论文标题
切换曲线的切线曲线
Tangent curves to degenerating hypersurfaces
论文作者
论文摘要
我们研究了在超表面退化下与超表面相切的理性曲线的行为。在对数Gromov-Witten理论的框架内工作，我们将退化公式扩展到对数奇异的环境，从而在地图空间上产生虚拟级别，直至变性纤维。然后，我们采用对数变形理论来表达该类别是普通稳定图的模量空间上的阻塞束组成部分。这产生了对数Gromov-witten不变性的新改进，并编码了切线曲线的变性行为。在平滑的平面立方体退化为圆磨边界的示例中，我们采用了定位和热带技术来计算这些改进。最后，我们利用这些计算来描述嵌入的曲线如何与立方体的平滑立方变性相切。获得的结果具有古典性质，但是证据使对数Gromov-witten理论的必不可少。
We study the behaviour of rational curves tangent to a hypersurface under degenerations of the hypersurface. Working within the framework of logarithmic Gromov-Witten theory, we extend the degeneration formula to the logarithmically singular setting, producing a virtual class on the space of maps to the degenerate fibre. We then employ logarithmic deformation theory to express this class as an obstruction bundle integral over the moduli space of ordinary stable maps. This produces new refinements of the logarithmic Gromov-Witten invariants, encoding the degeneration behaviour of tangent curves. In the example of a smooth plane cubic degenerating to the toric boundary we employ localisation and tropical techniques to compute these refinements. Finally, we leverage these calculations to describe how embedded curves tangent to a smooth cubic degenerate as the cubic does; the results obtained are of a classical nature, but the proofs make essential use of logarithmic Gromov-Witten theory.
