A fascinating problem in Peter Cameron’s tribute to Atiyah

Tribute here: https://cameroncounts.wordpress.com/2019/01/12/michael-atiyah/

“The highlight of the lectures was a surprising theorem. Given a “generic” polynomial f of degree 6 over the complex numbers, in how many ways can you write f = g^(2)+h^(3), where g has degree 3 and h degree 2? Clearly multiplying h by a cube root of unity or g by −1 doesn’t change anything, so ignore this.

The answer is 40. Atiyah proved this by writing down a curve with a horrendous singularity at one point; after dealing with that point, the rest was well-behaved, and he could come up with the answer.”

Anybody can point towards books which explain this in more detail? Why did he mention the fact that multiplying by -1 or cube root of unity won’t change anything? Can this be found in e.g. Hartshorne?

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