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
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i2 : monomialCurveIdeal(R,{3,5})
5 2 3
o2 = ideal(b - a c )
o2 : Ideal of R
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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
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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 ,
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2 2 3
b d - a*e*f, b c - a*d*f, b - a*c*f)
o4 : Ideal of R
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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
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