  While the splinter property is a local property for Noetherian schemes in
characteristic zero, Bhatt observed that it imposes strong conditions on the
global geometry of proper schemes in positive characteristic. We show that if a
proper scheme over a field of positive characteristic is a splinter, then its
Nori fundamental group is trivial and its Kodaira dimension is negative. In
another direction, Bhatt also showed that any splinter in positive
characteristic is a derived splinter. We ask whether the splinter property is a
derived-invariant for projective varieties in positive characteristic and give
a positive answer for Gorenstein projective varieties with big anticanonical
divisor. For that purpose, we introduce the notion of O-equivalence and show
that the derived splinter property for schemes of finite type and separated
over a fixed Noetherian base is preserved under O-equivalence. Finally, we show
that global F-regularity is a derived-invariant for normal Gorenstein
projective varieties in positive characteristic.
