THE MODULI SPACE OF TWISTED CANONICAL DIVISORS
THE MODULI SPACE OF TWISTED CANONICAL DIVISORS
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扭曲正则除数的模空间
DOI：
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发表时间：
2015
影响因子：
0.9
通讯作者：
R. Pandharipande
中科院分区：
文献类型：
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作者：
G. Farkas;R. Pandharipande
非奇异曲线上规范因子（有规定的零点和极点）的模空间是不紧的，因为曲线可能简并。我们在$\overline{{\mathcal{M}}}_{g,n}$中定义了一个扭曲正则因子的固有模空间，其中正则因子的空间是一个开子集。该理论导致正则除数模空间闭包的几何/组合约束。当微分至少有一个极点时（严格亚纯情况），在$\overline{{\mathcal{M}}}_{g,n}$中，格$g$曲线上的扭曲正则因子的模空间是纯余维$g$。除了正则因子在非奇异曲线上的闭包外，模空间还具有虚分量。在附录A中，我们提出了一个关于所有成分（具有内在复数）的基本类和与Pixton公式的完整建议。结果得到了扭曲正则子模空间的加权基类在重言环上的一个精确而显式的猜想。作为这个猜想的结果，我们确定了非奇异曲线上正则因子模空间闭包的类（在全纯和亚纯情况下）。
The moduli space of canonical divisors (with prescribed zeros and poles) on nonsingular curves is not compact since the curve may degenerate. We define a proper moduli space of twisted canonical divisors in $\overline{{\mathcal{M}}}_{g,n}$ which includes the space of canonical divisors as an open subset. The theory leads to geometric/combinatorial constraints on the closures of the moduli spaces of canonical divisors. In case the differentials have at least one pole (the strictly meromorphic case), the moduli spaces of twisted canonical divisors on genus $g$ curves are of pure codimension $g$ in $\overline{{\mathcal{M}}}_{g,n}$ . In addition to the closure of the canonical divisors on nonsingular curves, the moduli spaces have virtual components. In the Appendix A, a complete proposal relating the sum of the fundamental classes of all components (with intrinsic multiplicities) to a formula of Pixton is proposed. The result is a precise and explicit conjecture in the tautological ring for the weighted fundamental class of the moduli spaces of twisted canonical divisors. As a consequence of the conjecture, the classes of the closures of the moduli spaces of canonical divisors on nonsingular curves are determined (both in the holomorphic and meromorphic cases).