Question

Hello,

I am aware of the related question "Minimal size of an open affine cover", but would like to ask more specifically:

Do you have some elementary (i.e. not using hard things like compactification and such) proof for one of the following (here "variety" is separated over alg. closed field):

(1) Let $X$ be a variety; Can you show that $X$ can be covered by $C \cdot dim(X) + D$ open affines, where $C,D$ are universal constants?

(2) Let $X$ be quasi-projective; Can you show (1) for it with $C=1,D=1$?

(3) Let $X$ be smooth quasi-projective, and char. = 0; Can you show (2) for it?

It is easy for a variety $X$ to find an open affine whose complement is of smaller dimension than $X$. But I don't see how given $Y$ closed in $X$, to find an affine open $U$ in $X$ such that $Y-U$ is of smaller dimension than $Y$.

Sasha

Answer 1

Here is a way to do (2) (and hence (3)):

Let $X$ be a quasi-projective variety, i.e., $X=Y\setminus W$, where $Y,W\subseteq \mathbb P^n$ are (closed) projective varieties. Consider the irreducible decomposition $Y=\cup_i Y_i$ and observe that $I_W\not\subseteq \cup I_{Y_i}$ where $I_T\subseteq k[x_o,\dots,x_N]$ denotes the ideal of the set $T\subseteq \mathbb P^N$. Pick a homogenous polynomial $f$ of degree $d$ such that $f\in I_W\setminus (\cup_i I_{Y_i})$. Let $H=Z(f)$. Then $\mathbb P^n\setminus H$ is affine and hence so is $Y\setminus H$.

By construction $Y\setminus H\subseteq Y\setminus W=X$ and $H\not\supseteq Y_i$ for any $i$ by the choice of $f$. Therefore $\dim (Y_i\cap H)<\dim Y_i$ so we may use induction on $\dim X$. Notice that the affine subset is obtained as an affine subset of the ambient projective space intersected with our variety, so the affine varieties obtained subsequently are restrictions of affine subvarieties of the original $X$.

Answer 2

I am not sure what you mean by "hard", but here is an answer to (2). I claim that if $X$ is quasiprojective of dimension $d$, there is an affine morphism $X\to\mathbb{P}^d$. This obviously implies (2): cover $X$ by the preimages of the $d+1$ standard affine charts.

Proof of claim: take a dense open immersion $j:X\hookrightarrow\overline{X}$, with $\overline{X}$ projective. Blowing up if necessary, you can assume that $\overline{X}\setminus X$ is a Cartier divisor. Then $j$ is an affine morphism (locally defined by inverting one function). On the other hand, by Noether normalization, there is a finite (hence affine) morphism $p:\overline{X}\to\mathbb{P}^d$, so the composite map $p\circ j$ is affine.

(In this argument, the only subtle point is the notin of affine morphism, in particular the fact that it is a local condition on the target.)

About the last question: for given $X$, a positive answer is equivalent to the "Chevalley property" that every finite subset of $X$ has an affine neighborhood. This clearly holds for quasiprojective $X$. If $X$ is smooth and complete, this property is equivalent to projectivity (Kleiman).