Hirzebruch's construction of toroidal compactification of Hilbert modular surfaces is explicit, namely one can explicitly choose rational polyhedral cone decomposition in a sort of optimal way using convex hull/continued fractions. Is such an explicit theory available for higher degree totally real fields? If so, is there a reference?
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$\begingroup$ Borisov and Gunnells consider partial toroidal resolutions for a particular example of a Hilbert modular threefold over a totally real cubic field in section 3 of On Hilbert modular threefolds of discriminant 49. They also mention Grundman's Explicit resolutions of cubic cusp singularities, which might be closer to what you're looking for. $\endgroup$– Viktor VaughnCommented Oct 28, 2021 at 8:51
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