monomialCurve

monomialCurve(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 = monomialCurve(R,{3,5})
     
     o2 = ideal | b5-a2c3 |
     
     o2 : Ideal

And a genus 2 curve with one singular point:

     i3 = monomialCurve(R,{3,4,5})
     
     o3 = ideal | c2-bd b2c-ad2 b3-acd |
     
     o3 : Ideal

Two singular points, genus = 7:

     i4 = monomialCurve(R,{6,7,8,9,11})
     
     o4 = ideal | de-bf e2-cf cd-be d2-ce c2-bd bce-af2 b2d-aef b2c-adf b3-acf |
     
     o4 : Ideal

Finally, the smooth rational quartic in P^3

     i5 = monomialCurve(R,{1,3,4})
     
     o5 = ideal | bc-ad c3-bd2 ac2-b2d b3-a2c |
     
     o5 : Ideal

Go to main index.

Go to concepts index.