The Logarithmic Quot space: foundations and tropicalisation

TL;DR

This paper develops a logarithmic analogue of the Hilbert and Quot schemes, introducing tropical supports and piecewise linear spaces to study degenerations and moduli of sheaves with new geometric and combinatorial tools.

Contribution

It constructs a logarithmic Quot space with a tropicalisation, generalizing classical moduli spaces and extending previous work to arbitrary rank and dimension.

Findings

The logarithmic Quot space is a separated and universally closed algebraic space.

The tropical support space is representable by piecewise linear spaces.

The construction extends logarithmic Donaldson--Thomas and Quot schemes to broader settings.

Abstract

We construct a logarithmic version of the Hilbert scheme, and more generally the Quot scheme, of a simple normal crossings pair. The logarithmic Quot space admits a natural tropicalisation called the space of tropical supports, which is a functor on the category of cone complexes. The fibers of the map to the space of tropical supports are algebraic. The space of tropical supports is representable by ``piecewise linear spaces'', which are introduced here to generalise fans and cone complexes to allow non--convex geometries. The space of tropical supports can be seen as a polyhedral analogue of the Hilbert scheme. The logarithmic Quot space parameterises quotient sheaves on logarithmic modifications that satisfy a natural transversality condition. We prove that our moduli space is a separated and universally closed logarithmic algebraic space. The logarithmic Hilbert space parameterizes families of proper monomorphisms, and in this way is exactly analogous to the classical Hilbert scheme. The new complexity of the space can then be viewed as stemming from the complexity of proper monomorphisms in logarithmic geometry. Our construction generalises the logarithmic Donaldson--Thomas space studied by Maulik--Ranganathan to arbitrary rank and dimension, and the good degenerations of Quot schemes of Li--Wu to simple normal crossings geometries.