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.

arXiv commentWe 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

References in corpus (5)

• Completely reducible hypersurfaces in a pencil
• Realization of finite Abelian groups by nets in P^2
• Multinets, parallel connections, and Milnor fibrations of arrangements
• Using Tropical Degenerations For Proving The Nonexistence Of Certain Nets
• Classification of k-nets

Cited by in corpus (1)

• $L^2$-type invariants and cohomology jump loci for complex smooth quasi-projective varieties