论文标题
对称群体的滚动不变性，syzygies和表示形式
Scrollar invariants, syzygies and representations of the symmetric group
论文作者
论文摘要
就对称群体$ s_d $的表示，我们给出了明确的最低分级的免费分辨率，该级别的$ \ m \ m m mathbb {p}^{d-2} $在$ \ mathbb {p}^{d-2} $中，在ring gromateTrizations的上下文中。当应用于相对规范的嵌入式曲线$ c $的简单分支度$ d $封面的几何通用纤维时，我们的构造对syzygy弹跳的分裂类型进行了新的解释，以其相对最小的分辨率出现。具体而言，我们的工作意味着所有这些分裂类型都由滚动覆盖物的滚动不变式组成。这大大概括了由于Casnati引起的先前观察，即，根据其立方体分辨率的滚子不变的syzygy套件$ 4 $覆盖的分裂。我们的工作还表明，Syzygy捆绑包的分裂类型，以及多组的滚动不变性，属于一类更大的多组不变性，可以将其附加到$ c \ of $ c \ to \ mathbb {p}^1 $：一个不可修复的代表$ s_d $ $ s_d $，i.e.e.e.e.，一个$ d $ d $ d $ d $ d $ d $。
We give an explicit minimal graded free resolution, in terms of representations of the symmetric group $S_d$, of a Galois-theoretic configuration of $d$ points in $\mathbb{P}^{d-2}$ that was studied by Bhargava in the context of ring parametrizations. When applied to the geometric generic fiber of a simply branched degree $d$ cover of $\mathbb{P}^1$ by a relatively canonically embedded curve $C$, our construction gives a new interpretation for the splitting types of the syzygy bundles appearing in its relative minimal resolution. Concretely, our work implies that all these splitting types consist of scrollar invariants of resolvent covers. This vastly generalizes a prior observation due to Casnati, namely that the first syzygy bundle of a degree $4$ cover splits according to the scrollar invariants of its cubic resolvent. Our work also shows that the splitting types of the syzygy bundles, together with the multi-set of scrollar invariants, belong to a much larger class of multi-sets of invariants that can be attached to $C \to \mathbb{P}^1$: one for each irreducible representation of $S_d$, i.e., one for each partition of $d$.
