[This][1] paper gives the following definition of a relatively minimal variety:a relatively minimal variety $X'$ over a base $Y$ is a projective variety with at most $\mathbb{Q}$-factorial terminal singularities which has an extremal ray contraction $\phi: X'\to Y$ of fiber type, i.e., $\dim Y<\dim X$.


  [1]: http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.nmj/1114631348