Kapfer, Simon Computing cup products in integral cohomology of Hilbert schemes of points on \(K3\) surfaces. (English) Zbl 1347.14005 LMS J. Comput. Math. 19, No. 1, 78-97 (2016). The Hilbert schemes of points on a \(K3\) surface are irreducible holomorphic symplectic manifolds. The cup products in the integral cohomology are studied. A computer algebra program is used to compute the cup products. The source code and an explanation how to use it is given in an appendix.Let \(S^{[3]}\) be the Hilbert scheme of three points on a projective \(K3\) surface. \(\mathrm{Sym}^kH^2(S^{[3]}, \mathbb{Z})\) can be identified with its image in \(H^{2k}(S^{[3]}, \mathbb{Z})\) under the cup product mapping. As a result of the computations the following theorem is obtained:\(H^4(S^{[3]}, \mathbb{Z})/\mathrm{Sym}^2H^2(S^{[3]}, \mathbb{Z})\simeq\mathbb{Z}/3\mathbb{Z}\oplus \mathbb{Z}^{23}\)\(H^6(S^{[3]}, \mathbb{Z})/H^2 (S^{[3]}, \mathbb{Z})\cup H^4(S^{[3]}, \mathbb{Z})\simeq(\mathbb{Z}/3\mathbb{Z})^{23}\). Reviewer: Gerhard Pfister (Kaiserslautern) Cited in 1 Document MSC: 14J10 Families, moduli, classification: algebraic theory 14Q15 Computational aspects of higher-dimensional varieties 05A17 Combinatorial aspects of partitions of integers 55N32 Orbifold cohomology Keywords:\(K3\) surface; Hilbert scheme of points; integral cohomology; cup product Software:HilbK3; Haskell; GHC PDFBibTeX XMLCite \textit{S. Kapfer}, LMS J. Comput. Math. 19, No. 1, 78--97 (2016; Zbl 1347.14005) Full Text: DOI arXiv References: [1] DOI: 10.2307/2373541 · Zbl 0176.18401 · doi:10.2307/2373541 [2] Ellingsrud, J. Algebra. Geom. 10 pp 81– (2001) [3] Dénes, Publ. Math. Inst. Hung. Acad. Sci. 4 pp 63– (1959) [4] DOI: 10.1112/jtopol/jtt002 · Zbl 1295.14023 · doi:10.1112/jtopol/jtt002 [5] DOI: 10.1007/BF02247112 · Zbl 0861.53069 · doi:10.1007/BF02247112 [6] DOI: 10.1007/s00222-002-0270-7 · Zbl 1035.14001 · doi:10.1007/s00222-002-0270-7 [7] DOI: 10.2307/2951818 · Zbl 0915.14001 · doi:10.2307/2951818 [8] DOI: 10.1007/978-3-642-88330-9 · doi:10.1007/978-3-642-88330-9 [9] DOI: 10.1142/S0129167X10005957 · Zbl 1184.14074 · doi:10.1142/S0129167X10005957 [10] DOI: 10.1016/j.aim.2006.03.006 · Zbl 1115.14036 · doi:10.1016/j.aim.2006.03.006 [11] DOI: 10.1007/s00208-004-0602-6 · Zbl 1081.14006 · doi:10.1007/s00208-004-0602-6 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.