Title:

Resolution and completion of algebraic varieties

Little is known concerning the resolution of the singular locus of an algebraic variety, apart from that they exist when the ground field has characteristic zero and in some other cases. We obtain results concerning the geometric structure of the resolution, if such exists, of any given singularity. Our results depend upon some preliminary material in Chapter I, and are valid for any characteristic. The main technique used is to study the vanishing locus of differential forms and its behavior under any given monoidal transformation. This is done in Chapter II. From this we obtain a range of results. Inter alia we show that certain places must appear in the resolution pf any given singular locus to a divisor. Our results do not exhaust the power of the technique. In Chapter III we study certain birational modifications, called toral, that can be defined from the normal crossings divisor. Singularities that can be resolved by such modification are used to give examples for the results of Chapter II. Also, we prove a result concerning the completion of algebraic varieties. Suppose that the complete and nonsingular variety M contains an algebraic torus T as the complement of normal crossing divisor D. Then the birational action of T on M is biregular just in case some differential of top degree regular on T has a pole of order one along each component of D. The results of Chapter III can be thought of as belonging to the theory of torus embedding. However, they are a special case of an as a yet unwritten theory of general toral modifications, the definition of which appears in the Introduction, and an example of which concludes the work. This general theory offers a chance to prove more generally valuable results that are presently known only in special cases, such as the finite generation of the canonical ring.
