The authors consider the space \(F_n\) of configurations of ordered \(n\) points in \(\mathbb{CP}^2\) satisfying the condition that no three of the points lie on a line. The symmetric group \(\mathfrak{S}\) acts on \(F_n\) by permuting coordinates, and the authors look at the quotient \(B_n:=F_n/\mathfrak{S}_n\).
The main results of this paper are to compute \(H^*(F_n;\mathbb{Q})\) and \(H^*(B_n;\mathbb{Q})\) for \(n=4, 5\) and \(n=6\). To determine \(H^*(B_n;\mathbb{Q})\), the authors determine \(H^i(F_n;\mathbb{Q})\) as an \(\mathfrak{S}\)-representation and use transfer.
The space \(F_n\) also comes equipped with a natural action of \(PGL_3(\mathbb{C})\), and denote the quotient by \(X_n\). In fact, \(F_n\cong PGL_3(\mathbb{C})\times X_n.\) The authors show that \(H^*(F_n)\cong H^*(PGL_3(\mathbb{C}))\otimes H^*(X_n)\) is true as representations of \(\mathfrak{S}_n\).
Below, \(U\) and \(V\) stand for the trivial and fundamental representations, respectively, of either \(\mathfrak{S}_5\) or \(\mathfrak{S}_6\). Other irreducibles are subscripted by the corresponding partitions. The authors also use the convention that \(H^{\mathbb{P}^1}\) has weight 1.
Theorem 1.1 With terminology as above and as \(\mathfrak{S}_6\)-representations, \[ H^*(X_6;\mathbb{Q})\cong \begin{cases} U &\text{ if } *=0,\\ S_{3,3}\oplus S_{4,2} &\text{ if }*=1,\\ V\oplus \wedge^2V^{\oplus 2}\oplus \wedge^3 V\oplus S_{3,3} \oplus S_{3,2,1}^{\oplus 2} &\text{ if }*=2\\ V\oplus \wedge^2 V^{\oplus 3}\oplus \wedge^3 V^{\oplus 3}\oplus S_{3,3}\oplus S_{2,2,2}\oplus S_{4,2}^{\oplus 2}\oplus S_{2,2,1,1}^{\oplus 2}\oplus S_{3,2,1}^{\oplus 3} &\text{ if }*=3,\\ U\oplus U^{'}\oplus V\oplus V^{'}\oplus \wedge^2 V\oplus \wedge^3 V^{\oplus 2}\oplus S_{3,3}^{\oplus 2}\oplus S_{2,2,2}^{\oplus 3}\oplus S_{4,2}^{\oplus 2}\oplus S_{2,2,1,1}\oplus S_{3,2,1}^{\oplus 3} &\text{ if }*=4\\ 0 &\text{ otherwise }\end{cases} \] For \(\mathfrak{S}_5\)-representations, the authors get the following
Remark 1.2 \[ H^*(X_5;\mathbb{Q})\cong \begin{cases} U &\text{ if }*=0,\\ S_{3,2} &\text{ if }*=1,\\ \wedge^2 V &\text{ if }*=2,\\ 0 &\text{ otherwise }\end{cases} \]
And for the \(H^*(B_n;\mathbb{Q})\), the authors obtain that
Corollary 1.3 With terminology as above, \[ H^*(B_5;\mathbb{Q})\cong H^*(PGL_3(\mathbb{C}))\cong \begin{cases} \mathbb{Q} &\text{ if }*=0,3,5,8\\ 0 &\text{ otherwise }\end{cases} \] \[ H^*(B_6;\mathbb{Q})\cong H^{*}(S^4\times PGL_3(\mathbb{C}))\cong \begin{cases} \mathbb{Q} &\text{ if }*=0,3,4,5,7,8,9,12\\ 0 &\text{ otherwise }\end{cases} \] The first isomorphism is induced by the orbit map and hence is an isomorphism of mixed Hodge structures. Similarly, the inclusion of \(H^*(PGL_3(\mathbb{C}))\) into \(H^*(B_6)\) preserves mixed Hodge structures, and the extra generator in \(H^4(B_6)\) has weight 4.
Other results about this configuration space of \(\mathbb{CP}^2\) can be seen in [S. Ashraf and B. Berceanu, Adv. Geom. 14, No. 4, 691–718 (2014; Zbl 1305.55009); Y. Feler, J. Eur. Math. Soc. (JEMS) 10, No. 3, 601–624 (2008; Zbl 1144.32010); V. L. Moulton, J. Pure Appl. Algebra 131, No. 3, 245–296 (1998; Zbl 0999.20027)]. More general configuration spaces of vector spaces and projective spaces can be seen in [J. Wang and X. Zhao, J. Knot Theory Ramifications 29, No. 13, Article ID 2043001, 20 p. (2020; Zbl 1510.55009)].