Finite $F$-representation type for homogeneous coordinate rings of non-Fano varieties
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917534516641792 |
|---|---|
| author | Mallory, Devlin |
| author_facet | Mallory, Devlin |
| contents | 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 $(\mathrm{Sym}^m Ω_X)^\vee$. By results of Takagi and Takahashi, this allows us to rule out FFRT for coordinate rings of varieties with $(\mathrm{Sym}^m Ω_X)^\vee$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_08966 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Finite $F$-representation type for homogeneous coordinate rings of non-Fano varieties Mallory, Devlin Commutative Algebra Algebraic Geometry 13A35 (Primary) 13N10, 14F10 (Secondary) 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 $(\mathrm{Sym}^m Ω_X)^\vee$. By results of Takagi and Takahashi, this allows us to rule out FFRT for coordinate rings of varieties with $(\mathrm{Sym}^m Ω_X)^\vee$ 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. |
| title | Finite $F$-representation type for homogeneous coordinate rings of non-Fano varieties |
| topic | Commutative Algebra Algebraic Geometry 13A35 (Primary) 13N10, 14F10 (Secondary) |
| url | https://arxiv.org/abs/2207.08966 |