The Fellegi-Sunter model for record linkage can be framed as a latent class model. As such, it is vulnerable to convergence to boundary estimates when attempting to maximize the likelihood. Boundary estimates are the result of attempting to maximize a likelihood that is not strictly convex. In the context of identifiable log-linear models, Fienberg and Rinaldo propose a theory for extending classic exponential models to allow the estimation of dimensionally-collapsed parameter spaces. We recast the ideas of Fienberg and Rinaldo in the context of the Fellegi-Sunter model and we suggest a related approach based on computing dimensionally-reduced toric varieties in the associated linear algebra. This approach leads to fully parameterized applications of the Fellegi-Sunter model in record-linkage situations where parameters are not nominally estimable.