The tangent developable of a rational normal curve

In this example, we find the equations of the tangent developable to the rational normal curve in . Recall that the rational normal curve is given by the image of the polynomial map , and that the tangent developable is the union of all the tangent lines of the curve.

The tangent line at the image point is given parametrically by

Thus the tangent developable is given parametrically by

In order to find the equations of the image we just embed using this map.

The following remarks refer to the Macaulay example below.

1. The variables of used in this example are a,b,c,d,e,f,g. Each polynomial of the map T has degree 6 in the variables x,y,s,t, and so we set the weight of each variable to be 6.
2. The next several commands compute the map T without having to write down the equations directly. The cat command provides a handy way to write down matrices of indeterminates.
3. We construct the polynomial map f by using the power command. This is simpler than typing in all the degree 6 monomials directly. More importantly, changing this example to compute the tangent developable in , for some other n becomes a much easier task.
4. It turns out that the ideal of the tangent developable is generated by degree 2 polynomials in . In the ring R, these polynomials have degree 12. We use the Macaulay parameters ``autocalc'' and ``autodegree'' to change the highest degree to which Macaulay will compute. After we are done with the standard basis, we set ``autocalc'' back to its previous value. Otherwise all subsequent computations will be done only to degree 12!
5. Finally, using the elim command we obtain the ideal of the tangent developable. We then place this ideal into a cleaner ring, and the compute its minimal finite free resolution. The betti command is used to display concisely the graded betti numbers of the resolution.

%  ring R
! characteristic (if not 31991)       ? <return>
! number of variables                 ?  11
!  11 variables, please               ?  stxya-g
! variable weights (if not all 1)     ?  1:4 6
! monomial order (if not rev. lex.)   ?  4 6
[252k];   largest degree of a monomial        : 512 512 512 

%  ; create 3 ideals (s,t), (x,y), and (a,b,c,d,e,f,g).
%  cat s st
! rows ?  0
! columns ?  0 1

%  cat x xy
! rows ?  0
! columns ?  0 1

%  cat a ag
! rows ?  0 
! columns ?  0..6

%  ; compute the polynomial map T
%  power st 6 f

%  type f
; s6 s5t s4t2 s3t3 s2t4 st5 t6 

%  jacob f Df s t

%  transpose Df Df

%  mult xy Df T

%  type T
; 6s5x 5s4tx+s5y 4s3t2x+2s4ty 3s2t3x+3s3t2y 2st4x+4s2t3y 

; t5x+5st4y 6t5y 

%  ; compute the ideal, T1, to eliminate.  Compute a standard basis and 
%  ; perform the elimination.
%  subtract ag T T1

%  type T1
; -6s5x+a -5s4tx-s5y+b -4s3t2x-2s4ty+c -3s2t3x-3s3t2y+d 

; -2st4x-4s2t3y+e -t5x-5st4y+f -6t5y+g 

%  set autocalc 1    ; see above remark

%  set autodegree 12 ; see above remark

%  std T1 T1
; 67.8.9.10.11.12.

%  set autocalc -1

%  elim T1 I

%  type I
; e2-4/3df+1/3cg de-3/2cf+1/2bg d2-9/5bf+4/5ag ce-8/5bf+3/5ag 

; cd-3/2be+1/2af c2-4/3bd+1/3ae 

%  ; place the result ideal I into a cleaner ring, and then compute
%  ; its resolution.
%  ring S
! characteristic (if not 31991)       ? <return>
! number of variables                 ?  7
!   7 variables, please               ?  a-g
! variable weights (if not all 1)     ? <return>
! monomial order (if not rev. lex.)   ? <return>
;   largest degree of a monomial        : 76 

%  fetch I I

%  nres I I
; 2..3...4.[315k]...5....6....
; computation complete after degree 6

%  betti I
; total:      1     6    10     6     1 
; --------------------------------------
;     0:      1     -     -     -     - 
;     1:      -     6     5     -     - 
;     2:      -     -     5     6     - 
;     3:      -     -     -     -     1 

%  pres I
; 
; ----------------------------------

; e2-4/3df+1/3cg de-3/2cf+1/2bg d2-9/5bf+4/5ag ce-8/5bf+3/5ag 

; cd-3/2be+1/2af c2-4/3bd+1/3ae 
; 
; ----------------------------------

; 0    a     b     c     d     0                0             
; a    0     -2/3c -8/9d -e    ce-8/5bf         de-6/5cf      
; -4b  -8/3c 0     8/9e  4/3f  12/5bg           -e2+9/5cg     
; 9/2b 3c    0     -e    -3/2f -de+3/2cf-16/5bg 3/2df-12/5cg  
; 3c   8/3d  2/3e  0     -1/3g -9/5cg           6/5ef-9/5dg   
; -3d  -3e   -f    -1/3g 0     9/5dg            -9/5f2+12/5eg 

; 0               0               0               
; 0               0               0               
; cd-3/2be+1/2af  -ce+8/5bf-3/5ag c2-4/3bd+1/3ae  
; 0               d2-9/5bf+4/5ag  0               
; -d2+9/5bf-4/5ag 0               0               
; 0               0               -d2+9/5bf-4/5ag 
; 
; ----------------------------------

; 0         -32/81dg 32/45eg          e2-4/3df-9/5cg    
; -d2+9/5bf 4/9cg    -32/45dg         -de+3/2cf+27/10bg 
; -9/5af    -cf      8/5df            -9/2ce-27/10ag    
; 0         bf-4/9ag -de              9/2be-3/2af       
; ad        0        ce-8/5bf+32/45ag ae                
; 0         -5/9d    e                -3c               
; a         5/9c     -8/9d            4b                
; -8/3d     0        -8/27g           -8/3e             
; 3c        -5/9e    4/3f             0                 
; 3e        5/27g    0                4f                

; -8/9d2+8/5bf 32/45cg       
; 0            -16/15bg      
; 0            4/3d2+16/15ag 
; -ad          -cd           
; ac           c2-4/3bd      
; a            c             
; 0            -4/3b         
; -8/3c        8/9e          
; 4b           0             
; 8/3d         -4/3f         
; 
; ----------------------------------

; 8/27de-4/9cf+4/27bg   
; 8/5cd-12/5be+4/5af    
; c2-4/3bd+1/3ae        
; -8/27d2+8/15bf-1896ag 
; -1/3e2+4/9df-1/9cg    
; -ce+8/5bf-3/5ag       
; 
; ----------------------------------