可积系统的奇异性与节曲线
Singularities of integrable systems and nodal curves
摘要：
可积系统与代数几何之间的关系自十九世纪以来就已为人所知。现代的方法是把一个可积系统表示成一个带谱参数的Lax方程。在这种方法中，系统的积分变成Lax矩阵的特征多项式$χ$的系数，并且解用与曲线$χ= 0$相关的θ函数表示。本文的目的是表明，可能性写一个可积系统的Lax形式，以及与这种可能性有关的代数几何技术，也可以适用于研究系统的定性特征，特别是它的奇点。
英文摘要：
The relation between integrable systems and algebraic geometry is known since the XIXth century. The modern approach is to represent an integrable system as a Lax equation with spectral parameter. In this approach, the integrals of the system turn out to be the coefficients of the characteristic polynomial $χ$ of the Lax matrix, and the solutions are expressed in terms of theta functions related to the curve $χ= 0$. The aim of the present paper is to show that the possibility to write an integrable system in the Lax form, as well as the algebro-geometric technique related to this possibility, may also be applied to study qualitative features of the system, in particular its singularities.
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