7 ms·
The curve they mean y = x^4 is irreducible but the genus is 0 since it’s isomorphic to the affine line.
by thornhill 6mo ago
The curve they mean y = x^4 is irreducible but the genus is 0 since it’s isomorphic to the affine line.
- jlev1 6mo agoThe correct description is “a smooth curve of genus at least 2”. https://en.wikipedia.org/wiki/Faltings%27_theorem 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.
- thornhill 6mo agoI think the theorem applies to any curve, if you take geometric genus.