We give examples of non-isotrivial K3 surfaces over complex function fields with Zariski-dense rational points and NéronSeveri rank one.

We review recent developments in the arithmetic of K3 surfaces. Our focus lies on aspects of modularity, Picard number and rational points. Throughout we emphasise connections to geometry.

Rabbit corneal epithelial cells cultured in the presence of 3T3 feeder cells undergo biochemical differentiation, as evidenced by their initial expression of K5 and K14 keratins characteristic of basal keratinocytes, followed by the subsequent expression of K3 and K12 keratin markers of corneal epithelial differentiation. Previous data established that mutations of an Sp1 site in a DNA element,...

The inositol pyrophosphate disphosphoinositol pentakisphosphate (PP-InsP(3)/InsP(7)) is formed in mammals by two recently cloned inositol hexakiphosphate kinases, InsP(6)K1 and InsP(6)K2 (Saiardi, A., Erdjument-Bromage, H., Snowman, A. M., Tempst, P., and Snyder, S. H. (1999) Curr. Biol. 9, 1323-1326). We now report the identification, cloning, and characterization of a third InsP(7) forming en...

We show how certain F 4 couplings in eight dimensions can be computed using the mirror map and K3 data. They perfectly match with the corresponding heterotic one-loop couplings, and therefore this amounts to a successful test of the conjectured duality between the heterotic string on T 2 and F -theory on K3. The underlying quantum geometry appears to be a 5-fold, consisting of a hyperkähler 4-f...

We study the heterotic string on the simplest (non-trivial) Calabi-Yau man-ifold, K3, and its orbifold limits. We set the background gauge connection equal to the spin connection. The massless spectrum of the field theory that is the low energy limit of the Es @ Es string is derived on K3. Some terms in the effective Lagrangian for the massless modes are shown to be determined by the topology o...

In this paper we present Fabry-Pérot high resolution spectroscopy in Hα, Hβ, [O iii], and [N ii], optical low resolution spectroscopy spanning the usable range 4700–6800 Å, along with a broad band direct image in Gunn-red for the planetary nebula (PN) K3-82. K3-82 turned out to be highly symmetrical, seen most probably edge-on. Two point-symmetrical knots appear in the light of Gunn-red. These ...

The vertex colouring problem is known to be NP-complete in the class of triangle-free graphs. Moreover, it remains NP-complete even if we additionally exclude a graph F which is not a forest. We study the computational complexity of the problem in (K3, F )-free graphs with F being a forest. From known results it follows that for any forest F on 5 vertices, the vertex colouring problem is polyno...

The analagous class for surfaces are the K3 surfaces. All K3 surfaces are homeomorphic: there is one underlying topological type. On the other hand, there are a large number of topological types of minimal Calabi-Yau threefolds, but it is an open question of whether there are a finite number of such types. A stronger question would be to ask whether there are a finite number of families of alge...