Optimal Degenerations of K-unstable Fano threefolds
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913025453195264 |
|---|---|
| author | Miao, Minghao Wang, Linsheng |
| author_facet | Miao, Minghao Wang, Linsheng |
| contents | We explicitly determine the optimal degenerations of Fano threefolds $X$ in family No 2.23 of Mori-Mukai's list as predicted by the Hamilton-Tian conjecture. More precisely, we find a special degeneration $(\mathcal{X}, ξ_0)$ of $X$ such that $(\mathcal{X}_0, ξ_0)$ is weighted K-polystable, which is equivalent to $(\mathcal{X}_0, ξ_0)$ admitting a Kähler-Ricci soliton (KRS) by \cite{HL23} and \cite{BLXZ23}. Furthermore, we study the moduli spaces of $(\mathcal{X}_0, ξ_0)$. The $\mathbf{H}$-invariant of $X$ divides the natural parameter space into two strata, which leads to different moduli spaces of KRS Fano varieties. We show that one of them is isomorphic to the GIT-moduli space of biconic curves $C\subseteq \mathbb{P}^1\times \mathbb{P}^1$, and the other one is a single point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13999 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal Degenerations of K-unstable Fano threefolds Miao, Minghao Wang, Linsheng Algebraic Geometry Differential Geometry 14J45, 32Q20, 14D20 We explicitly determine the optimal degenerations of Fano threefolds $X$ in family No 2.23 of Mori-Mukai's list as predicted by the Hamilton-Tian conjecture. More precisely, we find a special degeneration $(\mathcal{X}, ξ_0)$ of $X$ such that $(\mathcal{X}_0, ξ_0)$ is weighted K-polystable, which is equivalent to $(\mathcal{X}_0, ξ_0)$ admitting a Kähler-Ricci soliton (KRS) by \cite{HL23} and \cite{BLXZ23}. Furthermore, we study the moduli spaces of $(\mathcal{X}_0, ξ_0)$. The $\mathbf{H}$-invariant of $X$ divides the natural parameter space into two strata, which leads to different moduli spaces of KRS Fano varieties. We show that one of them is isomorphic to the GIT-moduli space of biconic curves $C\subseteq \mathbb{P}^1\times \mathbb{P}^1$, and the other one is a single point. |
| title | Optimal Degenerations of K-unstable Fano threefolds |
| topic | Algebraic Geometry Differential Geometry 14J45, 32Q20, 14D20 |
| url | https://arxiv.org/abs/2401.13999 |