--Worksheet 3, Mathematics 535, Fall 2002


--Projections
--Veronese projections
--Harris projections pg 36 

--Goals: understanding ideals of projections
 

--  Equations for projections of toric varieties:
--  The projection of any variety given by the Zariski closure of monomials is
--  also of that form, so the ideal of the variety may be computed by previous methods


--Equation of the Steiner Roman Surface 


R=QQ[x_0..x_3]

S=QQ[u_0,u_1,u_2]

romanpar={u_0^2+u_1^2+u_2^2,u_1*u_2,u_0*u_2,u_0*u_1}

roman=ker map(S,R,romanpar)

-- Another Steiner surface

tacnode=ker map(S,R,{u_0^2-u_1^2+u_2^2,u_2^2-u_1^2,u_1*u_2,u_0*u_1})

--The ellipsoid as projection of the Veronese surface      

ellipsoidpar={u_0^2+u_1^2+u_2^2,u_0*u_2,u_0*u_1,u_0^2-u_1^2-u_2^2}

ellipsoid=ker map(S,R,ellipsoidpar)

--Projection of Z(I) from linear space given by linear forms L_i
--If I is an ideal of R=K[x_0...x_n]  and L_i n-k linear polynomials in R 
--giving linear space to project from then let R2 have variables w_0,...w_{n-k-2} 
--The kernel of the map from R2 to R/I given by w_i->L_i is the Zariski closure
--of the projection of Z(I) from the linear space

--Harris problem 3.8, page 36 - projection of twisted cubic from [1,0,0,1]
--projection from linear space given by zeros of x_0-x_3,x_1,x_2
S1=QQ[s,t]
twistedcubic=ker map(S1,R,{t^3,t^2*s,t*s^2,s^3})
     
R2=QQ[w_0..w_2]
ker map(R/twistedcubic,R2,{x_0-x_3,x_1,x_2})



--PROBLEMS
--1.  Find the equations of the Cayley ruled cubic which is the closure of 
--        cayleycubicpar={u_0*u_1,u_0*u_2-u_1^2,u_1*u_2,u_2^2}
--2.  Find the equations of the Steiner surface given by the closure of
--        {u_1^2-u_2^2,u_1*u_2,u_0*u_1,u_0*u_2}
--3.  Find the projection of the twisted cubic from the point [0,1,0,0]