Some elementary examples of non-liftable schemes

Abstract

We present some simple examples of smooth projective varieties in positive characteristic, arising from linear algebra, which do not admit a lifting neither to characteristic zero, nor to the ring of second Witt vectors. Our first construction is the blow-up of the graph of the Frobenius morphism of a homogeneous space. The second example is a blow-up of ${\mathbb{P}}^3$ in a 'purely characteristic-$p$' configuration of points and lines.

10 pages, 1 figure, Comments welcome