We introduce the logarithmic Quot space which generalises the Quot schemes of a variety X to the geometry of a pair (X,D) with D a simple normal crossing divisor on X. Such logarithmic Quot spaces interact well with degeneration and gluing formulae. We prove that the logarithmic Quot space is a separated and universally closed logarithmic algebraic space, which in nice cases is bounded. A special case, called the logarithmic Hilbert space, parameterises families of proper monomorphisms, and in this way is exactly analogous to the classical Hilbert scheme. The new complexity in the logarithmic 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. The logarithmic Quot space admits a tropicalisation called the space of tropical supports, which is defined in terms of a natural moduli functor on the category of rational polyhedral cones. The space of tropical supports, a polyhedral analogue of the Hilbert scheme, is representable by ``piecewise linear spaces'' which are introduced here to generalise cone complexes to allow non--convex geometries. The key to studying geometry of logarithmic Quot spaces is the natural map from logarithmic Quot space to the space of tropical supports. A central example of a Quot scheme is a linear system. We characterise logarithmic linear systems as toric stacks and describe the associated fans in terms of the secondary polytopes of Galfand, Kapranov, and Zelevinksy. We use this characterisation to study logarithmic Pandharipande--Thomas spaces of toric surfaces. We calculate the Euler-Satake characteristics of logarithmic Pandharipande--Thomas spaces in a number of basic examples. Consider a surjection of sheaves O_X^n -> on a torus G_m^n. In establishing boundedness of the logarithmic Quot space, we give a tropical characterisation of the toric compactifications of G_m^n for which the closure of F is logarithmically flat. Our tropical characterisation gives an algebro--geometric interpretation to the Gr\"obner complex, refines a celebrated result of Tevelev, and illustrates how the structure of a piecewise linear space appears in nature.
algebraic geometry, logarithmic geometry, mathematics
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