  In 2009, de Fernex and Hacon proposed a generalization of the notion of the
singularities to normal varieties that are not Q-Gorenstein. Based on their
work, we generalize Kleiman's transversality theorem to subvarieties with log
terminal or log canonical singularities. We also show that some classical
varieties, such as generic determinantal varieties, W^r_d for general smooth
curves, and certain Schubert varieties in G(k, n) are log terminal.
