Families of Varieties of General Type: the Shafarevich Conjecture and Related Problems

Consider the curves on the (x; y)-plane parametrized by ¸ and given by the equation, y2 = x5 ¡ 5¸x+ 4¸: An easy computation shows that these curves are non-singular for ¸ 2 C n f0; 1g. On the other hand the projective closure of each of these curves is singular at the point [0: 1: 0]. Blowing up this point and another one in¯nitely near this one yields a resolution, C¸. The collection of C ¸ forms a family of smooth projective curves over the base C n f0; 1g. Exercise 0.1. Prove that the projection of C ¸ onto the y = 0 axis is the only degree 2 morphism onto P1 that C ¸ admits. (Hint: See [28], IV.5.3). Conclude that for ¸1 6 = ¸2, C¸1 6 ' C¸2. A family of curves is called isotrivial if any two general members are isomorphic. For example the blow up of the projective plane at a single point, considered as a P1-bundle over P1, is an isotrivial family since all of its members are isomorphic to the projective line. Since the genus of a curve in a family is constant, and there is only one curve of genus zero, one cannot expect much more in this case. However, for higher genus curves one can have non-isotrivial families as the above example shows. Exercise 0.2. Compute the genus of C¸. Find another non-isotrivial family of smooth curves of the same genus over C n f0; 1g. Make a guess: Is there an in¯nite sequence of pairwise di®erent families of that kind? At the 1962 ICM in Stockholm, Shafarevich conjectured that there exist only ¯nitely many isomorphism classes of non-isotrivial families of smoot