monomialCurveIdeal -- make the ideal of a monomial curve

Synopsis

Usage:

I = monomialCurveIdeal(R, a)
Inputs:
- R, a polynomial ring, over a field
- a, a list, containing integers to be used as exponents in the parametrization of a rational curve
Outputs:
- I, an ideal, corresponding to the projective curve given parametrically on an affine piece by $t \mapsto (t^{a_0}, \dots, t^{a_n})$

Description

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 used.

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

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

A genus 7 curve with two singular points

i4 : monomialCurveIdeal(R, {6, 7, 8, 9, 11})

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

The smooth rational quartic in $\PP^3$

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

For the programmer

The object monomialCurveIdeal is a function closure.