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Let $Y=\{(t,t^2,t^3)\mid t\in k\}$ be the twisted cubic curve. I'm trying to prove this curve is a variety, i.e., it's irreducible and affine algebraic set.

The easier part is to prove the twisted cubic curve is an affine algebraic set $(Y=Z(x^2-y,x^3-z))$.

I don't know how to prove that $Y$ is irreducible, I'm trying to prove that $(x^2-y,x^3-z)$ is prime, I think if I do this I proved what I want, but I found this hard to prove.

I need help to finish this question.

Thanks a lot.

user75086
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    Please use the word "prove" appropriately. In what you call "the easy part", you didn't prove anything; you applied a trivial definition. – Fly by Night Nov 15 '13 at 20:03
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    Show that the quotient is an integral domain by exhibiting an explicit isomorphism with the polynomial ring in one variable. – Zhen Lin Nov 15 '13 at 20:35
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    @ZhenLin Do you mean $k[x,y,z]/(x^2-y,x^3-z)\cong k[x,x^2,x^3]\cong k[t]$? – user75086 Nov 15 '13 at 21:23
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    How do we know $Y = Z(x^2-y, x^3-z)$? – user5826 Aug 24 '20 at 17:52
  • @user46372819 In case you are still wondering three years later: a point in $Y$ obviously kills both $x^2-y$ and $x^3-z$; on the other hand, a point $(a,b,c)$ that kills both equations is constrained to $b=a^2$ and $c=a^3$, so it has the form $(t,t^2,t^3)$. – Marc-André Brochu May 17 '23 at 18:02

4 Answers4

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Let's first prove that ideal $I:=(x^2-y, x^3-z)$ is prime. Suppose $f\cdot g \in I$. Using obvious isomorphisms $k[x,y,z] \cong (k[x,y])[z]$, $k[x,y] \cong (k[x])[y]$ and division algorithm, we have $$ f(x,y,z)=(x^3-z)f_1(x,y,z) + (x^2-y)f_2(x,y) + f_3(x), $$ $$ g(x,y,z)=(x^3-z)g_1(x,y,z) + (x^2-y)g_2(x,y) + g_3(x). $$ Now we have $f_3(x) \cdot g_3(x) \in I$, therefore $$f_3(x) \cdot g_3(x)=(x^3-z)h_1(x,y,z) + (x^2-y)h_2(x,y,z).$$ Insert $(x,y,z)=(t,t^2,t^3)$ and get $f_3(t) \cdot g_3(t) =0$ for all $t \in k$. If $k$ is algebraically closed (therefore infinite), we have $f_3 \cdot g_3 = 0$, so $f_3 = 0$ or $g_3= 0$. Then $f \in I$ of $g \in I$, so $I$ is prime (and therefore radical). We have $I(Y)=I(V(I)) = \operatorname{Rad}(I) = I$, which is prime. So $Y$ is irreducible.

