Nodal quintic surfaces and lines on cubic fourfolds

TL;DR

This paper investigates the geometry of nodal quintic surfaces with 16 nodes, drawing analogies to singular Kummer surfaces, and explores the structure of lines on cubic fourfolds using advanced algebraic geometry techniques.

Contribution

It provides new insights into the structure of nodal quintic surfaces and their relation to Fano varieties of lines on cubic fourfolds, with refined proofs and perspectives.

Findings

Characterization of nodal quintic surfaces with 16 nodes

Interpretation of the double cover as a Fano variety of lines

Refinements of known results with new proofs

Abstract

We study nodal quintic surfaces with an even set of 16 nodes as analogues of singular Kummer surfaces. The interpretation of the natural double cover of an even 16-nodal quintic as a certain Fano variety of lines could be viewed as a replacement for the additive structure of the cover of a singular Kummer surface by its associated abelian surface. Most of the results in this article can be seen as refinements of known facts and our arguments rely heavily on techniques developed by Beauville, Murre, and Voisin. Results due to Izadi and Shen are particularly close to some of the statements. In this sense, the text is mostly expository (but with complete proofs), although our arguments often differ substantially from the original sources.