On Sun, Jul 30, 2023 at 2:00 PM Kwankyu Lee <ekwan...@gmail.com> wrote:
>
> It seems related with the fact that the curve is reducible.
>

Could be, but some reducible curves pass the test:

sage: Kz.<x,y,z>=GF(3)[]
sage: f2=(x+y)*(x+y-1);g2=x+y+z+1
sage: C=Curve([f2,g2])
sage: C.irreducible_components()
[
Closed subscheme of Affine Space of dimension 3 over Finite Field of
size 3 defined by:
  z + 1,
  x + y,
Closed subscheme of Affine Space of dimension 3 over Finite Field of
size 3 defined by:
  z - 1,
  x + y - 1
]
sage: C.projective_closure()
Projective Curve over Finite Field of size 3 defined by -x0^2 + x3^2,
x0 + x1 + x2 + x3

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