- Let
`X=Z(xy-1)⊂A`. Then the function field of^{2}`A(X)=k[x,x`is clearly isomorphic to^{-1}]`k(X)`. In fact, projection on the`x`-coordinate defines a morphism`X→A`that induces an isomorphism of function fields.^{1} - Let
`X=Z(y`. In the previous chapter, we constructed a morphism^{2}-x^{3}-x^{2})⊂A^{2}`A`given by the equations^{1}→ X

and`x = m`^{2}-1

This morphism defines an injection`y = m`^{3}-m.

and hence induces a map`A(X) → k[m] ⊂ k(m),``k(X)→ k(m)`. The inverse of this map on function fields is given by`m = x/y`. - Suppose the characteristic of the field
`k`differs from 3. Then the curve`Z(x`is^{3}+y^{3}-1)**not**a rational curve. To see this, suppose that it is rational. Then we could write

where`x = p(t)/r(t), and y = q(t)/r(t),``p`,`q`,`r∈ k[t]`are relatively prime polynomials. Now the equationsp ^{3}+ q^{3}-r^{3}= 0 3p ^{2}p' + 3q^{2}q' -3r^{2}r'=0 Sincep q -r p' q' -r' p ^{2}q ^{2}r ^{2}= 0 0 . `p≠0`, this system is equivalent towhose general solution isp q -r 0 pq'-qp' rp'-pr' u v w = 0 0 , By symmetry, we can assume thatu v w = C rq'-qr' pr'-rp' pq'-qp' .

Since`deg(r)≤ deg(q) ≤ deg(p).``p`,^{2}`q`and^{2}`r`are relatively prime, we conclude that^{2}

Therefore,`p`^{2}divides rq'-qr'.

This contradicts the assumption we made about the degrees, and so no such polynomials can exist. Note that this proof immediately generalizes to show that`2deg(p) ≤ deg(r) + deg(q) -1.``Z(x`is not a rational curve if^{n}+y^{n}-1)`n>2`and the characteristic of`k`is relatively prime to`n`. - The surface
`Z(x`in^{3}+y^{3}+z^{3}-1)`A`is rational. To see this, consider the lines^{3}L = Z(z-1, x+y) M = Z(z-ω, x+ω y), `ω`is a primitive cube root of unity. Then`L`and`M⊂ X`, but`L∩ M=∅`. Now choose a plane`E⊂A`that contains neither^{3}`L`nor`M`. Every point`x∈A`uniquely determines a line^{3}\ L\ M`N(x, L, M)`that contains`x`and has a nonempty intersection with both`L`and`M`. Thus, one gets a morphism

by`A`^{3}\ L \ M → A^{2}

This map restricts to a well-defined morphism everywhere on`x |→ N(x, L, M) ∩ E.``X`. Since`N(x, L, M) ∩ X`generically determines a unique point of`X\ L \ M`, the map induces an isomorphism of function fields`k(X)≈ k(A`.^{2})

` g(φ _{1}(x), …, φ_{m}(x)) = 0
`

`φ ^{#} : A(Y) → A(U) ⊂
k(X).`

`φ ^{*} : k(Y) → k(X).`

` (ψ(y _{1}), …, ψ(y_{m}))`

`k(X) = k(t _{1}, …, t_{n})[s]/( minimal
polynomial of s).`

`A(Y) = k[t _{1}, …, t_{n}, s]/( minimal
polynomial of s).`

Comments on this web site should be addressed to the author:

Kevin R. CoombesDepartment of Biomedical Informatics

The Ohio State University

Columbus, Ohio 43210