We give a classification of the counterexamples of the Kawamata-Viehweg vanishing on a geometrically ruled surface in terms of the Tango invariant of the base curve.

Isometric surfaces share the same geometric structure also known as the ‘first fundamental form’. For example, all possible bending of a given surface, that include all length preserving deformations without tearing or stretching the surface, are considered to be isometric. We present a method to construct a bending invariant canonical form for such surfaces. This invariant representation is an...

We show that every supersingular K3 surface in characteristic 2 with Artin invariant ≤ 2 is obtained by the Kummer type construction of Schröer.

For a Veech surface (X, ω), we characterize Aff + (X, ω) invariant subspaces of X n and prove that non-arithmetic Veech surfaces have only finitely many invariant subspaces of very particular shape (in any dimension). Among other consequences we find copies of (X, ω) embedded in the moduli-space of translation surfaces. We study illumination problems in (pre-)lattice surfaces. For (X, ω) prelat...