0
$\begingroup$

I have a very simple question about quasiprojective varieties. I'm reading "Basic Algebraic Geometry" of Shafarevich.

Definition: A quasiprojective variety is an open subset of a closed projective set. A quasiprojective sub-variety $ Y \subset X$ is a subset $ Y \subset X$ such that $Y$ is itself a quasiprojective variety.

I want to prove that $ Y \subset X $ is a questiprojective subvariety if and only if $Y = Z - Z_a $ where $ Z,Z_a \subset X$ are closed projective sets.

Well obviously one side is direct. I have troubles with proving that if $ Y \subset X $ is a quasiprojective subvariety , then I can write $Y$ in that form.

I have the following : $ X = Z_1 \cap U_1 $ and $ Y = Z_2 \cap U_2 $ where $Z_i$ are closed projective sets and $U_i$ are open projective sets. And I also know that $ Y \subset X $ . Well... how can I write $Y$ in the form $Y = Z-Z_a $ where $Z, Z_a \subset X $ and closed projective sets? Please help me )=!

  • 2
    It says $Y=Z−Z_1$ with $Z,Z_1 \subset X$ closed subsets. I think it means that $Z,Z_1$ are closed subsets of $X$, not closed subsets of the ambient projective space, in which case the assertion is trivially true.2012-12-30

1 Answers 1

1

The correct statement is given in comments by Makoto.

In general it is not possible to write $Y=Z\setminus Z_a$ as you described. Indeed, take $Y=X$. Then $Z=X$. So $Z$ is never projective if $X$ is not projective.