On rigidity of Pham-Brieskorn surfaces
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866909270041165824 |
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| author | Gupta, Neena Pal, Ananya |
| author_facet | Gupta, Neena Pal, Ananya |
| contents | It is well known that, over an algebraically closed field $k$ of characteristic zero, for any three integers $a,b,c\geq 2$, any Pham-Brieskorn surface $B_{(a,b,c)}:= k[X,Y,Z]/(X^a + Y^b + Z^c)$ is rigid when at most one of $a,b,c$ is 2 and stably rigid when $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\leq 1$. In this paper we consider Pham-Brieskorn domains over an arbitrary field $k$ of characteristic $p\geq 0$ and give sufficient conditions on $(a,b,c)$ for which any Pham-Brieskorn domain $B_{(a,b,c)}$ is rigid. This gives an alternative approach to showing that there does not exist any non-trivial exponential map on $k[X,Y,Z,T]/(X^mY+T^{p^rq} + Z^{p^e})= k[x,y,z,t]$, for $m,q>1$, $p\nmid mq$ and $e>r\geq 1$, fixing $y$, a crucial result used in the paper "On the cancellation problem for the affine space $\mathbb{A}^3$ in characteristic $p$" by first author, to show that the Zariski Cancellation Problem (ZCP) does not hold for the affine $3$-space.
We also provide a sufficient condition for $B_{(a,b,c)}$ to be stably rigid. Along the way we prove that for integers $a,b,c\geq 2$ with $gcd(a,b,c) = 1$ and for $F(Y)\in k[Y]$, the ring $k[X,Y,Z]/(X^aY^b + Z^c+ F(Y))$ is a rigid domain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_01864 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On rigidity of Pham-Brieskorn surfaces Gupta, Neena Pal, Ananya Algebraic Geometry Commutative Algebra 14R20 (Primary) 13A50, 13A02 (Secondary) It is well known that, over an algebraically closed field $k$ of characteristic zero, for any three integers $a,b,c\geq 2$, any Pham-Brieskorn surface $B_{(a,b,c)}:= k[X,Y,Z]/(X^a + Y^b + Z^c)$ is rigid when at most one of $a,b,c$ is 2 and stably rigid when $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\leq 1$. In this paper we consider Pham-Brieskorn domains over an arbitrary field $k$ of characteristic $p\geq 0$ and give sufficient conditions on $(a,b,c)$ for which any Pham-Brieskorn domain $B_{(a,b,c)}$ is rigid. This gives an alternative approach to showing that there does not exist any non-trivial exponential map on $k[X,Y,Z,T]/(X^mY+T^{p^rq} + Z^{p^e})= k[x,y,z,t]$, for $m,q>1$, $p\nmid mq$ and $e>r\geq 1$, fixing $y$, a crucial result used in the paper "On the cancellation problem for the affine space $\mathbb{A}^3$ in characteristic $p$" by first author, to show that the Zariski Cancellation Problem (ZCP) does not hold for the affine $3$-space. We also provide a sufficient condition for $B_{(a,b,c)}$ to be stably rigid. Along the way we prove that for integers $a,b,c\geq 2$ with $gcd(a,b,c) = 1$ and for $F(Y)\in k[Y]$, the ring $k[X,Y,Z]/(X^aY^b + Z^c+ F(Y))$ is a rigid domain. |
| title | On rigidity of Pham-Brieskorn surfaces |
| topic | Algebraic Geometry Commutative Algebra 14R20 (Primary) 13A50, 13A02 (Secondary) |
| url | https://arxiv.org/abs/2310.01864 |