--Worksheet 2, Mathematics 535, Fall 2002


--Veronese varieties
--Segre varieties  
--Both of above are toric varieties, so we may find equations by methods
--of worksheet 1

--Goals: Finding the equations of varieties under the Veronese or Segre map
--

S=QQ[z_0..z_5]
R=QQ[x_0..x_2]
--computation of the ideal of the Veronese surface as toric variety
ver25={x_0^2,x_1^2,x_2^2,x_0*x_1,x_0*x_2,x_1*x_2} 
ker map(R,S,ver25)


I=ideal(x_0^3+x_1^3+x_2^3)  --Harris Example 2.7, page 24

--Now find equations satisfied by Veronese of Z(I)
ker map (R/I,S,ver25)
--see Harris page 24 
gens oo  --note 9 quadrics for generators

R=QQ[x_0..x_3]
S=QQ[z_0..z_9]
-- the Veronese threefold in P^9
ker map(R,S,{x_0^2,x_1^2,x_2^2,x_3^2,x_0*x_1,x_0*x_2,x_0*x_3,x_1*x_2,x_1*x_3,x_2*x_3})

-- the equations of P^4 x P^1 as a subvariety of P^9
R=QQ[x_0..x_4,y_0,y_1]
segre={x_0*y_0,x_0*y_1,x_1*y_0,x_1*y_1,x_2*y_0,x_2*y_1,x_3*y_0,x_3*y_1,x_4*y_0,x_4*y_1}
ker map(R,S,segre)


--PROBLEMS
--1. Find the equations of the Segre 3 fold
--2. Find the equations of the twisted cubic in P^3 under the
---- Veronese map to P^9