摘要翻译:
设$X$为光滑拟射影代数曲面,$L$为$X$上的线丛。设$X^{[n]}$在$X$和$L^{[n]}上的$n$点的Hilbert方案$在$X^{[n]}上的重言丛$与$X$上的线丛$L$自然相关联。对于Bridgeland-King-Reid等价性$\bkrh:\b{D}^b(x^{[n]})\ra\b{D}^b_{\perm_n}(x^n)$,我们明确地计算了$\b{D}^b_{\perm_n}(x^n)$中$\perm_n}(x^n)$中的$\comp{\mc{C}}_l$-等价束的象$\bkrh(L^{[n]})$。此外,我们给出了象$\bkrh(L^{[n]}\tens...\tens L^{[n]})$的一个刻划,即与复数$\comp{\mc{C}}_l$的导出的$k$-折叠张量幂相关的超导出谱序列$e^{p,q}_1$。通过对这种谱序列的$\perm_n$-不变量的研究,得到了Hilbert-Chow态射的重言丛的双张量幂和一般的$k$-倍外幂的导出直接像,并给出了这两种情况下的Danila-Brion型公式。这样很容易计算$x^{[n]}$与$l^{[n]}\tens l^{[n]}$和$\lambda^k l^{[n]}$中的值的上同调。
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英文标题:
《Cohomology of the Hilbert scheme of points on a surface with values in
representations of tautological bundles》
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作者:
Luca Scala
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最新提交年份:
2008
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分类信息:
一级分类:Mathematics 数学
二级分类:Algebraic Geometry 代数几何
分类描述:Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇,叠,束,格式,模空间,复几何,量子上同调
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一级分类:Mathematics 数学
二级分类:Commutative Algebra 交换代数
分类描述:Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics
交换环,模,理想,同调代数,计算方面,不变理论,与代数几何和组合学的联系
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一级分类:Mathematics 数学
二级分类:Combinatorics 组合学
分类描述:Discrete mathematics, graph theory, enumeration, combinatorial optimization, Ramsey theory, combinatorial game theory
离散数学,图论,计数,组合优化,拉姆齐理论,组合对策论
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英文摘要:
Let $X$ a smooth quasi-projective algebraic surface, $L$ a line bundle on $X$. Let $X^{[n]}$ the Hilbert scheme of $n$ points on $X$ and $L^{[n]}$ the tautological bundle on $X^{[n]}$ naturally associated to the line bundle $L$ on $X$. We explicitely compute the image $\bkrh(L^{[n]})$ of the tautological bundle $L^{[n]}$ for the Bridgeland-King-Reid equivalence $\bkrh : \B{D}^b(X^{[n]}) \ra \B{D}^b_{\perm_n}(X^n)$ in terms of a complex $\comp{\mc{C}}_L$ of $\perm_n$-equivariant sheaves in $\B{D}^b_{\perm_n}(X^n)$. We give, moreover, a characterization of the image $\bkrh(L^{[n]} \tens ... \tens L^{[n]})$ in terms of of the hyperderived spectral sequence $E^{p,q}_1$ associated to the derived $k$-fold tensor power of the complex $\comp{\mc{C}}_L$. The study of the $\perm_n$-invariants of this spectral sequence allows to get the derived direct images of the double tensor power and of the general $k$-fold exterior power of the tautological bundle for the Hilbert-Chow morphism, providing Danila-Brion-type formulas in these two cases. This yields easily the computation of the cohomology of $X^{[n]}$ with values in $L^{[n]} \tens L^{[n]}$ and $\Lambda^k L^{[n]}$.
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PDF链接:
https://arxiv.org/pdf/0710.3072