Let $C\subset\mathbb{P}^2$ be a smooth plane curve of degree six. On $C$ there are $21$ points given as the intersection points of two lines choosen among a set of seven lines. More precisely there are distinct lines $L_1,\dots,L_7$ such that $p_{i,j} = L_i\cap L_j \in C$ for all $i,j = 1,\dots,7$.
Does anyone know if these $21$ points have a special meaning from the point of view of $C$? And if so can they be recovered from the equation of $C$?
A set of points in the plane is called a star configuration of type $\ell$ if it is the set of pairwise intersections of some $\ell$ lines, no three concurrent. If the lines are defined by $L_1,\dotsc,L_\ell$, it's clear that each product $\hat{L}_j = L_1 \dotsm \widehat{L_j} \dotsm L_\ell = (\prod L_i)/L_j$ vanishes on the set of points $X$. In fact, these generate the ideal of $X$, $I_X = (\hat{L}_1,\dotsc,\hat{L}_\ell)$.
A star configuration $X$ lies on a curve $C$ defined by $F$ if and only if $F \in I_X$. In the case you are asking about, a star configuration of type $7$ lying on a sextic curve, $F$ has the same degree as all the $\hat{L}_j$, so they correspond to an expression $F = a_1 \hat{L}_1 + \dotsb + a_7 \hat{L}_7$, for some constants $a_1,\dotsc,a_7$.
Starting from $X$ there's evidently a family of sextics containing $X$. Conversely, a general sextic $C$ contains no such star configuration, by counting constants (the dimension of the space of sextics is way bigger than the dimension of the space of star configurations). For more see Carlini-Van Tuyl.
It seems reasonable to guess that if a sextic $C$ contains a star configuration of type $7$, then that star configuration should be unique. But that fails for some cases, e.g., if $F = L_1 \dotsm L_6$ is a product of $6$ general lines (then $F$ contains a family of star configurations given by any general choice of a seventh line $L_7$). Perhaps it holds for all irreducible $C$ or all smooth $C$, but I don't know.
But there's at least a partial answer to your second question: from the equation $F$ of $C$, if you can write $F$ as a sum of products of $6$ linear forms, drawn from the same set of $7$ linear forms, then it recovers a star configuration contained in $C$.
