  We study point-line configurations through the lens of projective geometry
and matroid theory. Our focus is on their realisation spaces, where we
introduce the concepts of liftable and quasi-liftable configurations, exploring
cases in which an $n$-tuple of collinear points can be lifted to a
non-degenerate realisation of a point-line configuration. We show that forest
configurations are liftable and characterise the realisation space of liftable
configurations as the solution set of certain linear systems of equations.
Moreover, we study the Zariski closure of the realisation spaces of liftable
and quasi-liftable configurations, known as matroid varieties, and establish
their irreducibility. Additionally, we compute an irreducible decomposition for
their corresponding circuit varieties. Applying these liftability properties,
we present a procedure generate some of the defining equations of the
associated matroid varieties. As corollaries, we provide a geometric
representation for the defining equations of two specific examples: the
quadrilateral set and the $3\times4$ grid. While the polynomials for the latter
were previously computed using specialised algorithms tailored for this
configuration, the geometric interpretation of these generators was missing. We
compute a minimal generating set for the corresponding ideals.
