Affine complements in the projective space with the trivial group of automorphisms

TL;DR

The paper studies automorphisms of affine complements ${\mathbb P}^M\setminus S$ with $M=2e+1\ge5$, where $S\subset{\mathbb P}^M$ is a rational hypersurface. It constructs a new class of such hypersurfaces via bi-homogeneous polynomials $q_{m,m+1}$ and $q_{m+1,m}$, then uses a blow-up along $P_1\cup P_2$ and a detailed birational analysis, including discrepancies and resolution of singularities, to prove rigidity results. The main theorem (Theorem 0.1) shows that (i) any isomorphism ${\mathbb P}\setminus S\cong{\mathbb P}\setminus S'}$ with a degree-$2m+1$ hypersurface ${S'}$ is induced by a projective automorphism taking $S$ to $S'$; (ii) under a general-position condition $(*)$ the automorphism group is trivial; and (iii) for the symmetric subfamily ${\cal S}_{\rm sym}$ the group is ${\mathbb Z}/2{\mathbb Z}$ generated by a specific involution ${\tau}$. This advances understanding of birational rigidity and produces affine complements with trivial automorphism groups.

Abstract

We construct a new class of affine complements ${\mathbb P}^M\setminus S$ with the trivial group of automorphisms, where $S\subset {\mathbb P}^M$ is a rational hypersurface, $M$ is odd and $M\geqslant 5$.