Finite $F$-representation type for homogeneous coordinate rings of non-Fano varieties

TL;DR

This paper investigates when homogeneous coordinate rings of certain varieties in positive characteristic lack finite F-representation type, linking differential operators, sheaf positivity, and algebraic properties.

Contribution

It establishes a connection between differential operators and sheaf positivity to identify classes of varieties without finite F-representation type in characteristic p.

Findings

Many classes of varieties, including abelian and Calabi–Yau, lack FFRT due to sheaf positivity conditions.

Provides examples of differential operator rings for non-F-pure varieties.

Connects sheaf semistability to the failure of FFRT in homogeneous coordinate rings.

Abstract

Finite $F$-representation type is an important notion in characteristic-$p$ commutative algebra, but explicit examples of varieties with or without this property are few. We prove that a large class of homogeneous coordinate rings in positive characteristic will fail to have finite $F$-representation type. To do so, we prove a connection between differential operators on the homogeneous coordinate ring of $X$ and the existence of global sections of a twist of $(\mathrm{Sym}^m \Omega_X)^\vee$. By results of Takagi and Takahashi, this allows us to rule out FFRT for coordinate rings of varieties with $(\mathrm{Sym}^m \Omega_X)^\vee$ not ``positive''. By using results positivity and semistability conditions for the (co)tangent sheaves, we show that several classes of varieties fail to have finite $F$-representation type, including abelian varieties, most Calabi--Yau varieties, and complete intersections of general type. Our work also provides examples of the structure of the ring of differential operators for non-$F$-pure varieties, which to this point have largely been unexplored.