On surfaces satisfying $q=0,p_g=0,c_1^2=9$
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916904569929728 |
|---|---|
| author | Joshi, Kirti |
| author_facet | Joshi, Kirti |
| contents | I consider the class of surfaces $X$ over algebraically closed fields with numerical invariants given in the title. In characteristic zero, this class contains fake projective planes which were introduced by David Mumford. I prove that in characteristic $p>0$ such surfaces are Hodge-Witt and also ordinary under additional assumptions. In particular, fake projective planes are Hodge-Witt (Theorem 3.1). I show that if $X$ is Frobenius split then $X\simeq \mathbb{P}^2$ (Theorem 4.1). I also establish a characteristic free characterization of the projective plane using the Nori fundamental group scheme (Theorem 5.1). Finally, I show that any fake projective plane over a number field has good ordinary reduction at all but finitely many primes and in particular fake projective planes exist in positive characteristics (Theorem 6.1). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_12339 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On surfaces satisfying $q=0,p_g=0,c_1^2=9$ Joshi, Kirti Algebraic Geometry Number Theory I consider the class of surfaces $X$ over algebraically closed fields with numerical invariants given in the title. In characteristic zero, this class contains fake projective planes which were introduced by David Mumford. I prove that in characteristic $p>0$ such surfaces are Hodge-Witt and also ordinary under additional assumptions. In particular, fake projective planes are Hodge-Witt (Theorem 3.1). I show that if $X$ is Frobenius split then $X\simeq \mathbb{P}^2$ (Theorem 4.1). I also establish a characteristic free characterization of the projective plane using the Nori fundamental group scheme (Theorem 5.1). Finally, I show that any fake projective plane over a number field has good ordinary reduction at all but finitely many primes and in particular fake projective planes exist in positive characteristics (Theorem 6.1). |
| title | On surfaces satisfying $q=0,p_g=0,c_1^2=9$ |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2508.12339 |