| Description |
In this thesis, we consider two different problems in birational geometry considered previously in the author's papers. The first problem concerns pluri-canonical maps in positive characteristic. We prove that for a smooth variety X of general type over an algebraically closed field k with positive characteristic, if X has maximal Albanese dimension and the Albanese map is separable, then |4KX| induces a birational map. The second problem is on the volume of isolated singularities over C. We give an equivalent definition of the local volume of an isolated singularity VolBdFF(X, 0) defined by Boucksom, de Fernex and Favre in the Q-Gorenstein case and we generalize it to the non-Q-Gorenstein case. We prove that there is a positive lower bound depending only on the dimension for the non-zero local volume of an isolated singularity if X is Gorenstein. We also give a non-Q-Gorenstein example with VolBdFF(X, 0) = 0, which does not allow a boundary such that the pair (X, ) is log canonical. |
