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Let A be a matrix in SL_2(Z) with | Tr A| > 2. Denote by λ^[±] the two eigenvalues, with |λ^+| > 1 and |λ^−| < 1. It is convenient to suppose further that λ^[±] > 0. The monomial map M_A : (C^* )^2 → (C^*)^2 is defined by M_A ( x y ) = (x^a y^b x^c y^d ) . The map MA : (C^* )^2 → (C^* )^2 is an isomorphism, but of course is not an isomorphism from P^2 → P^2 : the three points [0 : 0 : 1], [0 : 1 : 0], [1 : 0 : 0] are points of indeterminacy for either M_A or M_A^−1 . We will make an infinite number of blow-ups in P^2 to make a compact space X_A in which (C^* )^2 is dense, and such that M_A extends to an “isomorphism” M_A : X_A → X_A. Our interest in monomial mappings largely springs from trying to understand the resolution of singularities at the cusps of Hilbert modular surfaces SK = (H × H)/P SL_2(O_K) where K is a real quadratic field and O_K is its ring of integers.