On proper splinters in positive characteristic

TL;DR

This paper investigates the properties of splinter schemes in positive characteristic, revealing their implications on fundamental groups and Kodaira dimension, and explores invariance of the splinter property under derived equivalences.

Contribution

It establishes that proper splinter schemes in positive characteristic have trivial Nori fundamental group and negative Kodaira dimension, and proves invariance of the splinter property and global F-regularity for certain varieties.

Findings

Proper splinter schemes have trivial Nori fundamental group.

Proper splinter schemes have negative Kodaira dimension.

Global F-regularity is a derived invariant for certain varieties.

Abstract

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 scheme 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 normal Gorenstein projective varieties with big anticanonical divisor. We also show that global F-regularity is a derived invariant for normal Gorenstein projective varieties in positive characteristic.