Jet schemes and truncated wedge schemes.

Abstract

In this thesis, we present some results on jet schemes and truncated wedge schemes of monomial schemes and determinantal varieties. We give a multiplicity formula for the irreducible components of the jet schemes of a reduced monomial hypersurface. We also conjecture and prove in certain cases a formula for the multiplicity of the irreducible jet schemes of a multiple point in the affine line. We give explicit generators for several primary components of the jet schemes of the simple monomial scheme Spec <italic>k</italic> [<italic> x, y</italic>]/(<italic>xy</italic>). Our study of jet schemes of determinantal varieties reveals that these schemes are not irreducible in general. We show that the odd jet schemes and the second jet scheme of essentially all determinantal varieties are reducible. In the special case of the variety of matrices of rank at most one, we give the number and dimensions of the irreducible components of its jet schemes. Furthermore, we develop the theory of truncated wedge schemes. We prove some basic properties and give an irreducibility criterion for truncated wedge schemes of a locally complete interesection variety. We show that the reduced subscheme of a truncated wedge scheme of any monomial scheme is again a monomial scheme. We give explicit generators of each of the minimal primes of any truncated wedge schemes of a monomial hypersurface. We also discover that truncated wedge schemes of a monomial scheme need not be equidimensional and may have embedded components. Lastly, we conjecture and prove in some cases that the multiplicity of a truncated wedge scheme of a reduced monomial hypersurface along each of its irreducible components is one.