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 Incoming links: Bijective monomial cusp parametrization with nonregular inverse

The parametrization lands on the curve because

(t^{3})^{4} = (t^{4})^{3}

im F^{#} = k [t^{3}, t^{4}] \subset k [t] = A (A^{1}) .

3 a + 4 b

for nonnegative integers

a, b

. No such exponent is one, so no linear combination of these monomials is

t

. Thus

F^{#}

is not surjective and $F$ is not an isomorphism.

It is nevertheless bijective on points. Away from the origin,

x \neq = 0

and

t = y / x

gives

t^{3} = x

and

t^{4} = y

using

x^{4} = y^{3}

; the origin comes only from

t = 0

. The rational inverse cannot extend regularly at the origin: if

t = P (t^{3}, t^{4}) / Q (t^{3}, t^{4})

with

Q (0) \neq = 0

, then

tQ (t^{3}, t^{4})

has a nonzero degree-one term, whereas

P (t^{3}, t^{4})

has none.

F is bijective, but its inverse is not regular at the cusp .

This distinguishes a set-theoretic bijection from an isomorphism of varieties in every characteristic.