Extending the Torelli map to toroidal compactifications of Siegel space
Extending the Torelli map to toroidal compactifications of Siegel space
复制标题
将 Torelli 映射扩展到 Siegel 空间的环形紧化
DOI：
--
复制
发表时间：
2011
期刊：
影响因子：
--
通讯作者：
Adrian Brunyate
中科院分区：
文献类型：
--
作者：
V. Alexeev;Adrian Brunyate
自 20 世纪 70 年代以来，人们就知道 Torelli 映射 M g→A g 与其雅可比行列式关联到平滑曲线，从 Deligne–Mumford 紧化 $overline {operatorname {M}}_{g}$ 扩展到第二个 Voronoi 紧化 $overline {operatorname {A}}_{g}^{mathrm {vor}}$ 的常规映射。我们证明了扩展的 Torelli 映射到完美圆锥（第一个 Voronoi）压缩 $overline {operatorname {A}}_{g}^{mathrm {perf}}$ 也是正则的，而且 $overline {operatorname {A}}_{g}^{mathrm {vor}}$ 和 $overline {operatorname {A}}_{g}^{mathrm {perf}}$ 共享图像的公共 Zariski 开邻域$overline {operatorname {M}}_{g}$。我们还表明，对于 g≥9，Igusa 幺半群变换（中心锥压缩）的映射是不规则的；这反驳了浪川 1973 年的猜想。
It has been known since the 1970s that the Torelli map M g→A g, associating to a smooth curve its Jacobian, extends to a regular map from the Deligne–Mumford compactification $overline {operatorname {M}}_{g}$ to the 2nd Voronoi compactification $overline {operatorname {A}}_{g}^{mathrm {vor}}$. We prove that the extended Torelli map to the perfect cone (1st Voronoi) compactification $overline {operatorname {A}}_{g}^{mathrm {perf}}$ is also regular, and moreover $overline {operatorname {A}}_{g}^{mathrm {vor}}$ and $overline {operatorname {A}}_{g}^{mathrm {perf}}$ share a common Zariski open neighborhood of the image of $overline {operatorname {M}}_{g}$. We also show that the map to the Igusa monoidal transform (central cone compactification) is not regular for g≥9; this disproves a 1973 conjecture of Namikawa.