论文标题
长度为2
A GIT construction of moduli spaces of sheaves of length 2
论文作者
论文摘要
令$ \ bbbk $为特征零的代数封闭字段。令$ \ mathrm {sch}/\ bbbk $表示有限类型的类别，超过$ \ bbbk $。令$ b $为$ \ bbbk $的连接投影方案，让$ \ mathcal l $成为$ b $的宽线捆绑包。令$τ$为长度2的较难的纳拉西汉类型，然后让$Δ\ in \ mathbb n $。我们说，$ b $上的纯纸条$ \ natcal e $是$（τ，δ）$ - 如果其难度更难的narasimhan过滤$ 0 = \ MATHCAL E _ {\ leq 0} \ subsetneq \ subsetneq \ subsetneq \ subsetneq \ Mathcal e _ _ {\ leq 1} \ leq 1} \ subsete e $ qu $ seq \ n Mathcal e $是$τ$，具有稳定的亚品质的类型，以及$δ= \ dim_ \ bbbk \ mathrm {hom} _ {\ Mathcal o_b}（\ Mathcal E_2，\ Mathcal E_2，\ Mathcal E_1）$ e _ {\ leq i-1} $。 我们定义一个Moduli functor $ \ mathbf m'_ {τ，δ} $分类$（τ，δ）$ - $ b $上的稳定纸巾，并通过非还原的几何学不变理论（GIT）来构建其粗型模量。我们在ARXIV中的非还原性GIT：1607.04181和ARXIV：1601.00340扩展到对非还原方案的线性动作，并应用我们的非还原性GIT来证明Sheaffification $（\ Mathbf M'_____ {_ {τ，δ}） $（\ mathrm {sch}/\ bbbk）_ {étale} $由quasi-projective方案表示。 我们的方法概括了杰克逊（Jackson）在Arxiv中的$（τ，δ）$稳定束带的模量空间的构造：2111.07428在品种类别中，以允许非还原模量方案。
Let $\Bbbk$ be an algebraically closed field of characteristic zero. Let $\mathrm{Sch}/\Bbbk$ denote the category of schemes of finite type over $\Bbbk$. Let $B$ be a connected projective scheme over $\Bbbk$ and let $\mathcal L$ be an ample line bundle on $B$. Let $τ$ be a Harder-Narasimhan type of length 2, and let $δ\in\mathbb N$. We say a pure sheaf $\mathcal E$ on $B$ is $(τ,δ)$-stable if its Harder-Narasimhan filtration $0=\mathcal E_{\leq 0}\subsetneq\mathcal E_{\leq 1}\subsetneq\mathcal E_{\leq 2}=\mathcal E$ is non-splitting, of type $τ$, with stable subquotients, and $δ=\dim_\Bbbk\mathrm{Hom}_{\mathcal O_B}(\mathcal E_2,\mathcal E_1)$ for $\mathcal E_i:=\mathcal E_{\leq i}/\mathcal E_{\leq i-1}$. We define a moduli functor $\mathbf M'_{τ,δ}$ classifying $(τ,δ)$-stable sheaves on $B$ and construct its coarse moduli space by non-reductive geometric invariant theory (GIT). We extend the non-reductive GIT in arXiv:1607.04181 and arXiv:1601.00340 to linear actions on non-reduced schemes, and apply our non-reductive GIT to prove that the sheafification $(\mathbf M'_{τ,δ})^\sharp$ on $(\mathrm{Sch}/\Bbbk)_{étale}$ is represented by a quasi-projective scheme. Our methods generalise Jackson's construction of moduli spaces of $(τ,δ)$-stable sheaves in arXiv:2111.07428 in the category of varieties, to allow non-reduced moduli schemes.
