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It is known that some GIT compactifications associated to moduli spaces ofeither points in the projective line or cubic surfaces are isomorphic toBaily-Borel compactifications of appropriate ball quotients. In this paper, weshow that their respective toroidal compactifications are isomorphic to modulispaces of stable pairs as defined in the context of the MMP. Moreover, we givea precise mixed-Hodge-theoretic interpretation of this isomorphism for the caseof eight labeled points in the projective line.

Geometric interpretation of toroidal compactifications of moduli of points in the line and cubic surfaces