The rational quartic in P3

Let be a polynomial map. We wish to compute the equations in defining the image of this map. Let , where .

The ideal I of the image of f consists of the polynomial relations on the polynomials , and therefore

We simply need to eliminate the variables from the ideal .

In Macaulay this is accomplished by first defining a ring containing both the ``x'' and ``y'' variables, and choosing either the elimination or product order which orders the ``x'' variables before the ``y'' variables. After computing a standard basis, we use the elim command to obtain the final answer.

In the example below we apply this technique to determine the ideal of the rational quartic curve in . This is defined to be the image of the polynomial map in .

The following remarks refer to the Macaulay session below.

1. All polynomials of J must be homogeneous (or quasi-homogeneous). This will be the case if we set the weight of each variable a, b, c, d to be four, and the weight of s and t to be one, by using the ring command. Notice that the last ``4'' is replicated so that every variable a,b,c,d has weight 4.
2. In this example we have chosen the product order by responding with ``2 4'' when asked for the monomial order.
3. Before defining the ring S, the matrix k contains the equations of the rational quartic which is the desired result. However, this ring has the extra variables s and t, and the weight of the variables is four. We define a ``cleaner'' ring S and then use the ring map command fetch to put the matrix k in this simpler ring. Next we compute its finite free resolution.

%  ;; Finding the image of the polynomial map (s4, s3t, st3, t4), and 
%  ;; its finite free resolution.
%  ring R
! characteristic (if not 31991)       ? <return>
! number of variables                 ?  6
!   6 variables, please               ?  sta-d
! variable weights (if not all 1)     ?  1 1 4
! monomial order (if not rev. lex.)   ?  2 4
;   largest degree of a monomial        : 512 512 

%  ideal j
! number of generators ?  4
! (1,1) ?  a-s4
! (1,2) ?  b-s3t
! (1,3) ?  c-st3
! (1,4) ?  d-t4

%  type j
; -s4+a -s3t+b -st3+c -t4+d 

%  std j j
; 45.6.7.8.9.10.11.12.13.14.15.16.
; computation complete after degree 16

%  elim j k
[126k]
%  type k
; c3-bd2 bc-ad b3-a2c ac2-b2d 

%  ; put this ideal into a ring with only variables a,b,c,d all
%  ; having weight one.
%  ring S
! characteristic (if not 31991)       ? <return>
! number of variables                 ?  4
!   4 variables, please               ?  a-d
! variable weights (if not all 1)     ? <return>
! monomial order (if not rev. lex.)   ? <return>
;   largest degree of a monomial        : 512 

%  fetch k k

%  ; at this point the base ring of "k" is "S".
%  ; find the minimal finite free resolution of "k"
%  nres k kres
; 2.3...4...5...
; computation complete after degree 5

%  pres kres  ; and display it
; 
; ----------------------------------

; bc-ad c3-bd2 b3-a2c ac2-b2d 
; 
; ----------------------------------

; ac bd b2 c2 
; 0  -a 0  -b 
; -d 0  -c 0  
; -b c  -a d  
; 
; ----------------------------------

; -c 
; -b 
; d  
; a  
; 
; ----------------------------------