A splinter is a notion of singularity that has seen numerous applications of late, especially in connection with the direct summand theorem, the mixed characteristic minimal model program, Cohen-Macaulayness of absolute integral closures and vanishing theorems. However, many basic questions about splinters remain elusive. One such problem is whether the splinter condition spreads from a point to an open neighborhood of a noetherian scheme. In this talk, we will address this question in prime characteristic and show that a locally noetherian scheme whose associated
absolute Frobenius is finite map has an open splinter locus. In particular,
all varieties over perfect fields of positive characteristic have open splinter loci. If time permits, we will show how our methods also give openness of splinter loci for a large class of schemes that do not necessarily have finite Frobenius. This talk is based on joint work in progress with Kevin Tucker.