The Debarre-de Jong conjecture predicts that the Fano variety of lines on a smooth Fano hypersurface in \(\mathbb {P}^n\) is always of the expected dimension. We generalize this conjecture to the case of smooth Fano complete intersections and prove that for a smooth Fano complete intersection \(X\subset \mathbb {P}^n\) of hypersurfaces whose degrees sum to at most 7, the Fano variety of lines on X has the expected dimension.

中文翻译：

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关于法诺完整交集上各种线的猜想

Debarre-de Jong猜想预测\（\ mathbb {P} ^ n \）中的光滑Fano超曲面上的Fano线总是具有预期的尺寸。我们将该猜想推广到光滑Fano完整交点的情况，并证明对于度数最大为7的超曲面的光滑Fano完整交点\（X \ subset \ mathbb {P} ^ n \），Fano各种线在X上具有预期的尺寸。