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

epiga:10868 - Épijournal de Géométrie Algébrique, December 6, 2023, Volume 7 - https://doi.org/10.46298/epiga.2023.10868
Finite $F$-representation type for homogeneous coordinate rings of non-Fano varietiesPreprint

Authors: Devlin Mallory 1

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.


Volume: Volume 7
Published on: December 6, 2023
Accepted on: August 2, 2023
Submitted on: January 30, 2023
Keywords: Mathematics - Commutative Algebra,Mathematics - Algebraic Geometry,13A35 (Primary) 13N10, 14F10 (Secondary)

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