Let X be any smooth prime Fano threefold of degree \(2g-2\) in \({\mathbb P}^{g+1}\), with \(g \in \{3,\ldots ,10,12\}\). We prove that for any integer d satisfying \(\left\lfloor \frac{g+3}{2} \right\rfloor \leqslant d \leqslant g+3\) the Hilbert scheme parametrizing smooth irreducible elliptic curves of degree d in X is nonempty and has a component of dimension d, which is furthermore reduced except for the case when \((g,d)=(4,3)\) and X is contained in a singular quadric. Consequently, we deduce that the moduli space of rank–two slope–stable ACM bundles \({\mathcal F}_d\) on X such that \(\det ({\mathcal F}_d)={\mathcal O}_X(1)\), \(c_2({\mathcal F}_d)\cdot {\mathcal O}_X(1)=d\) and \(h^0({\mathcal F}_d(-1))=0\) is nonempty and has a component of dimension \(2d-g-2\), which is furthermore reduced except for the case when \((g,d)=(4,3)\) and X is contained in a singular quadric. This completes the classification of rank–two ACM bundles on prime Fano threefolds. Secondly, we prove that for every \(h \in {\mathbb Z}^+\) the moduli space of stable Ulrich bundles \({\mathcal E}\) of rank 2h and determinant \({\mathcal O}_X(3h)\) on X is nonempty and has a reduced component of dimension \(h^2(g+3)+1\); this result is optimal in the sense that there are no other Ulrich bundles occurring on X. This in particular shows that any prime Fano threefold is Ulrich wild.

设X为\({\mathbb P}^{g+1}\)中任意平滑素数法诺三倍\(2g-2\ ) ，其中\(g \in \{3,\ldots ,10,12 \}\)。我们证明，对于任何满足\(\left\lfloor \frac{g+ 3 }{2} \right\rfloor \leqslant d \leqslant g+3\) 的整数 d，希尔伯特方案参数化d次光滑不可约椭圆曲线X是非空的并且具有维度d的分量，除了\((g,d)=(4,3)\)且X包含在奇异二次曲面中的情况外，该分量进一步减小。因此，我们推论出二阶斜率稳定的模空间ACM在X上捆绑\({\mathcal F}_d\)使得\(\det ({\mathcal F}_d)={\mathcal O}_X(1)\) , \(c_2({\mathcal F} _d)\cdot {\mathcal O}_X(1)=d\)和\(h^0({\mathcal F}_d(-1))=0\)是非空的并且具有维度为\(2d -g-2\) ，除了\((g,d)=(4,3)\)且X包含在奇异二次曲面中的情况外，它还会进一步减少。这就完成了素数 Fano 上的二级ACM束的分类。其次，我们证明对于每个\(h \in {\mathbb Z}^+\)，稳定乌尔里希丛的模空间\({\mathcal E}\)X上的秩 2 h和行列式\({\mathcal O}_X(3h)\)是非空的，并且具有维度\(h^2(g+3)+1\)的缩减分量；这个结果是最优的，因为X上没有其他乌尔里希束出现。这尤其表明任何素数法诺三重都是乌尔里希野。