Complete description of determinantal representations of smooth irreducible curves
Complete description of determinantal representations of smooth irreducible curves
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DOI：
10.1016/0024-3795(89)90035-9
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发表时间：
1989-12
影响因子：
1.1
通讯作者：
V. Vinnikov
中科院分区：
文献类型：
--
作者：
V. Vinnikov
代数曲线的行列式表示本身就很有趣，它们的分类等价于矩阵三元组的同时分类。本文给出了代数闭域上光滑不可约曲线的行列式表示的完整描述。我们使用的概念的类因子的向量丛对应的行列式表示;我们证明了两个行列式表示的光滑curveFare等价当且仅当类因子的相应向量丛一致。我们给出了一个精确的特征，这些类的因子所产生的向量丛对应的行列式表示的F。然后，我们得到了一个参数化的行列式表示的F，直到等价的，由点的雅可比簇Fnot的某些例外的子簇。特别是它遵循任何光滑曲线的顺序3或更大拥有无限数量的非等价行列式表示。我们还专门研究我们的结果对称和自伴表示。
Determinantal representations of algebraic curves are interesting in themselves, and their classification is equivalent to the simultaneous classification of triples of matrices. We present a complete description of determinantal representations of smooth irreducible curves over any algebraically closed field. We used the notion of the class of divisors of the vector bundle corresponding to a determinantal representation; we prove that two determinantal representations of a smooth curveFare equivalent if and only if the classes of divisors of the corresponding vector bundles coincide. We give a precise characterization of those classes of divisors that arise from vector bundles corresponding to determinantal representations ofF. Then we obtain a parametrization of determinantal representations ofF, up to equivalence, by the points of the Jacobian variety ofFnot on some exceptional subvariety. In particular it follows that any smooth curve of order 3 or greater possesses an infinite number of nonequivalent determinantal representations. We also specialize our results to symmetrical and self-adjoint representations.