Determinantal hypersurfaces
Determinantal hypersurfaces
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行列式超曲面
DOI：
10.1307/mmj/1030132707
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发表时间：
1999
期刊：
影响因子：
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通讯作者：
Bill Fulton
中科院分区：
文献类型：
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作者：
Arnaud Beauville To;Bill Fulton
(0.1) 我们在本文中讨论 P 上的齐次形式可以写成具有齐次项（可能对称）的矩阵的行列式，或斜对称矩阵的 pfaffian。这个问题已经在各种具体案例中得到了考虑（见下面的历史评论），我们相信一般结果是专家们众所周知的；但我们一直无法在文献中找到它。本文的目的就是填补这一空白。我们将首先讨论一般结构定理；粗略地说，他们表明将齐次形式 F 表示为行列式（分别是普法夫）相当于在超曲面 F = 0 上产生某种类型的线丛（分别是二阶向量丛）。本文的其余部分由应用程序组成。我们将注意力限制在光滑的超曲面上；事实上，我们对 P 中 d 的通用形式可以写成上述形式之一的情况特别感兴趣。在这种情况下，对 (X,E) 的模空间（其中 X 是 P 中的 d 次光滑超曲面，E 是满足适当条件的 1 或 2 阶向量丛）出现为某个矩阵向量空间的开子集的商；特别是，这个模空间是非有理的。例如，平面曲线的雅可比行列式万能族（Cor. 3.6）或三次三次的中间雅可比行列式（Cor. 8.8）就是这种情况。不幸的是，这种情况并不经常发生：我们将证明只有曲线和三次曲面一般才允许行列式方程。对于 pfaffian 来说，情况稍好一些：任何阶数的平面曲线、阶数≤ 15 的曲面以及阶数 ≤ 5 的三重曲面都可以一般地由线性 pfaffian 定义。
(0.1) We discuss in this paper which homogeneous form on P can be written as the determinant of a matrix with homogeneous entries (possibly symmetric), or the pfaffian of a skew-symmetric matrix. This question has been considered in various particular cases (see the historical comments below), and we believe that the general result is well-known from the experts; but we have been unable to find it in the literature. The aim of this paper is to fill this gap. We will discuss at the outset the general structure theorems; roughly, they show that expressing a homogeneous form F as a determinant (resp. a pfaffian) is equivalent to produce a line bundle (resp. a rank 2 vector bundle) of a certain type on the hypersurface F = 0 . The rest of the paper consists of applications. We have restricted our attention to smooth hypersurfaces; in fact we are particularly interested in the case when the generic form of degree d in P can be written in one of the above forms. When this is the case, the moduli space of pairs (X,E) , where X is a smooth hypersurface of degree d in P and E a rank 1 or 2 vector bundle satisfying appropriate conditions, appears as a quotient of an open subset of a certain vector space of matrices; in particular, this moduli space is unirational. This is the case for instance of the universal family of Jacobians of plane curves (Cor. 3.6), or of intermediate Jacobians of cubic threefolds (Cor. 8.8). Unfortunately this situation does not occur too frequently: we will show that only curves and cubic surfaces admit generically a determinantal equation. The situation is slightly better for pfaffians: plane curves of any degree, surfaces of degree ≤ 15 and threefolds of degree ≤ 5 can be generically defined by a linear pfaffian.