论文标题
在Goulden-Jackson-Vakil的猜想上
On the Goulden-Jackson-Vakil conjecture for double Hurwitz numbers
论文作者
论文摘要
Goulden，Jackson和Vakil观察到了一部分双方双Hurwitz编号的多项式结构，该结构列举了$ \ Mathbb {cp}^1 $的分支封面，其中具有$ \ infty $的规定后的分支概况，这是0次，在0上唯一的预先映射，在0中，以及简单的分支。这使他们猜想了模量空间和重言式阶级的存在，这些空间和重言式阶级的相交理论产生了单一赫维兹数字的著名ELSV公式的类似物。 在本文中，我们提出了三个公式，这些公式将一部分双Hurwitz数字表示为某些模量空间上的交点号。第一个涉及稳定地图的模量空间上的Hodge类以分类空间。第二个涉及自旋曲线模量空间上的chiodo类。第三个涉及稳定曲线模量空间上的重言式类别。我们继续讨论这些公式的优点，以与古尔登，杰克逊和瓦基尔阐明的所需财产列表。我们的公式导致稳定曲线的模量空间上的重言式相交数量之间的非平凡关系，并在Chiodo类基础的进一步结构上进行了提示。本文以我们的结果概括为旋转Hurwitz数字的概括。
Goulden, Jackson and Vakil observed a polynomial structure underlying one-part double Hurwitz numbers, which enumerate branched covers of $\mathbb{CP}^1$ with prescribed ramification profile over $\infty$, a unique preimage over 0, and simple branching elsewhere. This led them to conjecture the existence of moduli spaces and tautological classes whose intersection theory produces an analogue of the celebrated ELSV formula for single Hurwitz numbers. In this paper, we present three formulas that express one-part double Hurwitz numbers as intersection numbers on certain moduli spaces. The first involves Hodge classes on moduli spaces of stable maps to classifying spaces; the second involves Chiodo classes on moduli spaces of spin curves; and the third involves tautological classes on moduli spaces of stable curves. We proceed to discuss the merits of these formulas against a list of desired properties enunciated by Goulden, Jackson and Vakil. Our formulas lead to non-trivial relations between tautological intersection numbers on moduli spaces of stable curves and hints at further structure underlying Chiodo classes. The paper concludes with generalisations of our results to the context of spin Hurwitz numbers.
