For a smooth projective quadric X , Alexander Vishik introduced the notion of connections . He discovered connections arising from the splitting pattern of X as well as the so-called excellent connections. Recently, Victor Petrov and Nikita Semenov found new connections arising from the J-invariant J ( X ), which is a layer of the elementary discrete invariant $$ED(X)$$ E D ( X ) , both further Vishik’s inventions. In this note, we give a new proof for Peterov–Semenov’s connections and extend them to the remaining layers of $$ED(X)$$ E D ( X ) . Also, we briefly discuss connections on nonsmooth quadrics.

中文翻译：

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扩展 Petrov–Semenov 的连接

对于平滑的投影二次曲面 X，Alexander Vishik 引入了连接的概念。他发现了由 X 的分裂模式产生的连接以及所谓的极好连接。最近，Victor Petrov 和 Nikita Semenov 发现了由 J-不变量 J ( X ) 产生的新联系，它是基本离散不变量 $$ED(X)$$ ED ( X ) 的一层，两者都进一步推动了 Vishik 的发明。在这篇笔记中，我们给出了 Peterov–Semenov 连接的新证明，并将它们扩展到 $$ED(X)$$ED ( X ) 的其余层。此外，我们简要讨论了非光滑二次曲面的连接。