# Cohomology rings and algebraic torus actions on hypersurfaces in the product of projective spaces and bounded flag varieties

## TL;DR

This paper investigates the maximal algebraic torus actions on Milnor hypersurfaces, computes their automorphism groups, classifies certain toric hypersurfaces, and describes their cohomology rings.

## Contribution

It provides the first comprehensive analysis of torus actions and cohomology structures for Milnor hypersurfaces in complex algebraic geometry.

## Key findings

- Maximum dimension of torus actions on Milnor hypersurfaces determined
- Automorphism groups of Milnor hypersurfaces computed
- Cohomology rings of classified hypersurfaces explicitly described

## Abstract

In this paper, for any Milnor hypersurface we find the largest dimension of effective algebraic torus actions on it. The proof of the corresponding theorem is based on the computation of the automorphism group for any Milnor hypersurface. We find all generalised Buchstaber-Ray and Ray hypersurfaces that are toric varieties. We compute the Betti numbers of these hypersurfaces and describe their integral singular cohomology rings in terms of the cohomology of the corresponding ambient varieties.