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The problem reads as follows:

Show that $[x_1:\ldots:x_n]\mapsto[x_1:\ldots:x_n:0]$ gives an isomorphism of $\mathbb P^{n-1}$ with $H_\infty\subset\mathbb P^n$. If a variety $V$ in $\mathbb P^n$ is contained in $H_\infty,V$ is isomorphic to a variety in $\mathbb P^{n-1}$. Any projective variety is isomorphic to a closed sub variety $V\subset\mathbb P^n$ (for some $n$) such that $V$ is not contained in any hyperplane in $\mathbb P^n$.

This statement is very similar to Proposition I.4.9 of Hartshorne, which states that any variety of dimension $r$ is birational to a hypersurface of $\mathbb{P}^{r+1}$. However, Fulton has not yet talked at all about dimensions of varieties or birational maps.

I am not sure how to approach this problem. Also, my intuition is failing me in seeing why or how the statement of the problem can be true. Any hints?

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  • $\begingroup$ Your parenthetical «i.e.» remark is not right, if the rest is to hold. $\endgroup$ – Mariano Suárez-Álvarez Feb 21 '15 at 20:50
  • $\begingroup$ @MarianoSuárez-Alvarez: You mean that the statement does not hold if $X$ is quasi-projective? I think i know where the problem is with my writing: Fulton actually says that $X$ should be isomorphic to a "closed subvariety". $\endgroup$ – Manos Feb 21 '15 at 21:02
  • $\begingroup$ You want to conclude that the variety is isomorphic to a projective variety satisfying certain conditions. It has to be projective to begin with! $\endgroup$ – Mariano Suárez-Álvarez Feb 21 '15 at 21:10
  • $\begingroup$ Pick an X which is affine variety, say an affine line. That is not isomorphic to any closed subvariety of any projective space. $\endgroup$ – Mariano Suárez-Álvarez Feb 21 '15 at 21:14
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    $\begingroup$ An affine line has nonconstant regular functions, and there are no nonconstant regular functions on a projective variety. $\endgroup$ – Mariano Suárez-Álvarez Feb 21 '15 at 21:34
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Hint. Consider a linear subspace of minimal dimension containing your variety.

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  • $\begingroup$ In particular, this has nothing to do with the result you quote from Hartshorne, really. $\endgroup$ – Mariano Suárez-Álvarez Feb 21 '15 at 20:52

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