  In this paper, the curvature structure of a (2+1)-dimensional black hole in
the massive-charged-Born-Infeld gravity is investigated. The metric that we
consider is characterized by four degrees of freedom which are the mass and
electric charge of the black hole, the mass of the graviton field, and a
cosmological constant. For the charged and neutral cases separately, we present
various constraints among scalar polynomial curvature invariants which could
invariantly characterize our desired spacetimes. Specially, an appropriate
scalar polynomial curvature invariant and a Cartan curvature invariant which
together could detect the black hole horizon would be explicitly constructed.
Using algorithms related to the focusing properties of a bundle of light rays
on the horizon which are accounted for by the Raychaudhuri equation, a
procedure for isolating the black hole parameters, as the algebraic
combinations involving the curvature invariants, would be presented. It will be
shown that this technique could specially be applied for black holes with zero
electric charge, contrary to the cases of solutions of lower-dimensional
non-massive gravity. In addition, for the case of massive (2+1)-dimensional
black hole, the irreducible mass, which quantifies the maximum amount of energy
which could be extracted from a black hole would be derived. Therefore, we show
that the Hawking temperatures of these black holes could be reduced to the pure
curvature properties of the spacetimes. Finally, we comment on the relationship
between our analysis and the novel roles it could play in numerical quark-gluon
plasma simulations and other QCD models and also black hole information paradox
where the holographic correspondence could be exploited.
