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 K3 surface are dense. We will also compute the entire Néron-Severi group of this surface and find all low degree curves on it.

We consider those projective bundles (or Brauer-Severi varieties) over an abelian variety that are homogeneous, i.e., invariant under translation. We describe the structure of these bundles in terms of projective representations of commutative group schemes; the irreducible bundles correspond to Heisenberg groups and their standard representations. Our results extend those of Mukai on semi-homo...

For every supersingular K3 surface X in characteristic 2, there exists a homogeneous polynomial G of degree 6 such that X is birational to the purely inseparable double cover of P defined by w = G. We present an algorithm to calculate from G a set of generators of the numerical Néron-Severi lattice of X. As an application, we investigate the stratification defined by the Artin invariant on a mo...

In this paper we compute the equivariant homotopy type of spaces of algebraic cycles on real Brauer-Severi varieties, under the action of the Galois group Gal(C/R). Appropriate stabilizations of these spaces yield two equivariant spectra. The first one classifies Dupont/Seymour’s quaternionic K-theory, and the other one classifies and equivariant cohomology theory Z∗(−) which is a natural recip...

We give a survey on certain results related to the cohomology of projective schemes with coefficients in coherent sheaves. In particular we present results on cohomological patterns, cohomological Hilbert functions and cohomological Hilbert polynomials. Bounding results for Castelnuovo-Mumford regularities, Severi coregularities and cohomological postulation numbers are discussed. Moreover, a n...

We prove that the degree of a nonconstant morphism from a smooth projective 3-fold X with Néron-Severi group Z to a smooth 3-dimensional quadric is bounded in terms of numerical invariants of X. In the special case where X is a 3-dimensional cubic we show that there are no such morphisms. The main tool in the proof is Miyaoka’s bound on the number of double points of a surface.

For a given sequence of integers (ni) ∞ i=1 we consider all the central simple algebras A (over all fields) satisfying the condition ind A = ni and find among them an algebra having the biggest torsion in the second Chow group CH of the corresponding Severi-Brauer variety (“biggest” means that it can be mapped epimorphically onto each other). We describe this biggest torsion in a way in general...

In this paper we study the geometry of 14 families K3 surfaces Picard number four with finite automorphism group, whose Neron—Severi lattices have been classified by È. B. Vinberg. We provide projective models, identify degrees a generating set Cox ring and in some cases prove unirationality associated moduli space.

Three lectures on elliptic surfaces and curves of high rank Noam D. Elkies Over the past two years we have improved several of the (Mordell–Weil) rank records for elliptic curves over Q and nonconstant elliptic curves over Q(t). For example, we found the first example of a curve E/Q with 28 independent points P i ∈ E(Q) (the previous record was 24, by R. Martin and W. McMillen 2000), and the fi...

T he golden age of the Italian school of algebraic geometry began with Antonio Luigi GaudenzioGiuseppeCremona and included among its main contributors Enrico Castelnuovo, Federigo Enriques, and Francesco Severi. The Italian school spanned nearly a century, from the unification of Italy in 1861 to Enriques’s posthumously published postWorld War II monograph on algebraic surfaces [Enrq 49]. In th...