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Macaulay2Doc :: monomialCurveIdeal

monomialCurveIdeal -- make the ideal of a monomial curve

Synopsis

Description

monomialCurveIdeal(R,a) yields the defining ideal of the projective curve given parametrically on an affine piece by t |---> (t^a1, ..., t^an).

The ideal is defined in the polynomial ring R, which must have at least n+1 variables, preferably all of equal degree. The first n+1 variables in the ring are usedFor example, the following defines a plane quintic curve of genus 6.

i1 : R = ZZ/101[a..f]

o1 = R

o1 : PolynomialRing
i2 : monomialCurveIdeal(R,{3,5})

            5    2 3
o2 = ideal(b  - a c )

o2 : Ideal of R
Here is a genus 2 curve with one singular point.
i3 : monomialCurveIdeal(R,{3,4,5})

             2         2       2   3
o3 = ideal (c  - b*d, b c - a*d , b  - a*c*d)

o3 : Ideal of R
Here is one with two singular points, genus 7.
i4 : monomialCurveIdeal(R,{6,7,8,9,11})

             2                    2                    2                   2 
o4 = ideal (e  - c*f, d*e - b*f, d  - c*e, c*d - b*e, c  - b*d, b*c*e - a*f ,
     ------------------------------------------------------------------------
      2            2            3
     b d - a*e*f, b c - a*d*f, b  - a*c*f)

o4 : Ideal of R
Finally, here is the smooth rational quartic in P^3.
i5 : monomialCurveIdeal(R,{1,3,4})

                        3      2     2    2    3    2
o5 = ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c)

o5 : Ideal of R