The reason for the confusion is that a smooth, projective plane curve of degree d has genus (d-1)(d-2)/2, which is 2 or greater starting at d=4. Hence the phrasing in the article, which is missing the “smooth, projective” hypothesis. The equation y = x^4 doesn’t define a smooth curve when extended to the projective plane, because it has a singularity at infinity.
Comments
The curve they mean y = x^4 is irreducible but the genus is 0 since it’s isomorphic to the affine line.
The correct description is “a smooth curve of genus at least 2”.
https://en.wikipedia.org/wiki/Faltings%27_theorem
The reason for the confusion is that a smooth, projective plane curve of degree d has genus (d-1)(d-2)/2, which is 2 or greater starting at d=4. Hence the phrasing in the article, which is missing the “smooth, projective” hypothesis. The equation y = x^4 doesn’t define a smooth curve when extended to the projective plane, because it has a singularity at infinity.
I think the theorem applies to any curve, if you take geometric genus.