# Automorphisms of certain affine complements in the projective space

## TL;DR

This paper characterizes automorphisms of certain affine complements in projective space, showing they extend to projective automorphisms under specific conditions on the hypersurface.

## Contribution

It proves that automorphisms of affine complements of hypersurfaces with a unique singular point extend to projective automorphisms, revealing the structure of their automorphism groups.

## Key findings

- Automorphisms of affine complements are restrictions of projective automorphisms.
- For general hypersurfaces, the automorphism group of the complement is trivial.
- The result applies to hypersurfaces with a specific singularity and degree conditions.

## Abstract

We prove that every biregular automorphism of the affine algebraic variety ${\mathbb P}^M\setminus S$, $M\geqslant 3$, where $S\subset {\mathbb P}^M$ is a hypersurface of degree $m\geqslant M+1$ with a unique singular point of multiplicity $(m-1)$, resolved by one blow up, is a restriction of some automorphism of the projective space ${\mathbb P}^M$, preserving the hypersurface $S$; in particular, for a general hypersurface $S$ the group $\mathop{\rm Aut} ({\mathbb P}^M\setminus S)$ is trivial.