林斯内托特殊叶状结构族的显式积分、双有理模型与对称性
Explicit integration, birational models and symmetries of Lins Neto's exceptional families of foliations
AI总结：
本文针对Lins Neto的三个次数为2、3、4的特殊叶状结构族，给出其双有理等价的显式公式、双有理变换群生成元、有理首次积分算法，还刻画了特征正域下的可积性并研究了约化的可积性。
AI中文摘要：
2002年，Lins Neto（林斯内托）在复射影平面上引入了三个显著的一维全纯曲线叶状结构族，次数分别为2、3、4，其性质表明庞加莱问题的一般形式无解。已知这些叶状结构与阿贝尔曲面上线性叶状结构的某些商双有理等价。我们给出了这些双有理等价的显式公式，特别得到了叶状结构叶片的参数化。对于次数为3和4的族，我们确定了保持它们的射影平面双有理变换群的显式生成元。对于这些族中具有有理首次积分的叶状结构，即参数为艾森斯坦有理数的那些，我们完整描述了一般积分曲线奇点的性质与位置，并给出了显式计算有理首次积分的算法。Lins Neto的叶状结构也可在特征为正的代数闭域上定义，在此背景下，我们刻画了那些代数可积的叶状结构。最后，我们研究了复族中部分不可积叶状结构到特征为正的域的约化的可积性。
英文摘要：
In 2002, Lins Neto introduced three remarkable one-parameter families of holomorphic foliations by curves on the complex projective plane, of degrees two, three, and four, whose properties established that the general form of the Poincaré Problem had no solution. These foliations are known to be birationally equivalent to certain quotients of linear foliations on abelian surfaces. We give explicit formulas for these birational equivalences, obtaining, in particular, parametrizations of the leaves of the foliations. For the families of degrees three and four, we determine explicit generators for the groups of birational transformations of the projective plane preserving them. For foliations in these families admitting a rational first integral, that is, for those whose parameter is an Eisenstein rational, we give a complete description of the nature and position of the singular points of a generic integral curve, and we present an algorithm that computes the rational first integral explicitly. Lins Neto's foliations can also be defined over algebraically closed fields of positive characteristic, and, in this setting, we characterize those that are algebraically integrable. Finally, we study the integrability of the reductions to fields of positive characteristic of some of the non-integrable foliations in the complex family.
