Real cohomology groups of the space of nonsingular curves of degree 5 in \(\mathbb{C}\mathbb{P}^2\) (Q816464)

Let \(\Pi_d\) denote the space of all homogeneous polynomials \(\mathbb C^3\to \mathbb C\) of degree \(d\) and we denote by \(P_d\) the subspace of \(\Pi_d\) consisting of all non-singular polynomials. Let \(\Sigma_d\) denote the complement \(\Sigma_d=\Pi_d \setminus P_d\), called the discriminant of \(P_d\), then there is an isomorphism \(H^i(P_d)\cong \overline{H}_{2D-1-i}(\Sigma_d)\) for \(0<i<2D-1-i\) (by Alexander duality), where \(D=\dim_{\mathbb C}\Pi_d\) and \(\overline{H}(\text{ })\) denotes Borel-Moore homology. A general method of calculating the cohomology of \(P_d\) was given by Vassiliev in [\textit{V. A. Vasil'ev}, Proc. Steklov Inst. Math. 225, 121--140 (1999; Zbl 0981.55008)] which consists in computing \(\overline{H}(\Sigma_d)\) by the spectral sequence induced from the resolution of the discriminant. Vassiliev also computed the real cohomology of it for \(d\leq 4\). In this paper, the author computes it for the case \(d=5\) and he shows that the Poincaré polynomial \(p_5(t)\) of the space \(P_5\) is equal to \(p_5(t)=(1+t)(1+t^3)(1+t^5)\). To prove this result, he uses mainly Vassiliev's method, but he simplifies the calculations.