# monomialCurveIdeal -- make the ideal of a monomial curve

## Synopsis

• Usage:
I = monomialCurveIdeal(R,a)
• Inputs:
• R, a ring,
• a, a list of integers to be used as exponents in the parametrization of a rational curve
• Outputs:

## 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```