论文标题
希尔伯特计划的精制Verlinde和Segre公式
Refined Verlinde and Segre formula for Hilbert schemes
论文作者
论文摘要
令$ \ mathrm {hilb} _ns $为光滑的投影表面$ s $上的$ n $点的希尔伯特方案。在K^0（s）$中$α\ suppory a in to introgological vector $α^{[n]} $ on $ \ mathrm {hilb} _ns $ and line bundle $ l _ {（n）} \ otimes e^{（n）} \ otimes e^{ $ r = \ mathrm {rk}（α）$。在本文中，我们为Segre类的生成函数提供了封闭的公式，$ \ int _ {\ Mathrm {hilb} _ns} s（α^{[n]}）$和Verlinde Numbers $ $χ（\ Mathrm {hilb} r}）$，对于任何表面$ s $和k^0（s）$中的任何类$α\。实际上，我们确定了$ k $的更通用的生成功能 - 希尔伯特（Hilbert）积分方案的不变性，其中包含Segre和Verlinde数字的公式作为专业。如果$ k_s^2 = 0 $，我们证明了这些公式。在不假设条件$ k_s^2 = 0 $的情况下，我们显示了约翰逊和玛丽安 - oprea-pandharipande的塞格尔佛经者的猜想，它们通过变量的显式更改将Segre和Verlinde生成序列相关联。
Let $\mathrm{Hilb}_nS$ be the Hilbert scheme of $n$ points on a smooth projective surface $S$. To a class $α\in K^0(S)$ correspond a tautological vector bundle $α^{[n]}$ on $\mathrm{Hilb}_nS$ and line bundle $L_{(n)}\otimes E^{\otimes r}$ with $L=\det(α)$, $r=\mathrm{rk}(α)$. In this paper we give closed formulas for the generating functions for the Segre classes $\int_{\mathrm{Hilb}_nS} s(α^{[n]})$, and the Verlinde numbers $χ(\mathrm{Hilb}_nS,L_{(n)}\otimes E^{\otimes r})$, for any surface $S$ and any class $α\in K^0(S)$. In fact we determine a more general generating function for $K$-theoretic invariants of Hilbert schemes of points, which contains the formulas for Segre and Verlinde numbers as specializations. We prove these formulas in case $K_S^2=0$. Without assuming the condition $K_S^2=0$, we show the Segre-Verlinde conjecture of Johnson and Marian-Oprea-Pandharipande, which relates the Segre and Verlinde generating series by an explicit change of variables.
