Logarithmic Quot spaces, boundedness, and K-tropicalizations

TL;DR

This paper develops the theory of logarithmic Quot spaces, establishing their boundedness and properness, introduces K-tropicalizations as scheme-structure-sensitive tropicalizations, and shows their parametrization by finite polyhedral complexes.

Contribution

It proves boundedness and properness of logarithmic Quot spaces and introduces K-tropicalizations, linking tropical geometry with K-theory and scheme structures.

Findings

Boundedness and properness of logarithmic Quot spaces established.

Introduction of K-tropicalizations sensitive to scheme structures.

K-tropicalizations with fixed numerics are parametrized by finite polyhedral complexes.

Abstract

Logarithmic Hilbert and Quot spaces are generalizations of their traditional versions adapted to study pairs and degenerations. The logarithmic Quot spaces of $(X,D)$ parameterize "algebraically transverse" (logarithmically flat) quotient sheaves on degenerations of $X$. We prove boundedness and deduce properness of logarithmic Quot spaces. The results complete the basic foundations of logarithmic Quot spaces and specialize to work of Li-Wu and Maulik and the second author in special cases. Boundedness relies on two results of independent interest. First, we show that for a simple normal crossing pair $(X, D)$ and a subscheme $Z$, there is a smallest logarithmic space ${X}^\flat$ modifying $X$ such that the strict transform of $Z$ is algebraically transverse. Precisely, given $Z$ in $X$, there is a canonical logarithmic space $X^\flat$ over $X$ with the following universal property - an snc logarithmic blowup $X'\to X$ makes the strict transform of $Z$ algebraically transverse if and only if $X'$ is a modification of $X^\flat$. Parallel results hold for arbitrary coherent sheaves. This proves boundedness for logarithmic quotients with fixed tropicalization. A logarithmic quotient sheaf defines a K-tropicalization, an enhancement of tropicalization that is sensitive to scheme structures. The K-tropicalization has the same relationship to K-theory as traditional tropicalization has to Chow, and is related to Gr\"obner theory and convex geometry via state and secondary polytopes. Using the K-theory of toric bundles, we derive a balancing condition for K-tropicalizations that imposes strong finiteness properties. The second key result is that K-tropicalizations with fixed numerics are parametrized by a finite-dimensional polyhedral complex.