Problem Set 20 Problem 1. Let F and G be two homogeneous polynomials in R = k[x0, x1, . . . , xn]. Summary: Problem Set 20 Problem 1. Let F and G be two homogeneous polynomials in R = k[x0, x1, . . . , xn]. Assume that F and G do not share any common factors. Suppose that A, B R and AF + BG = 0. Prove that there exists an H R such that A = -HG and B = HF. (In other words prove that every syzygy of F G is of the form H -G F .) Suppose the Hilbert Polynomial of R/I is F(X) = arXr + ar-1Xr-1 + · · · + a1X + a0. Associated to I is a scheme (not necessarily equidimensional). The dimension of the scheme is equal to r. The degree of the scheme is ar/r!. If F(X) = a1X + a0 then the scheme is one dimensional and has degree a1. Let G = 1 - a0. If the scheme is one dimensional then G is called the arithmetic genus. Problem 2. a) Use the previous problem to determine the Hilbert Polynomial of k[x0, x1, x2, x3]/(F, G) where F is irreducible of degree 3 and G is irreducible of degree 5. Collections: Mathematics