论文标题
关于决定性超表面的对称呈现的数量
On the number of symmetric presentations of a determinantal hypersurface
论文作者
论文摘要
A hypersurface $H$ in $\mathbb{P}^r$ of degree $n$ is called determinantal if it is the zero locus of a polynomial of the form $\operatorname{det}(x_0 A_0 + \ldots + x_r A_r)$ for some $(r+1)$-tuple of $n \times n$ matrices $A = (A_0, \ ldots，a_r）$。我们将$ a $称为$ h $的介绍。另一个演示文稿$ b =（b_0，b_1，\ ldots，b_r）$ $ h $可以通过选择$ g_1，g_1，g_2 \ in \ operatatorName {gl} _n $和设置$ b_i = g_1 a_i g_2 $，每$ i = 0，1，i = 0，1，1，1，1，1，1，1，1，1，1，1，1，1，1，1，1，1，1，1，1，1，y = ldots rdots rdots，r $ ldots，r $。在这种情况下，$ a $和$ b $称为同等产品。第二作者和A. vistoli表明，对于$ r \ geq 3 $，一般的决定性超出表面仅承认有限的许多演示文稿，直到等效。在本文中，我们证明了每$ r \ geq 2 $的对称演示的结果类似。在这里，需要对称矩阵$ a_0，\ ldots，a_r $和两个$（r+1）$ - $ n \ times n $ n $对称矩阵$ a =（a_0，a_1，a_1，\ ldots，a_r），a_r）$ a_1 $ a =（b_0，b_0，b_0，b_1，b_1，b_1，b_ e equive a e equive， \ operatatorName {gl} _n $，这样每$ i = 0，\ ldots，r $ $ b_i = g^{\ rm transpose} a_i g $。
A hypersurface $H$ in $\mathbb{P}^r$ of degree $n$ is called determinantal if it is the zero locus of a polynomial of the form $\operatorname{det}(x_0 A_0 + \ldots + x_r A_r)$ for some $(r+1)$-tuple of $n \times n$ matrices $A = (A_0, \ldots, A_r)$. We will refer to $A$ as a presentation of $H$. Another presentation $B = (B_0, B_1, \ldots, B_r)$ of $H$ can be obtained by choosing $g_1, g_2 \in \operatorname{GL}_n$ and setting $B_i = g_1 A_i g_2$ for every $i = 0, 1, \ldots, r$. In this case $A$ and $B$ are called equivalent. The second author and A. Vistoli have shown that for $r \geq 3$ a general determinantal hypersurface admits only finitely many presentations up to equivalence. In this paper we prove a similar result for symmetric presentations for every $r \geq 2$. Here the matrices $A_0, \ldots, A_r$ are required to be symmetric, and two $(r+1)$-tuples of $n \times n$ symmetric matrices $A = (A_0, A_1, \ldots, A_r)$ and $B = (B_0, B_1, \ldots, B_r)$ are considered equivalent if there exists a $g \in \operatorname{GL}_n$ such that $B_i = g^{\rm transpose} A_i g$ for every $i = 0, \ldots, r$.
