论文标题
计数$ \ Mathbb {p}^1 \ times \ Mathbb {p}^1 $中的热带曲线
Counting tropical curves in $\mathbb{P}^1\times\mathbb{P}^1$: computation & polynomiality properties
论文作者
论文摘要
$ \ mathbb {p}^1 \ times \ mathbb {p}^1 $带有固定的触点顺序的曲线计数，带有固定的触点订单，并可以通过Mikhalkin的热带方法来确定令人满意的点条件。如果我们要求我们的曲线仅在接触点$ 1 $中与零和无限部分相交，但允许零和无限纤维的任意联系顺序，相应的数字显示出精美的结构属性，例如分段多项式性，类似于$ \ \ \ \ ramifie的$ \ ramifie的范围，类似于双hurwitz数字的范围，均与ramifie的范围c}^1^1^1^1^1^1^1布鲁加利。使用落地图方法来计算热带曲线，从而获得了此结果。 在这里，我们扩展了热带工具，以确定$ \ mathbb {p}^1 \ times \ mathbb {p}^1 $中的曲线计数。我们提供了一个计算工具（在Gawrilow和Joswig的Polymake上建立），该工具通过直接概括Mikhalkin的晶格路径算法来确定任何属的热带曲线和任何接触顺序。该工具也可以用于其他曲面表面。为了手工启用有效的计算，我们引入了一种新的计数工具（对于与无穷大截面的横向接触的有理曲线的情况），可以看作是地板图和晶格路径方法的组合：地下图。我们同时使用计算工具和地下图来揭示这些计数的结构特性。 我们在$ \ mathbb {p}^1 \ times \ mathbb {p}^1 $中获得（分段）多项式结构的（分段）多项式结构，并在零和Infinity和Infinity section的零纤维和限制性选择上进行任意联系顺序。
Counts of curves in $\mathbb{P}^1\times\mathbb{P}^1$ with fixed contact order with the toric boundary and satisfying point conditions can be determined with tropical methods by Mikhalkin. If we require that our curves intersect the zero- and infinity-section only in points of contact order $1$, but allow arbitrary contact order for the zero- and infinity-fiber, the corresponding numbers reveal beautiful structural properties such as piecewise polynomiality, similar to the case of double Hurwitz numbers counting covers of $\mathbb{P}^1$ with special ramification profiles over zero and infinity by Ardila and Brugallé. This result was obtained using the floor diagram method to count tropical curves. Here, we expand the tropical tools to determine counts of curves in $\mathbb{P}^1\times\mathbb{P}^1$. We provide a computational tool (building on Polymake by Gawrilow and Joswig) that determines such numbers of tropical curves for any genus and any contact orders via a straightforward generalization of Mikhalkin's lattice path algorithm. The tool can also be used for other toric surfaces. To enable efficient computations also by hand, we introduce a new counting tool (for the case of rational curves with transverse contacts with the infinity section) which can be seen as a combination of the floor diagram and the lattice path approach: subfloor diagrams. We use both our computational tool and the subfloor diagrams for experiments revealing structural properties of these counts. We obtain first results on the (piecewise) polynomial structure of counts of rational curves in $\mathbb{P}^1\times\mathbb{P}^1$ with arbitrary contact orders on the zero- and infinity-fiber and restricted choices for the contact orders on the zero- and infinity-section.
