Given a smooth projective variety $X$ of dimension $l$, we denote with $F(X,n)$ the configuration space of points $$ F(X,n):=\{(x_{1}, \dots, x_{n})\in X^{n}\: : \: x_{i}\neq x_{j}\text{ for each }i,j \} $$ In https://www.jstor.org/stable/2946581?seq=1#page_scan_tab_contents there is an explicit rational dg algebra $E(n)$ such that $H^{\bullet}(E(n))\cong H^{\bullet}(X, \mathbb{Q})$ (a model). In particular , as an algebra, $E(n)$ is isomorphic to a free $H^{\bullet}(X^{n})$ algebra with generators $G_{ab}$ for $1\leq a,b \leq l$, modulo some easy relations. The same is true for the differential. Now, choose $k$ distinct points $y_{1}, \dots y_{k}$ on $X$. We consider the configuration space of $X-\{y_{1}, \dots , y_{k}\}$

$$F(X, n; y_{1}, \dots , y_{k}):=F(X-\{y_{1}, \dots , y_{k}\},n)$$

Here my question: do you know a model $E(n, k )$ that compute the rational cohomology of $F(X, n; y_{1}, \dots , y_{k})$? What are generators, relations and the differential?

I think that such a model can be obtained from $E(n)$ by the fact that $F(X, n; y_{1}, \dots , y_{k})$ is isomoprhic to the fiber at the point $(y_{1}, \dots, y_{k})$ of the projection $$ F(n+k,X)\rightarrow F(k, X) ,$$ but I wonder if there is something in the literature. I am interested to the case $dimX=1$.

Given a smooth projective variety $X$ of dimension $l$, we denote with $F(X,n)$ the configuration space of points $$ F(X,n):=\{(x_{1}, \dots, x_{n})\in X^{n}\: : \: x_{i}\neq x_{j}\text{ for each }i,j \} $$ In https://www.jstor.org/stable/2946581?seq=1#page_scan_tab_contents there is an explicit rational dg algebra $E(n)$ such that $H^{\bullet}(E(n))\cong H^{\bullet}(X, \mathbb{Q})$ (a model). In particular , as an algebra, $E(n)$ is isomorphic to a free $H^{\bullet}(X^{n})$ algebra with generators $G_{ab}$ for $1\leq a,b \leq l$, modulo some easy relations. The same is true for the differential. Now, choose $k$ distinct points $y_{1}, \dots y_{k}$ on $X$. We consider the configuration space of $X-\{y_{1}, \dots , y_{k}\}$

$$F(X, n; y_{1}, \dots , y_{k}):=F(X-\{y_{1}, \dots , y_{k}\},n)$$

Here my question: do you know a model $E(n, k )$ that compute the rational cohomology of $F(X, n; y_{1}, \dots , y_{k})$? What are generators, relations and the differential?

I think that such a model can be obtained from $E(n)$ by the fact that $F(X, n; y_{1}, \dots , y_{k})$ is isomoprhic to the fiber at the point $(y_{1}, \dots, y_{k})$ of the projection $$ F(n+k,X)\rightarrow F(k, X) ,$$ but I wonder if there is something in the literature. I am interested to the case $dimX=1$.

Edit: I don't know if this is an open problem. I am interested to find a 1-minimal model for the case $dimX=1$, $k=1$. Is there a way procedure to compute $H^{1}( F(n, X;y) $ and the subspace $V\subset H^{2}( F(n, X;y)$ generated by $H^{1}( F(n, X;y) $ via the wedge product?