We construct a logarithmic version of the Quot scheme called the logarithmic Quot space. The associated tropical object, called the space of tropical supports, is a moduli space in its own right. We restrict attention to sheaves on logarithmic modifications satisfying a transversality condition first identified by Tevelev. The special case of the logarithmic Hilbert scheme is a moduli space of logarithmically flat proper monomorphisms in the category of logarithmic schemes. Our construction generalises the logarithmic Donaldsonâ€“Thomas space studied by Maulik and Ranganathan to arbitrary rank and dimension, and is amenable to gluing formulas.

We determine the rational class and Picard groups of the moduli space of stable logarithmic maps in genus zero, with target projective space relative a hyperplane. For the class group we exhibit an explicit basis consisting of boundary divisors. For the Picard group we exhibit a spanning set indexed by piecewise-linear functions on the tropicalisation. In both cases a complete set of boundary relations is obtained by pulling back the WDVV relations from the space of stable curves. Our proofs hinge on a controlled method for manufacturing test curves in logarithmic mapping spaces, for which we outline a range of future applications.

We study Maulik and Ranganathan's definition of logarithmic stable pairs spaces for toric surfaces relative the toric boundary. We provide a simplified and canonical construction of these spaces.

As a special case we construct the logarithmic linear system: a toric stack whose combinatorics is related to the secondary polytope construction of Gelfand, Kapranov and Zelevinsky. We identify logarithmic stable pairs spaces with the zero set of a section of an explicit vector bundle inside a toric stack, thus providing an explicit description of their virtual fundamental class. The canonicity of our moduli space allow us to perform a number of explicit computations in logarithmic Pandharipande--Thomas theory. These computations give an indication of the range of phenomena in this subject.