--Worksheet 7, Mathematics 535, Fall 2002




--Hilbert polynomials , resolutions


--Goals: computing Hilbert functions, polynomials
 
 

R=QQ[x_0..x_2]
I1=ideal(x_0,x_1) ---- the point [0,0,1]
I2=ideal(x_0,x_2) ---- the point [0,1,0]
I3=ideal(x_2,x_1) ---- the point [1,0,0]
I=intersect(I1,I2,I3)  --- the ideal of the three points

hilbertSeries I       --- the power series generating function 

hilbertSeries (I,Order =>8)  ---first 8 terms of Hilbert function values

hilbertPolynomial (I,Projective => false) -- the Hilbert polynomial

I4=ideal(x_0-x_1,x_2)

Iline=intersect(I2,I3,I4)  -- three points on a line
hilbertSeries (Iline,Order=>8)
J=intersect(I,Iline)   --- three points on a line , one off 
hilbertSeries (oo,Order=>8)

hilbertSeries (intersect(I,ideal(x_0-2*x_1,x_0-3*x_2)),Order=>8)  --4 non-collinear points
 
resolution I     -- free resolution of ideal of three non-collinear points
betti oo         -- degrees of generators of syzygies 
resolution Iline
betti oo         -- note that the degrees are as large as 4


--PROBLEMS
--1. Find the Hilbert polynomial and Hilbert function of the ideal
--   I=
	     

--  
--2.  Find the resolution of the module S(X) when X is is the scroll
--     given by the following radical ideal defining a rational scroll in P^4

--     S=QQ[z_0..z_4]
---    scroll=ideal(z_1*z_3-z_2*z_4,z_2*z_3-z_0*z_4,z_0*z_1-z_2^2)     	  

--3.  Find the Hilbert function and Hilbert polynomials for the scroll
--    of problem 2



--4.  Is the rational normal scroll in P^4 a complete intersection?