On p‐power freeness in positive characteristic

Basepoint freeness for nef and big line bundles in positive characteristic

A necessary and sufficient condition is given for semi-ampleness of a numerically effective (nef) and big line bundle in positive characteristic. One application is to the geometry of the universal stable curve over Mg, specifically, the semi-ampleness of the relative dualizing sheaf, in positive characteristic. An example is given which shows this and the semi-ampleness criterion fail in chara...

On Stable Fields in Positive Characteristic

On Davenport-Stothers inequalities in positive characteristic

We show that the Davenport-Stothers inequality fails in any characteristic p > 3. The proof uses elliptic surfaces over P and inseparable base change. We also comment on corrected inequalities. Our ideas are related to supersingular surfaces (in Shioda’s sense). For characteristic 2 and 3, we achieve a similar result in terms of the maximal singular fibres of elliptic surfaces over P.

On zeroth Poisson homology in positive characteristic

A Poisson algebra is a commutative algebra with a Lie bracket {, } satisfying the Leibniz rule. Such algebras appear in classical mechanics. Namely, functions on the phase space form a Poisson algebra, and Hamilton’s equation of motion is df dt = {f, H}, where H is the Hamiltonian (energy) function. Moreover, the transition from classical to quantum mechanics can be understood in terms of defor...

In Positive Characteristic

We establish Noether’s inequality for surfaces of general type in positive characteristic. Then we extend Enriques’ and Horikawa’s classification of surfaces on the Noether line, the so-called Horikawa surfaces. We construct examples for all possible numerical invariants and in arbitrary characteristic, where we need foliations and deformation techniques to handle characteristic 2. Finally, we ...