We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index $e$ , then the degree of irrationality of a very general complex Fano hypersurface of index $e$ and dimension n is bounded from below by a constant times $\sqrt{n}$ . To our knowledge, this gives the first examples of rationally connected varieties with degrees of irrationality greater than 3. The proof follows a degeneration to characteristic $p$ argument, which Kollár used to prove nonrationality of Fano hypersurfaces. Along the way, we show that in a family of varieties, the invariant ‘the minimal degree of a dominant rational map to a ruled variety’ can only drop on special fibers. As a consequence, we show that for certain low-dimensional families of varieties, the degree of irrationality also behaves well under specialization.

中文翻译：

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具有任意大非理性程度的 FANO 超曲面

我们表明，复杂的 Fano 超曲面可以具有任意程度的非理性。更准确地说，如果我们修复一个 Fano 索引 $e$ , 那么指数的一个非常一般的复数 Fano 超曲面的非理性程度 $e$ 并且维度 n 从下方以恒定时间为界 $\sqrt{n}$ . 据我们所知，这给出了非理性度大于 3 的理性连接变体的第一个例子。证明遵循特征退化 $p$ Kollár 用它来证明 Fano 超曲面的非理性。一路走来，我们表明，在一个品种家族中，不变的“占主导地位的理性映射到被统治的品种的最小程度”只能落在特殊的纤维上。因此，我们表明，对于某些低维变体家族，非理性程度在专业化下也表现良好。