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Real cohomology groups of the space of nonsingular curves of degree 5 in $$\mathbb{C}\mathbb{P}^2$$. (English) Zbl 1088.55012
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 [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.

##### MSC:
 55R80 Discriminantal varieties and configuration spaces in algebraic topology 55S15 Symmetric products and cyclic products in algebraic topology
Zbl 0981.55008
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##### References:
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