We study the deformations of a curve C on an Enriques-Fano 3-fold $X\subset {\mathbb{P}}^n$, assuming that C is contained in a smooth hyperplane section $S\subset X$, that is a smooth Enriques surface in X. We give a sufficient condition for C to be (un)obstructed in X, in terms of half pencils and $(-2)$-curves on S. Let ${\mathrm{Hilb}}^{sc}X$ denote the Hilbert scheme of smooth connected curves in X. By using the Hilbert-flag scheme of X, we also compute the dimension of ${\mathrm{Hilb}}^{sc}X$ at $[C]$ and give a sufficient condition for ${\mathrm{Hilb}}^{sc}X$ to contain a generically non-reduced irreducible component of Mumford type.

中文翻译：

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Enriques-Fano 3折上的曲线变形的障碍

我们研究Enriques-Fano 3倍曲线C的变形$X\subset {\mathbb{P}}^{ñ}$，假设C包含在光滑的超平面截面中$\mathrm{小号}\subset X$，即X中的光滑Enriques曲面。我们给出了一个充分条件Ç的阻碍是（联合国）X，在半铅笔的条款和$（-2）$-curves上小号。让${\mathrm{希伯}}^{sC}X$表示X中的光滑连接曲线的希尔伯特方案。通过使用X的Hilbert-flag方案，我们还计算了${\mathrm{希伯}}^{sC}X$ 在 $[C]$ 并为 ${\mathrm{希伯}}^{sC}X$ 包含一般不可还原的Mumford类型不可约成分。