论文标题
平面曲线的平滑限制和马尔可夫数字
Smooth limits of plane curves of prime degree and Markov numbers
论文作者
论文摘要
对于尺寸的质量高度至少3个，莫里询问每个平滑的适当极限是否仍然是一个超表面。有趣的是，在维度1和2中，情况并非如此。例如，格里芬（Griffin）构建了在维度1中给出反例的五重曲线的显式家族（Horikawa在维度2中构建了类似的Quintics示例）。在本文中，我们提出了一个猜想，该猜想使用黑客和普罗科罗夫（Prokhorov）在投射平面的Q-Gorenstein限制方面的工作进行了解释。特别是，如果P是Markov数字的质数，则我们猜想P度P的平面曲线的任何平滑射击极限都是平面曲线。主要的结果是证明该猜想的7度曲线，并将格里芬的反例扩展到也是马尔可夫数字的所有质数。
For prime degree hypersurfaces of dimension at least 3, Mori asked if every smooth proper limit is still a hypersurface. Interestingly in dimensions 1 and 2, this is not the case. For example, Griffin constructed explicit families of quintic curves that give counterexamples in dimension 1 (Horikawa constructed similar examples for quintics in dimension 2). In this paper we propose a conjecture explaining these examples using Hacking and Prokhorov's work on Q-Gorenstein limits of the projective plane. In particular, if p is a prime number that is not a Markov number, we conjecture that any smooth projective limit of plane curves of degree p is a plane curve. The main results are to prove this conjecture for degree 7 curves and to extend Griffin's counterexamples to all prime numbers that are also Markov numbers.
