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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].

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

Citations:

Zbl 0819.14003
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