A rationality criterion for projective surfaces - partial solution to   Kollar's conjecture
  Koll\'ar's conjecture states that a complex projective surface $S$ with quotient singularities and with $H^2(S,\bbQ)\cong \bbQ$ should be rational if its smooth part $S^0$ is simply connected.   We confirm the conjecture under the additional condition that the exceptional divisor in a minimal resolution of $S$ has at most 3 components over each singular point of $S$. 
JongHae Keum