Automorphism groups and cylindricity of weighted hypersurface Fano threefolds

TL;DR

This paper investigates the automorphism groups and cylindricity properties of specific weighted hypersurface Fano threefolds, focusing on 20 families with particular geometric features, expanding understanding of their symmetries and rationality.

Contribution

It provides a detailed analysis of automorphism groups and cylindricity for the remaining 20 families of weighted hypersurface Fano threefolds, which were not previously studied.

Findings

Determined automorphism groups for all members in 20 families.

Established cylindricity properties for these families and their forms.

Extended knowledge on the geometric structure and symmetries of these Fano threefolds.

Abstract

It is well known that there are totally 130 deformation families of quasi-smooth terminal weighted hypersurface Fano threefolds and all members belonging to 95 families of Fano indices one are birationally rigid. Among remaining $35$ families, $15$ families have the property that general members are irrational, in particular, any of them has a finite automorphism group and is not cylindrical. In the present paper, we will observe the cylindricity and the full automorphism groups of every member in the remaining $20$ families. Moreover, we deal with the cylindricity of their forms.