R^4中Hessian秩为2的仿射齐次超曲面S^3的分类
Classification of Affinely Homogeneous Hessian Rank 2 Hypersurfaces S^3 in R^4
AI总结：
本文确定了R^4中Hessian具有常数秩2的所有仿射齐次超曲面S^3，发现34个不等价终端分支并利用幂级数等价方法通过显式Lie代数描述了这些齐次模型。
摘要：
我们确定了R^4中所有Hessian（不变地）具有常数秩2的仿射齐次超曲面S^3，包括单可递的情形。我们发现了34个不等价的终端分支，每个分支都产生一个非空的R^4中超曲面S^3的齐次模型模空间，有时由某个复杂的代数簇参数化，特别是对于34个中的15个单可递模型族。我们采用了幂级数等价方法，该方法在原点捕捉不变量、创建分支并将计算无穷小化。遵循Lie最初的分类精神，我们通过列出显式无穷小变换Lie代数来描述所发现的齐次模型，有时由满足某些代数方程的绝对不变量参数化。
英文摘要：
We determine all affinely homogeneous hypersurfaces S^3 in R^4 whose Hessian is (invariantly) of constant rank 2, including the simply transitive ones. We find 34 inequivalent terminal branches yielding each to a nonempty moduli space of homogeneous models of hypersurfaces S^3 in R^4, sometimes parametrized by a certain complicated algebraic variety, especially for the 15 (over 34) families of models which are simply transitive. We employ the power series method of equivalence, which captures invariants at the origin, creates branches, and infinitesimalizes calculations. In Lie's original classification spirit, we describe the found homogeneous models by listing explicit Lie algebras of infinitesimal transformations, sometimes parametrized by absolute invariants satisfying certain algebraic equations.
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