  We give a canonical synthetic construction of the mirror family to a pair
(Y,D) of a smooth projective surface with an anti-canonical cycle of rational
curves, as the spectrum of an explicit algebra defined in terms of counts of
rational curves on Y meeting D in a single point. In the case D is
contractible, the family gives a smoothing of the dual cusp, and thus a proof
of Looijenga's 1981 cusp conjecture.
