Some elementary examples of non-liftable schemes

TL;DR

This paper provides elementary examples of smooth projective varieties in positive characteristic that cannot be lifted to characteristic zero or to the second Witt vectors, highlighting limitations in lifting properties.

Contribution

It introduces new explicit examples of non-liftable smooth projective varieties in positive characteristic using linear algebra and blow-up constructions.

Findings

Examples of non-liftable varieties from Frobenius graph blow-up

Construction of non-liftable varieties from special point-line configurations

Illustration of limitations in lifting smooth projective varieties

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.