Four-Dimensional Projective Orbifold Hypersurfaces

ABSTRACT

We classify four-dimensional quasismooth weighted hypersurfaces with small canonical class and verify a conjecture of Johnson and Kollár 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.

Keywords:

• Fano
• Calabi-Yau
• threefold
• fourfold
• orbifold
• weighted hypersurface

2000 AMS Subject Classification:

• Primary 14J35
• Secondary 14J70, 14J30

Acknowledgments

We are grateful to Miles Reid and Olof Sisask who explained this algorithm to us in the first place, and whose re-implementation of it in 2005 to recalculate the 7555 Calabi–Yau threefolds for the database at [Brown and Kasprzyk 02] was our starting point, and also to Jennifer Johnson and János Kollár for discussion of their conjecture. Our thanks to John Cannon for providing Magma for use on the Imperial College mathematics cluster and to Andy Thomas for technical support.