On the geography of $3$-folds via asymptotic behavior of invariants
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910426842791936 |
|---|---|
| author | Torres-Nova, Yerko |
| author_facet | Torres-Nova, Yerko |
| contents | Roughly speaking, the problem of geography asks for the existence of varieties of general type after we fix some invariants. In dimension $1$, where we fix the genus, the geography question is trivial, but already in dimension $2$, it becomes a hard problem in general. In higher dimensions, this problem is essentially wide open. In this paper, we focus on geography in dimension $3$. We generalize the techniques which compare the geography of surfaces with the geography of arrangements of curves via asymptotic constructions. In dimension $2$ this involves resolutions of cyclic quotient singularities and a certain asymptotic behavior of the associated Dedekind sums and continued fractions. We discuss the general situation with emphasis on dimension $3$, analyzing the singularities and various resolutions that show up, and proving results about the asymptotic behavior of the invariants we fix. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_10516 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the geography of $3$-folds via asymptotic behavior of invariants Torres-Nova, Yerko Algebraic Geometry Combinatorics Roughly speaking, the problem of geography asks for the existence of varieties of general type after we fix some invariants. In dimension $1$, where we fix the genus, the geography question is trivial, but already in dimension $2$, it becomes a hard problem in general. In higher dimensions, this problem is essentially wide open. In this paper, we focus on geography in dimension $3$. We generalize the techniques which compare the geography of surfaces with the geography of arrangements of curves via asymptotic constructions. In dimension $2$ this involves resolutions of cyclic quotient singularities and a certain asymptotic behavior of the associated Dedekind sums and continued fractions. We discuss the general situation with emphasis on dimension $3$, analyzing the singularities and various resolutions that show up, and proving results about the asymptotic behavior of the invariants we fix. |
| title | On the geography of $3$-folds via asymptotic behavior of invariants |
| topic | Algebraic Geometry Combinatorics |
| url | https://arxiv.org/abs/2307.10516 |