Open affine neighborhood of points

$X$ is a variety and there are $m$ points $x_1,x_2,\cdots,x_m$ on $X$. Can we find an open affine set which contains all $x_i$s?

A such variety is sometimes called FA-scheme (finite-affine). Quasi-projective schemes over affine schemes (e.g. quasi-projective varieties over a field) are FA.

On the other hand, there are varieties which are not FA. Kleiman proved that a propre smooth variety over an algebraically closed field is FA if and only if it is projective.

Some more details can be found in § 2.2 in this paper.

There is an easy proof for projective varieties $X$ over a field. Just take a homogeneous polynomial $F$ which doesn't vanish at any of $x_1,\dots, x_m$. Then the principal open subset $D_+(F)$ is an affine open subset containing the $x_i$'s. The existence of $F$ is given by the graded version of the classical prime avoidance lemma:

Edit

Let $R$ be a graded ring, let $I$ be a homogeneous ideal generated by elements of positive degrees. Suppose that any homogenous element of $I$ belongs to the union of finitely many prime homogeneous ideals $\mathfrak p_1, \dots, \mathfrak p_m$. Then $I$ is contained in one of the $\mathfrak p_i$'s.

A (sketch of) proof can be found in Eisenbud, § 3.2. For the above application, take $\mathfrak p_i$ be the prime homogeneous ideal corresponding to $x_i$ and $J=R_+$ be the (irrelevant) ideal generated by the homogeneous elements of positive degrees. As $R_+$ is not contained in any $\mathfrak p_i$, the avoidance lemma says there exists a homogeneous $F\in R_+$ not in any of the $\mathfrak p_i$'s.

This method can be used to prove that any quasi-projective variety $X$ is FA (embed $X$ into a projective variety $\overline{X}$ and take $J$ be a homogeneous ideal defining the closed subset $\overline{X}\setminus X$. let $F\in J$ be homogeneous and not in any $\mathfrak p_i$, then $D_+(F)$ is affine, contains the $x_i$'s and is contained in $X$).