多重线性级数与多重线性元
Multigraded linear series and recollement
摘要：
给出一个带有全局生成向量丛集合$E_1, \dots, E_n$的$Y$，研究了从$Y$到$E:=\mathcal{O}_Y\oplus E_1\oplus\cdots \oplus E_n$的自同态代数上循环模的细模空间$\mathcal{M}(E)$的泛态射.这将经典态射推广到无基点线丛的线性级数。我们描述了态射的象，并给出了满射在模范畴中的一个替换元上的充要条件。当态射是满射时，这给出了图像的精细模空间解释，并且作为应用，我们表明对于小的有限子群$G\subset \text{GL}(2,k)$，$\mathbb{A}^2_k/G$的每个次最小部分分辨率同构于精细模空间$\mathcal{M}(E_C)$，其中$E_C$是定义重建代数的束$E$的和项。我们还考虑应用Gorenstein仿射三倍，里德的配方揭示了一些光的代数类，从中可以重建一个给定的crepant决议。
英文摘要：
Given a scheme $Y$ equipped with a collection of globally generated vector bundles $E_1, \dots, E_n$, we study the universal morphism from $Y$ to a fine moduli space $\mathcal{M}(E)$ of cyclic modules over the endomorphism algebra of $E:=\mathcal{O}_Y\oplus E_1\oplus\cdots \oplus E_n$. This generalises the classical morphism to the linear series of a basepoint-free line bundle on a scheme. We describe the image of the morphism and present necessary and sufficient conditions for surjectivity in terms of a recollement of a module category. When the morphism is surjective, this gives a fine moduli space interpretation of the image, and as an application we show that for a small, finite subgroup $G\subset \text{GL}(2,k)$, every sub-minimal partial resolution of $\mathbb{A}^2_k/G$ is isomorphic to a fine moduli space $\mathcal{M}(E_C)$ where $E_C$ is a summand of the bundle $E$ defining the reconstruction algebra. We also consider applications to Gorenstein affine threefolds, where Reid's recipe sheds some light on the classes of algebra from which one can reconstruct a given crepant resolution.
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