As a variant of Shokurov's criterion of toric surface, we give a criterion of two new classes of normal projective surfaces, called pseudo-toric surfaces of defect one and half-toric surfaces. A typical example of pseudo-toric surface of defect one is a projective toric surface blown up at a non-singular point of the boundary divisor. A half-toric surface is the quotient of a projective toric surface by an almost free involution preserving the boundary divisor. The structure of pseudo-toric surface of defect one and that of half-toric surface are also studied in detail.
Digital Object Identifier: 10.2969/aspm/07510287