Regina Lectures on Fat Points

Let K be an algebraically closed field. For n ≥ 0, let An denote Kn, and let A = K[An] denote K[X1, . . . , Xn]. We refer to An as affine n-space. For any subset S ⊆ An, let I(S) ⊆ A denote the ideal of all polynomials that vanish on S. (For those familiar with Spec, the affine scheme associated to S is Spec(A/I(S)). Note that any ideal I ⊆ A defines an affine subscheme of Spec(A), and ideals I and J define the same affine subscheme if and only if I = J .) For n ≥ 0, let Pn denote equivalence classes of nonzero (n + 1)-tuples, where (a0, . . . , an) and (b0, . . . , bn) are equivalent if there is a 0 6= t ∈ K such that (a0, . . . , an) = t(b0, . . . , bn). Let R = K[Pn] denote K[x0, . . . , xn]. We refer to Pn as projective n-space. For any subset S ⊆ Pn, we obtain an associated homogeneous ideal (i.e., an ideal generated by homogeneous polynomials, also called forms) I(S) ⊆ R, the ideal generated by all homogeneous polynomials that vanish on S, where we regard R as being a graded ring with each variable having degree 1 and constants having degree 0. For those familiar with Proj, the projective scheme associated to S is Proj(R/I(S)). If M = (x0, . . . , xn), any homogeneous ideal I ⊆M ⊂ R defines a subscheme Proj(R/I) ⊆ Proj(R) = Pn, and homogeneous ideals I ⊆ M and J ⊆ M define the same subscheme if and only if It = Jt for t 0 (or equivalently, if and only if I ∩ M t = J ∩ M t for t 0), where It and Jt are the homogeneous components of the ideals of degree t. (Thus It is the vector space span of the elements of I of degree t. This applies in particular to R, so Rt is the K-vector space span of the homogeneous polynomials in R of degree t, and we have It = Rt ∩ I.) Given a homogeneous ideal I, among all homogeneous ideals J such that It = Jt for t 0 there is a largest such ideal contained in M which contains all of the others, called the saturation of I, denoted sat(I). Thus given homogeneous ideals I ⊆ M and J ⊆ M , we have Proj(R/I) = Proj(R/J) if and only if sat(I) = sat(J). We say an ideal is saturated if it is equal to its saturation. Thus geometrically we are most interested in homogeneous ideals which are saturated. We can regard An ⊂ Pn via the inclusion (a1, . . . , an) 7→ (1, a1, . . . , an). We have an isomorphism of function fields K(X1, . . . , Xn) = K(An) ∼= K(Pn) = K(x1/x0, . . . , xn/x0) defined by Xi 7→ xi x0 . Given any ideal 0 6= I ⊆ A, define α(I) to be the degree of the nonzero element of I of least degree. If 0 6= J ⊆ R is a homogeneous ideal, then α(J) again is the degree of a nonzero element of J of least degree. (Such an element is necessarily homogeneous if J is homogeneous.)