Kummer's Quartic Surface as a Wave Surface

A Quartic Surface of Integer Hexahedra

We prove that there are infinitely many sixsided polyhedra in R3, each with four congruent trapezoidal faces and two congruent rectangular faces, so that the faces have integer sides and diagonals, and also the solid has integer length diagonals. The solutions are obtained from the integer points on a particular quartic surface. A long standing unsolved problem asks whether or not there can be ...

Slowness Surface Wave Surface

Homological Mirror Symmetry for the Quartic Surface

Warning. This preprint is in finished form, and should be readable by itself. However, the argument uses certain general properties of Fukaya categories, the proofs of which will appear in [61] (under preparation). I apologize for this unsatisfactory state of affairs.

Manin’s conjecture on a nonsingular quartic del Pezzo surface

Given a nonsingular quartic del Pezzo surface, Manin's conjecture predicts the density of rational points on the open subset of the surface formed by deleting the lines. We prove that this prediction is of the correct order of magnitude for a particular surface.

Manin's conjecture for a singular quartic del Pezzo surface

We prove Manin’s conjecture for a split singular quartic del Pezzo surface with singularity type 2A1 and eight lines. This is achieved by equipping the surface with a conic bundle structure. To handle the sum over the family of conics, we prove a result of independent interest on a certain restricted divisor problem for four binary linear forms.