论文标题
立方三倍的Pfaffian表示
Pfaffian representations of cubic threefolds
论文作者
论文摘要
给定一个立方体超出表面$ x \ subset \ mathbb {p}^4 $，我们研究了$ x $的pfaffian表示的存在，即$ 6 \ times 6 $ skew-skew-quesw-quesw-quest-mmetterric of Linearear forms $ m $ $ m $，这样$ x $ $ x $由等式$ PF（M）= 0 $。众所周知，每当$ x $平滑时，这种矩阵总是存在。在这里，我们证明，每当$ x $都是单数时，同样的成绩也是如此，因此每个立方三倍都是pfaffian。
Given a cubic hypersurface $X\subset \mathbb{P}^4$, we study the existence of Pfaffian representations of $X$, namely of $6\times 6$ skew-symmetric matrices of linear forms $M$ such that $X$ is defined by the equation $Pf(M)=0$. It was known that such a matrix always exists whenever $X$ is smooth. Here we prove that the same holds whenever $X$ is singular, hence that every cubic threefold is Pfaffian.
