Derived $F$-zips

TL;DR

This paper introduces derived $F$-zips, explores their properties and connections to classical theory, and applies the framework to study the moduli of Enriques-surfaces in positive characteristic.

Contribution

It defines derived $F$-zips for proper, smooth morphisms, analyzes their stack structure, and applies the theory to Enriques-surfaces, providing new insights in characteristic 2.

Findings

Derived $F$-zips associated to smooth morphisms are constructed and analyzed.

The stack of derived $F$-zips reveals new geometric properties.

Application to Enriques-surfaces offers new approaches to classical problems.

Abstract

We define derived versions of $F$-zips and associate a derived $F$-zip to any proper, smooth morphism of schemes in positive characteristic. We analyze the stack of derived $F$-zips and certain substacks. We make a connection to the classical theory and look at problems that arise when trying to generalize the theory to derived $G$-zips and derived $F$-zips associated to lci morphisms. As an application, we look at Enriques-surfaces and analyze the geometry of the moduli stack of Enriques-surfaces via the associated derived $F$-zips. As there are Enriques-surfaces in characteristic $2$ with non-degenerate Hodge-de Rham spectral sequence, this gives a new approach, which could previously not be obtained by the classical theory of $F$-zips.