The only ball-quotient structures on P^2 with smooth pairwise normal-crossing branch divisor are the Deligne–Mostow complete quadrilateral and the degree-9 dual Hesse arrangement.
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The complex projective plane as a ball quotient
The only ball-quotient structures on P^2 with smooth pairwise normal-crossing branch divisor are the Deligne–Mostow complete quadrilateral and the degree-9 dual Hesse arrangement.