One higher dimensional analog of jet schemes

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.