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Let k be an algebraically closed field, and let k[x] denote the polynomial ring in n variables x_{1}, x_{2},..., x_{n} over k. We describe an elementary construction of the Hilbert scheme of d points of affine n-space A = Spec(k[x]). The k-points of this Hilbert scheme correspond to the closed subschemes of A that are defined by ideals I in k[x] such that k[x]/I is k-free of dimension d, the colength of I. The construction is accomplished by patching together a finite collection of affine schemes which correspond to the order ideals of size d in the variables x. An order ideal (of size d) is a set B of d monomials in the variables x such that if m is in B and monomial m' divides m, then m' is in B. Given such a B, we define U_{B} (as a k-point set) to be the set of points in the Hilbert scheme corresponding to ideals I such that the quotient k[x]/I is k-free with basis B. The subscheme U_{B} represents a sub-functor of the Hilbert functor that is represented by the entire Hilbert scheme, and the existence of U_{B} (as an object representing this sub-functor) can be established using only polynomial arithmetic, nothing more advanced, such as Grassmannians. One obtains a concrete description of the coordinate ring of U_{B} as a quotient of a polynomial ring. The entire Hilbert scheme is then obtained by patching the U_{B} together along overlaps. In fact, the construction of U_{B} is carried out over an arbitrary (commutative and unitary) ground ring.
In addition to describing this construction, we will endeavor to set the stage with a brief introduction to the theory of Hilbert schemes of points.
Refreshments at 4:15 pm in Harder 202.
This is the last CRANTS of the semester so we plan to have dinner after the talk. The exact time/place will be announced later.