Four-dimensional projective orbifold hypersurfaces

TL;DR

This paper classifies four-dimensional quasismooth weighted hypersurfaces with small canonical class, verifies a conjecture on infinite series with anticanonical sections, and classifies canonical, Calabi-Yau, and Fano fourfolds arising from quotient singularities.

Contribution

It provides a comprehensive classification of four-dimensional quasismooth weighted hypersurfaces and verifies a conjecture for these cases, extending understanding of their geometric properties.

Findings

Classified four-dimensional quasismooth weighted hypersurfaces with small canonical class.

Verified Johnson and Kollar's conjecture for fourfolds with anticanonical hyperplane sections.

Identified conditions for hypersurfaces to be canonical, Calabi-Yau, or Fano in four dimensions.

Abstract

We classify four-dimensional quasismooth weighted hypersurfaces with small canonical class, and verify a conjecture of Johnson and Kollar on infinite series of quasismooth hypersurfaces with anticanonical hyperplane section in the case of fourfolds. By considering the quotient singularities that arise, we classify those weighted hypersurfaces that are canonical, Calabi-Yau, and Fano fourfolds. We also consider other classes of hypersurfaces, including Fano hypersurfaces of index greater than 1 in dimensions 3 and 4.