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These congruences include the $\\mathrm{G}_2$-variety for $n=6$ and the variety of reductions of projected $\\mathbb{P}^2 \\times \\mathbb{P}^2$ for $n=7$.\n  We compute the degree of $X_\\omega$ as the $n$-th Fine number and study the Hilbert scheme of these congruences proving that the choice of $\\omega$ bijectively corresponds to $X_\\omega$ except when $n=5$. 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