We compute the nef cone of the Hilbert scheme of points on a general rational elliptic surface. As a consequence of our computation, we show that the Morrison–Kawamata cone conjecture holds for these nef cones.

This article has two purposes. In [15] we showed that the FIC (Fibered Isomorphism Conjecture for pseudoisotopy functor) for a particular class of 3-manifolds (we denoted this class by C) is the key to prove the FIC for 3-manifold groups in general. And we proved the FIC for the fundamental groups of members of a subclass of C. This result was obtained by showing that the double of any member of this subclass is either Seifert fibered or supports a nonpositively curved metric. In this article we prove that for any M ε C there is a closed 3-manifold P such that either P is Seifert fibered or is a nonpositively curved 3-manifold and π1(M) is a subgroup of π1(P). As a consequence it is obtained that the FIC is true for any B-group (see definition 4.2 in [15]). Therefore, the FIC is true for any Haken 3-manifold group and hence for any 3-manifold group (using the reduction theorem of [15]) provided we assume the Geometrization conjecture. The above result also proves the FIC for a class of 4-manifold groups (see [14]).

The second aspect of this article is to relax a condition in the definition of strongly poly-surface group ([13]) and define a new class of groups (we call them weak strongly poly-surface groups). Then using the above result we prove the FIC for any virtually weak strongly poly-surface group. We also give a corrected proof of the main lemma of [13].

In this article we will show that there are infinitely many symmetric, integral 3 × 3 matrices, with zeros on the diagonal, whose eigenvalues are all integral. We will do this by proving that the rational points on a certain non-Kummer, singular $\text{K3}$ surface are dense. We will also compute the entire Néron–Severi group of this surface and find all low degree curves on it.

The purpose of this note is to present a somewhat unexpected relation between diophantine approximations and the geometric invariant theory. The link is given by Mumford's degree of contact. We show that destabilizing flags of Chow-unstable projective varieties provide systems of diophantine approximations which are better than those given by Schmidt's subspace theorem, and we give examples of these systems.