A transversality theorem for some classical varieties

In 2009, de Fernex and Hacon [10] 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, Wdr for general smooth curves, and certain Schubert varieties in G(k,n) are log terminal in de Fernex and Hacon's notion, and canonical with some suitable boundary in the classical sense.