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 $C$?



Configuration of points on a plane curve
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](https://www.ams.org/journals/proc/2011-139-12/S0002-9939-2011-11204-8/S0002-9939-2011-11204-8.pdf).


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 $C$.


