Symmetries of Fano varieties
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916595302924288 |
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| author | Esser, Louis Ji, Lena Moraga, Joaquín |
| author_facet | Esser, Louis Ji, Lena Moraga, Joaquín |
| contents | We study Fano varieties endowed with a faithful action of a symmetric group, as well as analogous results for Calabi--Yau varieties, and log terminal singularities. We show the existence of a constant $m(n)$, so that every symmetric group $S_k$ acting on an $n$-dimensional Fano variety satisfies $k \leq m(n)$. We prove that $m(n)> n+\sqrt{2n}$ for every $n$. On the other hand, we show that $\lim_{n\to \infty} m(n)/(n+1)^2 \leq 1$. However, this asymptotic upper bound is not expected to be sharp.
We obtain sharp bounds for certain classes of varieties. For toric varieties, we show that $m(n)=n+2$ for $n\geq 4$. For Fano quasismooth weighted complete intersections, we prove the asymptotic equality $\lim_{n\to \infty} m(n)/(n+1)=1$. Among the Fano weighted complete intersections, we study the maximally symmetric ones and show that they are closely related to the Fano--Fermat varieties, i.e., Fano complete intersections in $\mathbb P^N$ cut out by Fermat hypersurfaces.
Finally, we draw a connection between maximally symmetric Fano varieties and boundedness of Fano varieties. For instance, we show that the class of $S_8$-equivariant Fano $4$-folds forms a bounded family. In contrast, the $S_7$-equivariant Fano $4$-folds are unbounded. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_12958 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Symmetries of Fano varieties Esser, Louis Ji, Lena Moraga, Joaquín Algebraic Geometry 14J45, 14J50 (Primary), 14E07, 14M10 (Secondary) We study Fano varieties endowed with a faithful action of a symmetric group, as well as analogous results for Calabi--Yau varieties, and log terminal singularities. We show the existence of a constant $m(n)$, so that every symmetric group $S_k$ acting on an $n$-dimensional Fano variety satisfies $k \leq m(n)$. We prove that $m(n)> n+\sqrt{2n}$ for every $n$. On the other hand, we show that $\lim_{n\to \infty} m(n)/(n+1)^2 \leq 1$. However, this asymptotic upper bound is not expected to be sharp. We obtain sharp bounds for certain classes of varieties. For toric varieties, we show that $m(n)=n+2$ for $n\geq 4$. For Fano quasismooth weighted complete intersections, we prove the asymptotic equality $\lim_{n\to \infty} m(n)/(n+1)=1$. Among the Fano weighted complete intersections, we study the maximally symmetric ones and show that they are closely related to the Fano--Fermat varieties, i.e., Fano complete intersections in $\mathbb P^N$ cut out by Fermat hypersurfaces. Finally, we draw a connection between maximally symmetric Fano varieties and boundedness of Fano varieties. For instance, we show that the class of $S_8$-equivariant Fano $4$-folds forms a bounded family. In contrast, the $S_7$-equivariant Fano $4$-folds are unbounded. |
| title | Symmetries of Fano varieties |
| topic | Algebraic Geometry 14J45, 14J50 (Primary), 14E07, 14M10 (Secondary) |
| url | https://arxiv.org/abs/2308.12958 |