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I have encountered a problem in Shafarevich's book and I have no clue:

Let $X$ be a hypersurface given by the equation $f_{m-1}(x_1,\cdots, x_n) + f_m(x_1,\cdots, x_n) = 0$ where $f_{m-1}$ and $f_m$ are non-zero homogeneous polynomials of degrees $m-1$ and $m$, respectively. Prove that if $X$ is irreducible, it is rational. (Shafarevich, Problem I.3.5.)

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@ Hezudao: I would be interested in a precise answer to this question. My only attempt was to consider the projection $X\to \mathbb A^{n-1}$ as a possible birational map, but I'm not able to conclude. Also, I looked at Shafarevich and there the exercise is stated with $m=n$, so what is the truth? – Brenin Oct 15 '12 at 19:36
I think you can set $x_2=t_1x_1,x_3=t_2x_1,\cdots,x_n=t_{n-1}x_1$, and the rest is easy. – Hezudao Oct 16 '12 at 1:50
up vote 6 down vote accepted

Why don't you try a concrete example first?

The curve $y^2 -x^3 = 0$ is an example of the kind of hypersurface that Shafarevic is considering. If you consider the intersection of the line $y = tx$ with this curve, you get $t^2 x^2 = x^3,$ which has the unique non-zero solution $x = t^2$. In this way one obtains the rational parameterization $t \mapsto (t^2,t^3)$ of the original curve. Can you see how to generalize this?

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