Finite $F$-representation type for homogeneous coordinate rings of non-Fano varieties
Épijournal de Géométrie Algébrique, Tome 7 (2023)
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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.
@article{10_46298_epiga_2023_10868,
author = {Mallory, Devlin},
title = {Finite $F$-representation type for homogeneous coordinate rings of {non-Fano} varieties},
journal = {\'Epijournal de G\'eom\'etrie Alg\'ebrique},
publisher = {mathdoc},
volume = {7},
year = {2023},
doi = {10.46298/epiga.2023.10868},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.46298/epiga.2023.10868/}
}
TY - JOUR AU - Mallory, Devlin TI - Finite $F$-representation type for homogeneous coordinate rings of non-Fano varieties JO - Épijournal de Géométrie Algébrique PY - 2023 VL - 7 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.46298/epiga.2023.10868/ DO - 10.46298/epiga.2023.10868 LA - en ID - 10_46298_epiga_2023_10868 ER -
%0 Journal Article %A Mallory, Devlin %T Finite $F$-representation type for homogeneous coordinate rings of non-Fano varieties %J Épijournal de Géométrie Algébrique %D 2023 %V 7 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.46298/epiga.2023.10868/ %R 10.46298/epiga.2023.10868 %G en %F 10_46298_epiga_2023_10868
Mallory, Devlin. Finite $F$-representation type for homogeneous coordinate rings of non-Fano varieties. Épijournal de Géométrie Algébrique, Tome 7 (2023). doi: 10.46298/epiga.2023.10868
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