On proper splinters in positive characteristic

A commutative ring is called a splinter if any finite-module ring extension splits. By the direct summand conjecture, now a theorem due to André, every regular ring is a splinter. The notion of splinter can naturally be extended to schemes. In that context, every smooth scheme in characteristic zero is a splinter. In contrast, Bhatt observed in his thesis that the global geometry of a proper scheme in positive characteristic plays a significant role as to whether or not it is a splinter; for instance, the structure sheaf of a proper splinter in positive characteristic is exceptional. I will report on on-going joint work with Johannes Krah where we describe further restrictions on the global geometry of proper splinters in positive characteristic.