Linkage of curves in

Let be a curve. Let be its homogeneous defining ideal. Suppose that are two homogeneous polynomials such that is a regular sequence. Then is also a curve, and , for some (possibly reducible) curve D. D is called the link of C via the complete intersection X.

Linkages are an important technique in the theory of curves. In this example, we compute the link D of the rational quartic curve via a complete intersection (f, g), where both f and g have degree three. This curve has degree five. After we find the equations of D, we then see whether the result is a nonsingular curve.

The ideal defining the curve D is the ideal quotient

This is accomplished in Macaulay using the quotient command.

We find the singular locus of any variety by using the following proposition (see Hartshorne, pg 36).

The following list of remarks refers to the Macaulay session below.

1. It is necessary to choose f and g sufficiently ``generic'' in order for the result to be a nonsingular curve. In this example we use polynomial expressions using the generators for the ideal .
2. Before determining the codimension of the ideal J, it is essential to concatenate the equations from to the equations from the minors of the Jacobian matrix. It is a common mistake to omit this step!

%  type i_C  ; the ideal of the rational quartic
; c3-bd2 bc-ad b3-a2c ac2-b2d 

%  ; first we choose two cubics in this ideal
%  ideal i_X
! number of generators ?  2
! (1,1) ?  (c3-bd2)+(a+b-c+d)*(bc-ad)
! (1,2) ?  (b3-a2c)+2*(ac2-b2d)+3d(bc-ad)

%  ; now we compute the ideal "i_D" of D
%  quotient i_X i_C i_D
; 0.[378k]1.2.3..4..
; computation complete after degree 4

%  putmat i_D
; 1
; 5
; a3+a2b+3/2ab2+b3-5/2a2c+3/2b2c+ac2-1/2bc2+2c3-a2d-4abd-1/2b2d \
;     +3/2acd+3/2bcd-1/2ad2-5bd2+9/2d3
; a2b+ab2+abc+2b2c+ac2-2bc2+2c3-4a2d-3abd+b2d+4acd+2bcd-4ad2 \
;     -4bd2+3cd2
; ab2+2b3-a2c-b2c+2ac2+bc2-2abd-3b2d+acd+5bcd-2c2d-3ad2-2bd2+3d3
; abc+b2c-bc2+c3-a2d-abd+acd+bcd-ad2-bd2
; b3-a2c+2ac2-2b2d+3bcd-3ad2

%  ; is this nonsingular?
%  ; first construct the ideal J
%  jacob i_D Df

%  wedge Df 2 J   ; 2 by 2 minors, since 2 = codim C

%  flatten J J

%  concat J i_D   ; don't forget to add the ideal i_D!!

%  ; now compute the codimension of J
%  std J J
; 34.5.
; computation complete after degree 5

%  codim J
; codimension : 4

%  ; the codimension is 4, and therefore this curve is nonsingular.