Deligne-Lusztig varieties whose canonical divisors have negativity
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914527795216384 |
|---|---|
| author | Görtz, Ulrich Schröer, Stefan |
| author_facet | Görtz, Ulrich Schröer, Stefan |
| contents | We investigate compactified Deligne-Lusztig varieties whose canonical divisor, when expressed as a linear combination of boundary divisors, has all coefficients strictly negative or zero. In dimension two we obtain explicit descriptions: The case $C_2$ gives the supersingular K3 surface with Artin invariant $σ=1$ in characteristic two. The arguments rely on properties of the Tutte-Coxeter graph; in this connection we also gain some insight into the arithmetic of quasi-elliptic Weierstrass equations and rational double points. The twisted cases ${}^2A_2$ and ${}^2C_2$ and ${}^2G_2$ yield particular ruled surfaces, attached in a canonical way to supersingular elliptic curves in characteristic two, or the Ree curve in characteristic three. The latter is a curve of genus fifteen with outstanding symmetries. In the former cases, the surfaces can also be expressed as symmetric squares. To obtain such results, we develop a new general framework for Deligne-Lusztig varieties that relies on the Isogeny Theorem, and works over prime fields rather than their algebraic closures, and includes the notorious Suzuki-Ree cases without effort. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02522 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Deligne-Lusztig varieties whose canonical divisors have negativity Görtz, Ulrich Schröer, Stefan Algebraic Geometry Representation Theory 14L15, 14M15, 14G17, 14J28, 14J26, 17B22 We investigate compactified Deligne-Lusztig varieties whose canonical divisor, when expressed as a linear combination of boundary divisors, has all coefficients strictly negative or zero. In dimension two we obtain explicit descriptions: The case $C_2$ gives the supersingular K3 surface with Artin invariant $σ=1$ in characteristic two. The arguments rely on properties of the Tutte-Coxeter graph; in this connection we also gain some insight into the arithmetic of quasi-elliptic Weierstrass equations and rational double points. The twisted cases ${}^2A_2$ and ${}^2C_2$ and ${}^2G_2$ yield particular ruled surfaces, attached in a canonical way to supersingular elliptic curves in characteristic two, or the Ree curve in characteristic three. The latter is a curve of genus fifteen with outstanding symmetries. In the former cases, the surfaces can also be expressed as symmetric squares. To obtain such results, we develop a new general framework for Deligne-Lusztig varieties that relies on the Isogeny Theorem, and works over prime fields rather than their algebraic closures, and includes the notorious Suzuki-Ree cases without effort. |
| title | Deligne-Lusztig varieties whose canonical divisors have negativity |
| topic | Algebraic Geometry Representation Theory 14L15, 14M15, 14G17, 14J28, 14J26, 17B22 |
| url | https://arxiv.org/abs/2605.02522 |