Relation between projective space and graded rings There's a definition from synthetic geometry that says that a projective space is (\textit{$P,L,I$}),
where $\textit{P}$ is the set of points, $\textit{L}$, set of lines and $\textit{I}$ is the incidence relation, that is, tells if a point is in a line or not, and should satisfy the following axioms:
$\\ $

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*For every pair of distinct points $A$, $B$ there is a unique line that's incident to both of them, we call it $AB$

*If $A$, $B$, $C$ and $D$ are distinct points such that the lines $AB$ and $CD$ have a common point
(that's a point which's incident to both of them), then $AC$ and $BD$ also have a common point.

*Each line is incident to at least 3 points.

But then we have the definition from projective geometry, that's, if $S$ is a graded ring, consider
$Proj(S)$ to be set of homogeneous prime ideals not contaning the irrelevant ideal. If $A$ is a ring
then we call the $Proj(A[x_0,...,x_n])$ to be the projective n-space over $A$, since rings of
polynomials are graded by homogeneous elements.
I would like to know what are the points and lines in $Proj(A[x_0,...,x_n])$.
 A: $\operatorname{Proj}(A[x_0,...x_n])$ is not a "projective space" in the sense of synthetic geometry.  If $k$ is an $A$-algebra that is a field, then the set of $k$-points of $\operatorname{Proj}(A[x_0,...x_n])$ (that is, the morphisms $\operatorname{Spec} k\to \operatorname{Proj}(A[x_0,...x_n])$ over $\operatorname{Spec} A$) is a projective space in the sense of synthetic geometry, since it can be naturally identified with the projective space of the vector space $k^{n+1}$.  Explicitly, a point $(a_0,\dots,a_n)\in k^{n+1}$ determines a homomorphism of graded $A$-algebras $A[x_0,\dots,x_n]\to k[t]$ which maps each $x_i$ to $a_it$, and this determines a morphism $\operatorname{Spec} k\cong \operatorname{Proj}(k[t])\to\operatorname{Proj}(A[x_0,...x_n])$.  It can be shown that this morphism is unchanged if you multiply $(a_0,\dots,a_n)$ by a scalar, and that every $k$-point of $\operatorname{Proj}(A[x_0,...x_n])$ arises in this way, so that $k$-points of $\operatorname{Proj}(A[x_0,...x_n])$ are naturally in bijection with lines through the origin in $k^{n+1}$.
To be explicit, the set of lines through the origin in $k^{n+1}$ forms a projective space in the sets of synthetic geometry in the following way.  A "point" is a line through the origin in $k^{n+1}$, and a "line" is a plane through the origin in $k^{n+1}$.  The incidence relation is just containment: a "point" is on a "line" if the corresponding line through the origin is a subset of the corresponding plane through the origin.
