| Two questions related to the moduli space Mh of stable schemes are investigated. The first one, Arakelov-Parshin rigidity, concerns varieties mapping rigidly to the smooth part Mh of Mh . On an open subspace of Mh , the locus KFh of iterated Kodaira fibrations, certain rigid classes are detected. Second question is related to the construction of Mh . It asks if the compatibility of the relative canonical sheaf with base change holds for families of normal varieties. It is shown that it fails in general. Furthermore we exhibit connections of this failure to the behavior of Serre's Sd property in families and on individual schemes. |
