Let $K$ be an algebraically closed field. Say that a subset $C$ of the affine plane $K^2$ is an irreducible curve if there is an irreducible polynomial $f\in K[X,Y]$ (where $X$ and $Y$ are indeterminates) such that $$ C=\{(x,y)\in K^2\ |\ f(x,y)=0\}. $$ Let $\mathcal C$ be the set of all irreducible curves, and let $\mathcal L\subset\mathcal C$ be the subset of all lines. Clearly $\mathcal L$ satisfies the four axioms at the beginning of this Wikipedia entry.
Is $\mathcal L$ the only subset of $\mathcal C$ satisfying these four axioms?
To avoid any misunderstanding let me rephrase the same question in a more formal way.
Let $\Lambda$ be a subset of $\mathcal C$ such that
$\bullet$ If $a$ and $b$ are distinct points of $K^2$, then there is a unique $C\in\Gamma$ such that $a,b\in C$.
$\bullet$ Given any $C\in\Gamma$ and any $a\in K^2\setminus C$ there is a unique $C'\in\Gamma$ such that $a\in C'$ and $C\cap C'=\varnothing$.
Is $\Gamma$ necessarily equal to $\mathcal L$?
[Note that two of the four axioms have been omitted because they are trivially true in the present context.]
EDIT A positive answer to the above question would imply:
If $K$ an algebraically closed field and $X$ and $Y$ are indeterminates, then $K$ can be recovered (up to isomorphism) from the topological space $\text{Spec}(K[X,Y])$.
Indeed, $K[X,Y]$ being noetherian, the topology of $\text{Spec}(K[X,Y])$ depends only on its poset structure (because any closed subset of $\text{Spec}(K[X,Y])$ is a finite union of closures of points). Moreover, this poset structure depends only on the inclusions of the form $(f)\subset(X-x,Y-y)$, where $f\in K[X,Y]$ is irreducible and $x,y\in K$. Clearly such an inclusion holds if and only if $f(x,y)=0$, that is, if and only if the point $(x,y)$ belongs to the irreducible curve defined by $f$.
In other words, the topology of $\text{Spec}(K[X,Y])$ depends only on the incidence relations between points and irreducible curves in $K^2$.
If we knew that this incidence structure depended only on the incidence relations between points and lines in $K^2$, we could invoke Section II.5 of Emil Artin's book Geometric Algebra, which shows that $K$ can be recovered (up to isomorphism) from these incidence relations between points and lines in $K^2$.