristic 2

In the early 1990s, Flynn gave an explicit description of the Jacobian of a genus 2 hyperelliptic curve in order to perform efficient arithmetic on these objects. In this paper, we give a generalization of Flynn’s work when the ground field has characteristic 2. More precisely, we give an explicit description of the Kummer surface. We also give and explain how we found, using symbolic computations, explicit formulas for the structure of the group law on the Jacobian preserved on the Kummer surface. Finally, we use these new objects to give a very fast scalar multiplication algorithm for hyperelliptic curve cryptography in characteristic 2.

Numerically trivial involutions of Kummer type of an Enriques surface

There are two types of numerically trivial involutions of an Enriques surface according to their period lattice. One is of U(2)⊥U(2)-type and the other is of (U(2)⊥U(2))-type. An Enriques surface with an involution of (U(2)⊥U(2))-type is doubly covered by a Kummer surface of product type, and such involutions are classified again into two types according to the parity of the corresponding Göpel subgroups. Involutions of odd (U(2)⊥U(2))-type are constructed from the standard Cremona involutions of the quadric surface and closely related with quartic del Pezzo surfaces.

The two faces of the twisted Kummer surface

The two faces of the twisted Kummer surface

Symplectic automorphisms on Kummer surfaces

Nikulin proved that the isometries induced on the second cohomology group of a K3 surface X by a finite abelian group G of symplectic automorphisms are essentially unique. Moreover he computed the discriminant of the sublattice of $${H^2(X,\mathbb{Z})}$$ which is fixed by the isometries induced by G. However for certain groups these discriminants are not the same as those found for explicit examples. Here we describe Kummer surfaces for which this phenomena happens and we explain the difference.

Factorization with genus 2 curves

The elliptic curve method (ECM) is one of the best factorization methods available. It is possible to use hyperelliptic curves instead of elliptic curves but it is in theory slower. We use special hyperelliptic curves and Kummer surfaces to reduce the complexity of the algorithm. Our implementation GMP-HECM is faster than GMP-ECM for factoring big numbers.

A Generalization of the Dual Kummer Surface

Given a hyperelliptic curve \({\mathcal{C}}\) of genus g, we construct a variety \({\mathcal{k}}^*\) with a hyperplane section Hm whose complement \({\mathcal{k}}^*_m\) parametrizes classes of divisors on \({\mathcal{C}}\) of degree g + 1, in general position, modulo the ± Y involution. The variety \({\mathcal{k}}^*\) is birational to the Kummer variety belonging to \({\mathcal{C}}\). It is a tool for studying rational classes and divisors. The variety \({\mathcal{k}}^*_m\) is birational to a quotient of an affine variety by an affine group.

Twisted homogeneous coordinate rings of abelian surfaces via mirror symmetry

In this paper we study Seidel's mirror map for abelian and Kummer surfaces. We find that mirror symmetry leads in a very natural way to the classical parametrization of Kummer surfaces in P 3 . Moreover, we describe a family of embeddings of a given abelian surface into noncommutative projective spaces.

The arithmetic of characteristic 2 Kummer surfaces and of elliptic Kummer lines

The purpose of this paper is a description of a model of Kummer surfaces in characteristic 2, together with the associated formulas for the pseudo-group law. Since the classical model has bad reduction, a renormalization of the parameters is required, that can be justified using the theory of algebraic theta functions. The formulas that are obtained are very efficient and may be useful in cryptographic applications. We also show that applying the same strategy to elliptic curves gives Montgomery-like formulas in odd characteristic that are faster than the classical ones, and we recover already known formulas by Stam in characteristic 2.

Inducing stability conditions

We study stability conditions induced by functors between triangulated categories. Given a finite group acting on a smooth projective variety, we prove that the subset of invariant stability conditions embeds as a closed submanifold into the stability manifold of the equivariant derived category. As an application, we examine stability conditions on Kummer and Enriques surfaces and we improve the derived version of the Torelli Theorem for the latter surfaces already present in the literature. We also study the relationship between stability conditions on projective spaces and those on their canonical bundles.

Enriques Surfaces Covered by Jacobian Kummer Surfaces

AbstractThis paper classifies Enriques surfaces whose K3-cover is a fixed Picard-general Jacobian Kummer surface. There are exactly 31 such surfaces. We describe the free involutions which give these Enriques surfaces explicitly. As a biproduct, we show that Aut(X) is generated by elements of order 2, which is an improvement of the theorem of S. Kondo.

Transcendental lattices of some K 3‐surfaces

AbstractIn a previous paper, [12], we described six families of K 3‐surfaces (over ℂ) 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)

The Brauer–Manin obstruction on del Pezzo surfaces of degree 2 branched along a plane section of a Kummer surface

AbstractThis paper discusses the Brauer–Manin obstruction on double covers of the projective plane branched along a plane section of a Kummer surface from both the practical and the theoretical points of view. Theoretical highlights include the determination of a complete set of generators for the Brauer group; on the practical side, we give several surfaces with a Brauer–Manin obstruction and verify that the Brauer–Manin obstruction is the only one for a collection of several thousand surfaces of this type.

The Hilbert scheme of points for supersingular abelian surfaces

We study the geometry of Hilbert schemes of points on abelian surfaces and Beauville’s generalized Kummer varieties in positive characteristics. The main result is that, in characteristic two, the addition map from the Hilbert scheme of two points to the abelian surface is a quasifibration such that all fibers are nonsmooth. In particular, the corresponding generalized Kummer surface is nonsmooth, and minimally elliptic singularities occur in the supersingular case. We unravel the structure of the singularities in dependence of p-rank and a-number of the abelian surface. To do so, we establish a McKay Correspondence for Artin’s wild involutions on surfaces. Along the line, we find examples of canonical singularities that are not rational singularities.

Equivariant autoequivalences for finite group actions

The familiar Fourier–Mukai technique can be extended to an equivariant setting where a finite group G acts on a smooth projective variety X. In this paper we compare the group of invariant autoequivalences Aut ( D b ( X ) ) G with the group of autoequivalences of D G ( X ) . We apply this method in three cases: Hilbert schemes on K3 surfaces, Kummer surfaces and canonical quotients.

Double covers of Kummer surfaces

Besides its construction as a quotient of an abelian surface, a Kummer surface can be obtained as the quotient of a K3 surface by a \({\mathbb{Z}}/2{\mathbb{Z}}\) -action. In this paper, we classify all such K3 surfaces. Our classification is expressed in terms of period lattices and extends Morrison’s criterion of K3 surfaces with a Shioda–Inose structure. Moreover, we list all the K3 surfaces associated to a general Kummer surface and provide very geometrical examples of this phenomenon.

VECTOR BUNDLES, DUALITIES AND CLASSICAL GEOMETRY ON A CURVE OF GENUS TWO

Let C be a curve of genus two. We denote by [Formula: see text] the moduli space of semi-stable vector bundles of rank 3 and trivial determinant over C, and by Jd the variety of line bundles of degree d on C. In particular, J1 has a canonical theta divisor Θ. The space [Formula: see text] is a double cover of ℙ8 = |3Θ| branched along a sextic hypersurface, the Coble sextic. In the dual [Formula: see text], where J1 is embedded, there is a unique cubic hypersurface singular along J1, the Coble cubic. We prove that these two hypersurfaces are dual, inducing a non-abelian Torelli result. Moreover, by looking at some special linear sections of these hypersurfaces, we can observe and reinterpret some classical results of algebraic geometry in a context of vector bundles: the duality of the Segre–Igusa quartic with the Segre cubic, the symmetric configuration of 15 lines and 15 points, the Weddle quartic surface and the Kummer surface.