The Only Complex 4-Net Is the Hesse Configuration

Abstract

It has been conjectured that the only nets realizable in ${\mathbb{CP}}^2$ are 3-nets and the Hesse configuration (up to isomorphism). We prove this conjecture.

We change the (previously wrong) construction of the multi-cyclic cover. In the new construction singularities appear and resolving them gives a contribution to the signature. This changes some of the calculations, however all results in the previous version concerning existence of nets remain true