monomialCurveIdeal -- make the ideal of a monomial curve

monomialCurveIdeal -- make the ideal of a monomial curve

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   2            2            3

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