A geometric realization of an abstract simplicial complex 
is a simplicial complex 
whose vertex
scheme is isomorphic to 
. Such a realization is uniquely determined up to a linear
isomorphism (Munkres 1993).
In the theory of point-line configurations, a geometric realization represents a combinatorial configuration by distinct points and distinct straight lines with the prescribed incidences. Usage differs concerning the ambient geometry. Bokowski and Pilaud (2014) define geometric configurations in the real projective plane, where points at infinity are permitted. Berman et al. (2024) use the term strong geometric realization for a realization by points and straight lines in Euclidean space, usually the Euclidean plane. Therefore, "geometric realization" does not by itself unambiguously exclude points at infinity.
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Polycyclic Configuration." Electron. J. Combin. 31, #P4.54, 2024.
-Configurations." Comput. Geom. 47,
175-186, 2014.