Jörg Jahnel

K3 surfaces of Picard rank one and degree two

We construct explicit examples of K3 surfaces over Q whichare of degree 2 and geometric Picard rank 1. We construct, particularly,examples of the form w2 = det M where M is a (3 × 3)-matrix of ternaryquadratic forms.

Experiments with General Cubic Surfaces

For general cubic surfaces, we test numerically the conjecture of Manin (in the refined form due to E. Peyre) about the asymptotics of points of bounded height on Fano varieties. We also study the behavior of the height of the smallest rational point versus the Tamagawa type number introduced by Peyre.

Kummer surfaces and the computation of the Picard group

We test R. van Luijk’s method for computing the Picard group of a K3 surface. The examples considered are the resolutions of Kummer quartics in P 3 . Using the theory of abelian varieties, the Picard group may be computed directly in this case. Our experiments show that the upper bounds provided by van Luijk’s method are sharp when suciently

The asymptotics of points of bounded height on diagonal cubic and quartic threefolds

For the families ax3 = by3 + z3 + v3 + w3, a, b = 1, ... ,100, and ax4 = by4 + z4 + v4 + w4, a, b = 1, ... ,100, of projective algebraic threefolds, we test numerically the conjecture of Manin (in the refined form due to Peyre) about the asymptotics of points of bounded height on Fano varieties.

On the characteristic polynomial of the Frobenius on étale cohomology

Let

The Diophantine Equation ⁴+2⁴=⁴+4⁴

X

New sums of three cubes

be a smooth proper variety of even dimension

Line bundles on arithmetic surfaces and intersection theory

d

Local singularities. filtrations and tangential flatness

over a finite field. We establish a restriction on the value at

Determinantal quartics and the computation of the Picard group

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