An elliptic curve on the Veronese in

Let be a polynomial map. It is useful to compute the image of a subvariety , rather than . The technique in the previous example is easily modified to handle this case.

Let be the ideal defining the variety X. The ideal defining the image of X in is

This is computed exactly as in the example above.

In this example, we find the image of the nonsingular cubic curve under the Veronese map to : . After placing the result into a ``clean'' ring S, we compute its standard basis and find its Hilbert function.

The following remarks refer to the Macaulay session below.

1. As in the previous example, we create a ring with variables x,y,z of weight one, and variables a,b,c,d,e,f of weight two. We choose the product order ``3 6'' which has the effect of eliminating the variables x,y,z.
2. The Hilbert series of the result is

   

   The curve has degree 6 and genus 1, as expected.

%  ring R
! characteristic (if not 31991)       ? <return>
! number of variables                 ?  9
!   9 variables, please               ?  xyza-f
! variable weights (if not all 1)     ?  1 1 1 2
! monomial order (if not rev. lex.)   ?  3 6
[189k];   largest degree of a monomial        : 512 235 

%  ideal J
! number of generators ?  7
! (1,1) ?  a-x2
! (1,2) ?  b-xy
! (1,3) ?  c-xz
! (1,4) ?  d-y2
! (1,5) ?  e-yz
! (1,6) ?  f-z2
! (1,7) ?  y2z-x3-xz2

%  ; now compute a standard basis, and then eliminate
%  std J J
; 23.4.5.6.7.8.
; computation complete after degree 8

%  elim J I

%  type I
; e2-df ce-bf cd-be c2-af bc-ae b2-ad ac+cf-df ab-de+bf a2-be+af 

%  ; now move I to a cleaner ring
%  ring S
! characteristic (if not 31991)       ? <return>
! number of variables                 ?  6
!   6 variables, please               ?  a-f
! variable weights (if not all 1)     ? <return>
! monomial order (if not rev. lex.)   ? <return>
;   largest degree of a monomial        : 117 

%  fetch I I

%  ; the base ring of I is now S.
%  ; Compute a standard basis and the Hilbert series of S/I
%  std I I
; 23.4.
; computation complete after degree 4

%  hilb I

;       1 t  0
;      -9 t  2
;      16 t  3
;      -9 t  4
;       1 t  6

;       1 t  0
;       4 t  1
;       1 t  2

; codimension = 4
; degree      = 6
; genus       = 1