The Hilbert curve of a complex polarized manifold (X,L) is the complex affine plane curve of degree dim(X) defined by the Hilbert-like polynomial χ(xKX + yL), where KX is the canonical bundle of X and x and y are regarded as complex variables. A natural expectation is that this curve encodes several properties of the pair (X,L). In particular, the existence of a fibration of X over a variety of smaller dimension induced by a suitable adjoint bundle to L translates into the fact that the Hilbert curve has a quite special shape. Along this line, Hilbert curves of special varieties like Fano manifolds with low coindex, as well as fibrations over low-dimensional varieties having such a manifold as general fiber, endowed with appropriate polarizations, are investigated. In particular, several polarized manifolds relevant for adjunction theory are completely characterized in terms of their Hilbert curves.

中文翻译：

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通过希尔伯特曲线表征一些极化的 Fano 纤维

复极化流形的希尔伯特曲线(X,大号)是度数的复仿射平面曲线暗淡(X)由类希尔伯特多项式定义χ(XķX + 是的大号)， 在哪里ķX是规范丛X和X和是的被视为复变量。一个自然的期望是这条曲线编码了该对的几个属性(X,大号). 特别是，存在纤维化X在由合适的伴随丛诱导的各种更小的维度上大号转化为希尔伯特曲线具有非常特殊的形状这一事实。沿着这条线，研究了特殊品种的希尔伯特曲线，如具有低共指数的 Fano 流形，以及具有像一般光纤这样的流形的低维品种上的纤维化，赋予适当的偏振。特别是，与附加理论相关的几个极化流形完全用它们的希尔伯特曲线来表征。