×

zbMATH — the first resource for mathematics

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
Citations:
Zbl 0981.55008
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] Arnold ( V.I. ). - On some topological invariants of algebraic functions , Transact. (Trudy) of Moscow Math. Society , 21 , p. 27 - 46 , 1970 . MR 274462 · Zbl 0225.14005
[2] Vassiliev ( V.A. ). - How to calculate homology groups of spaces of nonsingular algebraic projective hypersurfaces in CPn , Proc. Steklov Math. Inst. , vol. 225 , p. 121 - 140 , 1999 . MR 1738399 | Zbl 0981.55008 · Zbl 0981.55008
[3] Vassiliev ( V.A. ). - Topology of complements of discriminants , Phasis , Moscow , 1997 (in Russian). · Zbl 0991.58014
[4] Peters ( C.A.M. ) , Steenbrink ( J.H.M. ). - Degeneration of the Leray spectral sequence for certain geometric quotients , Moscow Math. J. , Vol. 3 , n^\circ 3 , 2003 , p. 1085 - 1095 . Zbl 1049.14035 · Zbl 1049.14035
[5] Shafarevich ( I.R. ). - Basic Algebraic Geometry 1 , Springer-Verlag , Berlin , 1994 . MR 1328833 | Zbl 0797.14001 · Zbl 0797.14001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.