复双曲等距轨迹
Complex hyperbolic equidistant loci
摘要：
我们描述和研究所谓的复双曲几何中距许多点等距的轨迹，即，在全纯$2$ -球$\Bbb B$的几何中。特别是，我们表明，二等分线（=从$2$点等距的轨迹）包含（光滑的实代数）曲线等距给定的$4$泛点形成一个真正的椭圆曲线和上述二等分线的焦点构成同构的椭圆曲线。我们将使用所获得的事实中的结构的（紧）的$\Bbb B$的离散群。利用类似的技术，我们还将合痕泛$3$维代数（即，双线性运算）在特征为$\ne2,3$的代数闭域$\Bbb K$上。简单地说，一个代数是由它的零因子的（平面投影）曲线$D$分类的，该曲线配备有$D$的非投影自同构。这种分类几乎等同于A. Bondal和A. Polishchuk在他们的研究非累积射影平面。
英文摘要：
We describe and study the loci equidistant from finitely many points in the so-called complex hyperbolic geometry, i.e., in the geometry of a holomorphic $2$-ball $\Bbb B$. In particular, we show that the bisectors (= the loci equidistant from $2$ points) containing the (smooth real algebraic) curve equidistant from given $4$ generic points form a real elliptic curve and that the foci of the mentioned bisectors constitute an isomorphic elliptic curve. We are going to use the obtained facts in constructions of (compact) quotients of $\Bbb B$ by discrete groups. With similar technique, we also classify up to isotopy generic $3$-dimensional algebras (i.e., bilinear operations) over an algebraically closed field $\Bbb K$ of characteristic $\ne2,3$. Briefly speaking, an algebra is classified by the (plane projective) curve $D$ of its zero divisors equipped with a nonprojective automorphism of $D$. This classification is almost equivalent to the classification of the so-called geometric tensors given in [BoP] by A. Bondal and A. Polishchuk in their study of noncummutative projective planes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
