Tikhomirov, A. S.; Troshina, T. L. Top Segre class of a standard vector bundle \({\mathcal E}^ 4_ D\) on the Hilbert scheme \(\text{Hilb}^ 4S\) of a surface \(S\). (English) Zbl 0819.14004 Tikhomirov, Alexander (ed.) et al., Algebraic geometry and its applications. Proceedings of the 8th algebraic geometry conference, Yaroslavl’, Russia, August 10-14, 1992. Braunschweig: Vieweg. Aspects Math. E 25, 205-226 (1994). This is a continuation of previous work of A. S. Tikhomirov [same conference, Aspects Math. E 25, 183-203 (1994; see the preceding review)] to which we refer for definitions and notations. The main result of the present paper is an explicit formula for the polynomial \(\delta_ 4\) of a nonsingular projective surface \(S\). According to the above mentioned article, the computation reduces to an enumerative problem: the determination of the number of 4-secant planes to a convenient embedding of \(S\) into \(\mathbb{P}^{10}\).For the entire collection see [Zbl 0793.00016]. Reviewer: I.Coandă (Bucureşti) Cited in 1 Document MSC: 14C05 Parametrization (Chow and Hilbert schemes) 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14N10 Enumerative problems (combinatorial problems) in algebraic geometry 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli Keywords:Hilbert scheme; Segre class; standard vector bundle; number of 4-secant planes Citations:Zbl 0819.14003 PDFBibTeX XMLCite \textit{A. S. Tikhomirov} and \textit{T. L. Troshina}, in: Algebraic geometry and its applications. Proceedings of the 8th algebraic geometry conference, Yaroslavl', Russia, August 10-14, 1992. Braunschweig: Vieweg. 205--226 (1994; Zbl 0819.14004)