GAAG seminar - On proper splinters in positive characteristic

A scheme X is a splinter if for any finite surjective morphism f: Y \to X the pullback map O_X \to f_* O_Y splits as O_X-modules. By the direct summand conjecture, now a theorem due to André, every regular Noetherian ring is a splinter. Whilst for affine schemes the splinter property can be viewed as a local measure of singularity, the splinter property imposes strong constraints on the global geometry of proper schemes over a field of positive characteristic. For instance, the structure sheaf of a proper splinter in positive characteristic has vanishing positive-degree cohomology. I will report on joint work with Charles Vial concerning further obstructions on the global geometry of proper splinters in positive characteristic