Infinite Staircases for Hirzebruch Surfaces
Infinite Staircases for Hirzebruch Surfaces
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Hirzebruch 表面的无限楼梯
DOI：
10.1007/978-3-030-80979-9_2
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
M. Weiler
中科院分区：
文献类型：
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作者：
M. Bertozzi;T. Holm;Emily L. Maw;D. Mcduff;Grace T. Mwakyoma;A. R. Pires;M. Weiler
本文考虑偏心率为z的椭球嵌入函数c_{H_B}（z）$到非平凡有理Hirzebruch曲面族H_B$中的辛嵌入，其辛形式为[0，1）$中的$B\。已知这个函数在单调情况下（当$B=0$和$B=1/3$）有一个无限阶梯。我们还知道，对于每个$B$，最多有一个z$的值可以是这样一个阶梯的累加点。在这篇手稿中，我们确定了三个序列的开放的，不相交的，阻塞的$B$-区间，组成的$B$-参数的椭球嵌入函数的$H_B$不包含一个无限的阶梯。在每个区间（0，1/5）、（1/5，1/3）和（1/3，1）中有一个序列。然后，我们建立六个相关无限阶梯序列，每个序列出现在被阻止的$B$-区间的每个端点处。阶梯数字是射影平面上斐波那契阶梯数字的变体（$B=0$的情况）。我们还表明，有没有在点$B=1/5$的楼梯，即使这个值是不阻塞。本文的重点是开发技术，图形和数字，允许识别潜在的楼梯，然后充分了解障碍物，以证明所谓的楼梯确实具有所需的属性。随后的论文将探讨更深入的一套$B$承认无限的楼梯。
We consider the ellipsoid embedding functions $c_{H_b}(z)$ for symplectic embeddings of ellipsoids of eccentricity $z$ into the family of nontrivial rational Hirzebruch surfaces $H_b$ with symplectic form parametrized by $b\in [0,1)$. This function was known to have an infinite staircase in the monotone cases (when $b=0$ and $b=1/3$). It was also known that for each $b$ there is at most one value of $z$ that can be the accumulation point of such a staircase. In this manuscript, we identify three sequences of open, disjoint, blocked $b$-intervals, consisting of $b$-parameters where the ellipsoid embedding function for $H_b$ does not contain an infinite staircase. There is one sequence in each of the intervals (0,1/5), (1/5,1/3), and (1/3,1). We then establish six sequences of associated infinite staircases, one occurring at each endpoint of the blocked $b$-intervals. The staircase numerics are variants of those in the Fibonacci staircase for the projective plane (the case $b=0$). We also show that there is no staircase at the point $b=1/5$, even though this value is not blocked. The focus of this paper is to develop techniques, both graphical and numeric, that allow identification of potential staircases, and then to understand the obstructions well enough to prove that the purported staircases really do have the required properties. A subsequent paper will explore in more depth the set of $b$ that admit infinite staircases.