finitely many points of $\mathbb{P}^r_k$ are contained in some $D(f)$

Question

How can one prove that given finitely many maximal points $Z=\{x\}$,there exists $f\in Γ(P^r,O(n))$ with $f(x)\neq 0$ for all $x\in Z$?I wonder if there is some solution using cohomology or something(e.g in the case of $P^1$ Riemann-Roch works),rather than commutative algebra as illustrated in illusie’s Topic in Algebraic Geometry as “Each $x \in Z $corresponds to some homogenous prime ideal $p_x ∈ ProjB. $ Since for each $x \in Z, B_+ = \sum_{n>0}B_n$ is not contained in $p_x$, we can find a homogenous element $f \in B_n $such that $f \in p_x $for all $x \in Z$ ([B] III.§1.4, Prop 8), then $f \in Γ(P,O(n)) $and $f(x)= 0 $for all $x.$”where [B] III.§1.4, Prop 8) is Bourbaki ‘s commutative algebra.

Answer 1

$\newcommand{\PSP}{\mathbb P}$$\newcommand{\Ohol}{{\mathcal O}}$$\newcommand{\sheaf}[1]{{\mathcal{#1}}}$ Let $Q_1,\ldots,Q_s$ be finitely many different maximal points of the projective space $X=\PSP^r_k$ with $k$ a field.

Let $\sheaf{I}_j$ be the ideal sheaf of the point $Q_i$. Then $\sheaf{I}_1 \cap \cdots \cap \sheaf{I}_s = \sheaf{I}$ is the ideal sheaf of $Q_1 \cup \cdots \cup Q_s$.

Consider the exact sequences

$$0 \to \sheaf{I}(d) \to \Ohol_X(d) \to k(Q_1) \times \cdots \times k(Q_s) \to 0$$

As $H^1(X,\sheaf{I}(d)) = 0$ for $d > d_0$ just choose an $f \in H^0(X,\Ohol_X(d))$ with image $1 \times \cdots \times 1$ in $H^0(X, k(Q_1) \times \cdots \times k(Q_s))$ for a $d > d_0$.

I think this comes quite close to a "solution using cohomology" and is quite useful to remember as a technique in more general situations, for example in Hartshorne's ex. III, 4.8 (d)

Answer 2

This does not answer your question but I just wanted to state this fact.

Claim: Finite set of points of $\mathbb{P}^n_k$ lie in a principal open subset.

Proof. Let $S$ be a finite set of points of $\mathbb{P}^n_k$. Now each point in $S$ corresponds to a homogeneous prime ideal of $k[T_0,\dots,T_n]$. We take their union $ \bigcup \mathfrak{p}_i$ and consider a homogeneous element of positive degree $f \in k[T_0,\dots,T_n] \setminus \bigcup \mathfrak{p}_i$. The fact that such an element exists is just the graded version of the prime avoidance lemma. Clearly $V_{+}(f) \cap S = \emptyset$.

Similar arguments show that if you have a finite set of points in any quasi-projective scheme over any ring, then they lie in an affine open subset.