Monday, April 22, 2013

HW #9

1. Let I be an ideal in C[x_1..x_n]. Let G_I be the sum of all the standard monomials. For example, if I = < x-y^2 > in C[x,y] with leading term x, then G_I = 1/(1-y). Whereas if the leading term is y^2, then G_I = (1+y)/(1-x). (Right?)

a. If I is homogeneous, what is the relation of G_I and the Hilbert series H_I?

b. Show that G_I is always a rational function.

c. If F is a subset of {1..n}, let t_F = product_{f in F} x_f / (1-x_f).
If I is generated by squarefree monomials, show G_I = sum_F t_F, where the sum is over those F such that product_{f in F} x_f is a standard monomial.

2. Consider pairs {(A,B) nxn matrices : AB, BA upper triangular}.

a. Use the "codim" command in Macaulay 2 to check that, for n=2,3, this is a complete intersection.

b. Assuming this for general n, what's its degree?

c. Decompose it (for n=2,3) and determine the degrees of the components. Check that they add up to the answer in (b).

3. In C[a,b,c,d], let I be the ideal of the ab-plane union the cd-plane.

a. What is that ideal?

b. Let J be the ideal < a + c - 1, b + d - 2 >. Describe V(I+J). Is it radical?

c. Let K be the ideal < a + c, b + d >. Describe V(I+J) and compute its degree.

d. Do you find (b) vs. (c) disturbing?