--Worksheet 4, Mathematics 535, Fall 2002


--Radical ideals
--Decompositon of varieties
--Grassmannians
--Fano varieties
--Goals: Decompose varieteis into a union of irreducible varieties
---------Equations for grassmannian varieties
---------Equations for Fano varieties 





--When working with affine varieties and coordinate rings it is
--important to work with radical ideals I contained in R such that
--R/I has no nilpotent elements.

R1=QQ[x,y,z]
I1=ideal(y-x^2,y)
radical I1
--note not radical suggested by second order contact at (0,0)
--some ideals
     
I=ideal(x,y)
J=ideal(x,z)
int=intersect (I^2, J^3)
primaryDecomposition int
decompose int


--every ideal is the intersection of ideals with prime radicals (primary)
--every radical ideal is the interesection of prime ideals



I=ideal(y-x^2,y-z*x)

radical oo


radical(ideal(y-x^2,y))


primaryDecomposition (ideal(y-x^2,y))


decompose (ideal(y-x^2,y))




Grassmannian(1,3)
Grassmannian(2,5)

clearAll

R=QQ[x_0..x_4]

I=ideal(x_0^2+x_1^2+x_2^2+x_3^2+x_4^2)

Fano(1,I)
dim oo
degree ooo
Fano(2,I)
dim oo

--note the smooth quadric in P^4 has no 2 planes contained in it 
--dimension 0 means pprojective dimension -1, or empty


---Fano varieties
clearAll
R=QQ[x,y,z,w]

I=ideal(x^3+y^3+z^3+w^3)  --- a smooth cubic surface
fanovar=Fano(1,I)  	  --- the variety of lines on the surface
dim fanovar	     	  --- the Fano variety is a finite set of points     	    
degree fanovar

--In fact every smooth cubic surface has exactly 27 lines on it




---Problems
--1.  Find the radical of  (x^2-2*x*y^4+y^6,y^3-y)
--2.  Find the prime decomposition of the radical of ideal in problem 1.
--3.  Find the equation of the Grassmannian of 3 planes in P^5
--4.  Find the Fano variety of 2-planes on the quadrics x^2+y^2+z^2+w^2=0  and
---   x^2+y^2 = 0 .  How many 2-planes are on these quadrics?