The degree of the top Segre class of the standard vector bundle on the Hilbert scheme $\operatorname{Hilb}^4S$ of an algebraic surface~$S$

T. L. Troshina

Izvestiya. Mathematics , Tome 43 (1994) no. 3, pp. 493-516

Résumé

In the present paper we compute the degree of the top Segre class $s_8(\mathscr E_D^4)$ of the standard vector bundle $\mathscr E_D^4=q_{\ast}p^{\ast}\mathscr O_s(D)$ on the Hilbert scheme $\operatorname{Hilb}^4S$ of an algebraic surface $S$, where $D$ is a divisor on $S$ and $S\stackrel{p}{\longleftarrow}Z_4\stackrel{q}{\longrightarrow}\operatorname{Hilb}^4S$ are the natural projections of the universal cycle $Z_4\subset S\times\operatorname{Hilb}^4S$. This degree is a polynomial with rational coefficients in invariants $x$, $y$, $z$, $w$ of the pair $(S,\mathscr O_S(D))$, where $x=(D^2)$, $y=(D\cdot K_S)$, $z=s_2(S)$, $w=(K^2_S)$.