# Automorphisms of singular hypersurfaces

Let $$X\subset\mathbb{P}^{n+1}$$ be an irreducible and reduced hypersurface of degree $$d$$.

A theorem by Matsumura and Monski asserts that if $$n\geq 2$$, $$d\geq 3$$, $$(n,d)\neq (2,4)$$ and $$X$$ is smooth then all the automorphisms of $$X$$ are induced by automorphisms of $$\mathbb{P}^{n+1}$$.

Could a similar statement hold for singular hypersurfaces? Perhaps assuming that $$X$$ is normal and its dualizing sheaf is not trivial?

According to Theorem 1.1 of

M. J. Bradley, H. J. D’Souza: On the orders of automorphism groups of complex projective hypersurfaces, Lanteri, A. (ed.) et al., Geometry of complex projective varieties. Proceedings of the conference, Cetraro, Italy, May 28-June 2, 1990. Rende: Mediterranean Press, Semin. Conf. 9, 75-88 (1993). ZBL0937.14031,

the non-singularity condition in Matsumura-Monsky result can be removed.

Here is the ZBMATH review of the paper:

From the introduction: Associated to any complex space $$M$$ is the group of morphisms from $$M$$ to $$M$$, that are biholomorphic away from the singular set of $$M$$. We denote by $$\text{Aut}(M)$$, this group of automorphisms of $$M$$. If moreover, $$M$$ is a projective variety then $$\text{Aut}(M)$$ has a subgroup $$\text{Lin}(M)$$ consisting of those automorphisms of the ambient projective space, which restrict to automorphisms of $$M$$.

H. Matsumura and P. Monsky [J. Math. Kyoto Univ. 3, 347-361 (1964; Zbl 0141.37401)] proved that if $$M$$ is an irreducible, non-singular hypersurface of degree $$d\ge 3$$, in $$\mathbb{P}^n$$, where $$n\ge 3$$, then for $$(d,n)\ne(4,3)$$, $$\text{Aut}(M)= \text{Lin} (M)$$. By an alternate proof we show that the condition of non-singularity can be removed.

A. Howard and A. J. Sommese [in: Manifolds and Lie groups, Pap. Honor Y. Matsushima, Prog. Math. 14, 145-158 (1981; Zbl 0483.32016)] proved that for the same collection of hypersurfaces, the order of $$\text{Aut}(M)$$ is bounded by a constant multiple of $$d^n$$. We give an alternate proof for this theorem, and generalize this result to two classes of singular hypersurfaces.\par Most of the results presented here are part of M. J. Bradley's author's doctoral thesis at the Univ. Notre Dame. For the entire collection see [Zbl 0930.00040].

• Thank you for the answer. However, I can not find the paper over the internet. When they say that in order to have $Aut(M) = Lin(M)$ the condition of non-singularity of $M$ can be removed do they mean that this holds without any assumption on the singularities of $M$? – user125056 Mar 7 '19 at 9:26
• Yes, they say precisely so ($M$ is just assumed irreducible). I have to say that I did not check the correctness of the paper. – Francesco Polizzi Mar 7 '19 at 9:40