We investigate the resolution of a general branched coverα : C→ P in its relative canonical embedding C ⊂ PE. We conjecture that the syzygy bundles appearing in the resolution are balanced for a general cover, provided that the genus is sufficiently large compared to the degree. We prove this for the Casnati-Ekedahl bundle, or bundle of quadrics F—the first bundle appearing in the resolution of the ideal of the relative canonical embedding. Furthermore, we prove the conjecture for all syzygy bundles in the resolution when the genus satisfies g = 1 mod d.

中文翻译：

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P1 的一般分支覆盖的不变量

我们研究了一般分支覆盖α : C→ P 在其相对规范嵌入 C ⊂ PE 中的分辨率。我们推测出现在分辨率中的 syzygy 束对于一般覆盖是平衡的，前提是该属与度相比足够大。我们为 Casnati-Ekedahl 丛或二次曲面 F 丛证明了这一点——第一个出现在相对规范嵌入的理想分辨率中的丛。此外，我们证明了当属满足 g = 1 mod d 时解析中所有 syzygy 丛的猜想。