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Let $k$ be a field and $P_{1}, ..., P_{r}$ be homogeneous polynomials. Let consider $S = Proj(k[x_0, ..., x_{n}/(P_{1}, ..., P_{r}))$ which is a scheme over $k$ (see here for the definition : https://en.wikipedia.org/wiki/Proj_construction#Proj_as_a_topological_space). Let $\mathcal{O}_{S}$ denotes its structural sheaf. For a point $x$, $k(x)$ will denote its residual field.

Is there a "good" way to characterize the rational points over $k$? (i.e. the set of points where the map $k \mapsto k(x)$, induces by the structure of $k$ scheme, is an isomorphism).

Even for $P_1 = ... = P_r = 0$ I don't have any "good" description.

Thank you in advance.

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    $\begingroup$ A $k$-point will be an homogeneous prime ideal of the form $\mathfrak{p}=(x_i-a_0x_0,\ldots, x_i-a_nx_n)$ with $a_j\in k,a_i=1$ and $x_i\not \in \mathfrak{p}$. $\endgroup$
    – reuns
    Dec 20, 2022 at 20:48
  • $\begingroup$ Hello and thanks. Do you have any references of this? Thanks a lot. $\endgroup$
    – Analyse300
    Dec 21, 2022 at 18:02

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