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.