Global Frobenius liftability II: surfaces and Fano threefolds

In this article, a sequel to Global Frobenius liftability I, we continue the development of a comprehensive theory of Frobenius liftings modulo p(2). We study compatibility of divisors and closed subschemes with Frobenius liftings, Frobenius liftings of blow-ups, descent under quotients by some group actions, stability under base change, and the properties of associated F-splittings. Consequently, we characterise Frobenius liftable surfaces and Fano threefolds in large characteristic, confirming the conjecture stated in our previous paper.