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Geometry and logic have been intertwined in the study of mathematical truth since at least the time of Euclid. In the modern era, logical proofs are treated as formal mathematical objects and studied through denotational semantics alongside structures such as vector spaces and coalgebras. Constructive logics, particularly linear logic, are of contemporary interest due to their deep connections with computer science. In linear logic, an atomic formula X must be used exactly once, whereas the nonlinear formula !X may be reused freely and, in our model, is represented by the "space of proofs of X". We formalise this moduli space by defining !X as the Hilbert scheme of the projective scheme associated with X. This construction yields a novel geometric model of linear logic, and offers new insights into the relationships between geometry, logic, and computer science