This is simpler
sage: psi = C.hom(liftedbasis, P2)
sage: psi.image()
Closed subscheme of Projective Space of dimension 2 over Rational Field
defined by:
x^2 + x*y + 2*y*z
On Monday, October 30, 2023 at 5:45:27 AM UTC+9 Nils Bruin wrote:
On Monday, 30 October 2023 at 00:19:47 UTC+13 Kwankyu wrote:
What is your code?
P2.<x,y,z> = ProjectiveSpace(QQ, 2)
f = 2*x^5 - 4*x^3*y*z + x^2*y*z^2 + 2*x*y^3*z + 2*x*y^2*z^2+ y^5
C = Curve(f)
kC = C.function_field()
D = kC(kC.base_field().gen(0)).differential().divisor()
L,m,s = (-D).function_space()
#the routine below is a bit of a shortcut based on how the affine patch for
kC
#is chosen. In more general code this would need to be a little more
sophisticated
def liftkC(u):
return sum([(m.numerator()(y/x))/(m.denominator()(y/x))*(z/x)^i for i,m
in enumerate(u.list())])
liftedbasis = [liftkC(m(b)) for b in L.basis()]
den = lcm([b.denominator() for b in liftedbasis])
liftedbasis = [parent(x)(b*den) for b in liftedbasis]
phi = P2.hom(liftedbasis,P2)
phi(C) # this fails
C._forward_image(phi,check=False) #this seems to work!
Of course, one could also do some linear algebra with (-D).function_space()
and (-2*D).function_space() to figure out this image.
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