# monomialCurveIdeal -- make the ideal of a monomial curve

## Synopsis

• Usage:
I = monomialCurveIdeal(R, a)
• Inputs:
• R, , 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 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

## 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 .