  We develop the theory of truncated wedge schemes, a higher dimensional analog
of jet schemes. We prove some basic properties and give an irreducibility
criterion for truncated wedge schemes of a locally complete intersection
variety analogous to Musta\c{t}\v{a}'s for jet schemes. We show that the
reduced subscheme structure of a truncated wedge scheme of any monomial scheme
is itself a monomial scheme in a natural way by explicitly describing
generators of each of the minimal primes of any truncated wedge scheme of a
monomial hypersurface. We give evidence that the irreducible components of the
truncated wedge schemes of a reduced monomial hypersurface all have
multiplicity one.