Rafael Mrden
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    There is no need to assume that $k$ is algebraically closed (and the OP didn't do it): in the last polynomial relation plug in $y=x^2$ and $z=x^3$ to find $f_3=0$ or $g_3=0$. –  Nov 23 '13 at 09:44
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    Element calculations make everything complicated ... $I$ is prime because obviously $k[x,y,z]/(I) = k[x]$. – Martin Brandenburg Nov 23 '13 at 16:40
  • Thank you for you answer, I have a question, why can you use division algorithm? – user75086 Dec 03 '13 at 11:04
  • First you use division algorithm in polynomials in one variable $z$ with coefficients in $k[x,y]$. Dividing $f$ with $x^3-z$ yields $f(x,y,z)=(x^3-z)\cdot f_1(x,y,z) + r(x,y,z)$. But (degree in $z$ of $r$) must be less then (degree in $z$ of $x^3-z$)=1, so actually $r$ does not depend on $z$, i.e. $r(x,y,z)=r(x,y)$. – Rafael Mrden Dec 03 '13 at 11:18
  • Now do the same of $r(x,y)$, divide it with $x^2-y$ as polynomials in one variable $y$ with coefficients in $k[x]$. – Rafael Mrden Dec 03 '13 at 11:20
  • yes, I understood that, but in order to use division algorithm in polynomials in one variable $z$ with coefficients in $k[x,y]$, $k[x,y]$ should be a field. – user75086 Dec 04 '13 at 02:40
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    Not necessarily. It is enough that $k[x,y]$ is a commutative ring, and that leading coefficient of $x^3-z$ is an unit in that ring. The leading coefficient is $-1$ (in variable $z$ of course). Proof is the same as in the case of fields. – Rafael Mrden Dec 04 '13 at 09:01
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    Here is a reference: http://books.google.hr/books?id=Fge-BwqhqIYC&printsec=frontcover&dq=lang+algebra&hl=en&sa=X&ei=L-6eUsbyPKvn7AaWzICICQ&ved=0CCoQ6AEwAA#v=onepage&q&f=false (page 173). – Rafael Mrden Dec 04 '13 at 09:01
  • Thank you for your very complete answer and for your comments! – user75086 Dec 04 '13 at 10:49
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    @MartinBrandenburg good idea, but to prove this isomorphism requires some calculations also. – user75086 Dec 04 '13 at 10:51
  • You are welcome. – Rafael Mrden Dec 04 '13 at 10:52
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    No, no calculation is needed. It is trivial to check that both sides represent the same functor, hence they are isomorphic (Yoneda). – Martin Brandenburg Dec 05 '13 at 00:53
  • @RafaelMrđen Why did you use those polynomials to define $I$? Why not $xy-z$ and $xz-y^2$? – user5826 Mar 31 '21 at 22:04
  • @user46372819 I do not remember exactly, but I guess I used it because it worked. :) Did you try to obtain the result using the relations you suggest? – Rafael Mrden Apr 01 '21 at 07:31
  • @RafaelMrđen when we say that $f_3\cdot g_3=0$, so $f_3=0$ or $g_3=0$ are we using the fact that $k[t]$ is an domain? –  Jun 21 '21 at 13:35
  • @Nat Yes, that's right. – Rafael Mrden Jun 21 '21 at 17:35
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There is an obvious isomorphism $Y \cong \mathbb{A}^1$. This proves everything else.

Martin Brandenburg
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Expand a little on Martin's answer.

Define $f:X \to Y$ as $$f(x) = (x, x^2, x^3)$$ Define $g:Y \to X$ as $$g(x,y,z) = x$$

Then $f^{-1} = g$, $f$ is an isomorphism between $A^1$ and Y.

Now by p.g.29 of Shafarecich's Basic algebraic Geometry 1, there exists an isomorphism between coordinate rings of $A^1$ and $ Y$, which means $$k[x] \cong k[x,y,z]/((x^2-y),x^3-z)$$

Since k[x] is an integral domain, $k[x,y,z]/((x^2-y),x^3-z))$ is an integral domain, which means that $((x^2-y),x^3-z))$ is irreducible.

LVMIMAOQ
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Here is a way to show that $Y$ is irreducible with little computation: note that since $Y$ is closed (as the set of zeros of two polynomials) we have $Y=Z(I(Y))$, where $I(Y)$ is the ideal of $Y$. The only thing to check is that $I(Y)$, is prime. But it is easy to see that $I(Y)$ is the kernel of a surjective ring homomorphism $\phi:k[x,y,z]\to k[t]$, namely the homomorphism such that $\phi(x)=t,\phi(y)=t^2,\phi(z)=t^3$. Let's check this in detail, even though this is a bit tautological: To say that $p\in k[x,y,z]$ belongs to $\ker\phi$ means that the polynomial $p(t,t^2,t^3)\in k[t]$ is the zero polynomial. This implies, obviously, that for all $t\in k$ we have $p(t,t^2,t^3)=0$, i.e. that $p\in I(Y)$. Conversely, if $p\in I(Y)$ then the polynomial $\phi(p)=p(t,t^2,t^3)\in k[t]$ has any $t\in k$ as a root, which means that $\phi(p)=0$. Hence $I(Y)=\ker\phi$.

It follows that the ring $A(Y)=k[x,y,z]/I(Y)$ is isomorphic to $k[t]$, which is integral, hence $I(Y)$ is prime, and $Y$ is an affine variety isomorphic to the affine line. Q.E.D.

Of course the ideal $I(Y)$ is the radical of $(x^2-y,x^3-z)$, and the above arguments show that in fact $I(Y)=(x^2-y,x^3-z)$, but the computation of $I(Y)$ is not needed.

jacaboul
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