Locally Trivial Deformations of Toric Varieties
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_ | 1866914561849819136 |
|---|---|
| author | Ilten, Nathan Robins, Sharon |
| author_facet | Ilten, Nathan Robins, Sharon |
| contents | We study locally trivial deformations of toric varieties from a combinatorial point of view. For any fan $Σ$, we construct a deformation functor $\mathrm{Def}_Σ$ by considering Čech zero-cochains on certain simplicial complexes. We show that under appropriate hypotheses, $\mathrm{Def}_Σ$ is isomorphic to $\mathrm{Def}'_{X_Σ}$, the functor of locally trivial deformations for the toric variety $X_Σ$ associated to $Σ$. In particular, for any complete toric variety $X$ that is smooth in codimension $2$ and $\mathbb{Q}$-factorial in codimension $3$, there exists a fan $Σ$ such that $\mathrm{Def}_Σ$ is isomorphic to $\mathrm{Def}_X$, the functor of deformations of $X$. We apply these results to give a new criterion for a smooth complete toric variety to have unobstructed deformations, and to compute formulas for higher order obstructions, generalizing a formula of Ilten and Turo for the cup product. We use the functor $\mathrm{Def}_Σ$ to explicitly compute the deformation spaces for a number of toric varieties, and provide examples exhibiting previously unobserved phenomena. In particular, we classify exactly which toric threefolds arising as iterated $\mathbb{P}^1$-bundles have unobstructed deformation space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02824 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Locally Trivial Deformations of Toric Varieties Ilten, Nathan Robins, Sharon Algebraic Geometry 14D15, 14M25 We study locally trivial deformations of toric varieties from a combinatorial point of view. For any fan $Σ$, we construct a deformation functor $\mathrm{Def}_Σ$ by considering Čech zero-cochains on certain simplicial complexes. We show that under appropriate hypotheses, $\mathrm{Def}_Σ$ is isomorphic to $\mathrm{Def}'_{X_Σ}$, the functor of locally trivial deformations for the toric variety $X_Σ$ associated to $Σ$. In particular, for any complete toric variety $X$ that is smooth in codimension $2$ and $\mathbb{Q}$-factorial in codimension $3$, there exists a fan $Σ$ such that $\mathrm{Def}_Σ$ is isomorphic to $\mathrm{Def}_X$, the functor of deformations of $X$. We apply these results to give a new criterion for a smooth complete toric variety to have unobstructed deformations, and to compute formulas for higher order obstructions, generalizing a formula of Ilten and Turo for the cup product. We use the functor $\mathrm{Def}_Σ$ to explicitly compute the deformation spaces for a number of toric varieties, and provide examples exhibiting previously unobserved phenomena. In particular, we classify exactly which toric threefolds arising as iterated $\mathbb{P}^1$-bundles have unobstructed deformation space. |
| title | Locally Trivial Deformations of Toric Varieties |
| topic | Algebraic Geometry 14D15, 14M25 |
| url | https://arxiv.org/abs/2409.02824 |