I'm trying to prove this equivalence which are also the definition of quasi-projective algebraic sets:

$X\subset \mathbb P^n$ is an open subset of its closure $\Leftrightarrow$ $X\subset \mathbb P^n$ is an open subset of a closed subset of $\mathbb P^n$.

The first side of this equivalence is easy, I need help with the converse.

I have another question:

The quasi-projective algebraic sets in Hartshorne's book are the open subsets of $\mathbb P^n$, I don't know why the definition above is equivalent to the definition in Hartshorne.

I really need help.

I would be very grateful if someone could help me

Thanks a lot.


Equivalences have directions, not sides, so I'm not sure which one you're asking about. Is it "$\Leftarrow$"?

If $X$ is open in $Y$, which is closed in $\mathbf{P}^n$, then $X$ is open in $Y\cap \overline{X}$. As $Y$ is closed and contains $X$, $Y$ contains $\overline{X}$, so $Y\cap \overline{X} = \overline{X}$

I don't know which Hartshorne book you're looking at, but the one I'm looking at defines on p. 10 a quasi-projective variety as an open subset of a projective variety, not projective space.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.