Intersection Multiplicities

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This proposal concerns questions related to intersection multiplicities and the study of divisor class groups. The intersection multiplicities studied by the investigator include Serre's multiplicity, Dutta's limit multiplicity and the Hilbert-Kunz multiplicity, all intimately related, most notably via the theory of localized Chern characters of Baum, Fulton, and MacPherson. Of particular interest are the following: the connection between characteristic p and characteristic zero situations, the possible misbehavior of Serre's multiplicity when both modules have finite projective dimension, and the question of the rigidity of Tor. Some of these projects will be joint work with Anurag Singh. The second project involves the study of how the divisor class group changes between a variety and a hypersurface inside that variety. The third project involves determining the divisor class groups of symmetric mixed ladder determinantal varieties.Commutative algebra is a crucial tool in developing the foundations of algebraic geometry. Together with number theory, these fields were in the center of the revolutionary new approaches and ways of thinking about classical problems which were eventually responsible for the modernization of much of mathematics in the middle of this last century. Noether, Krull, Weil, Grothendieck, Serre and Zariski were among those who brought about this revolution. Especially central was the modern development of intersection theory and it continues to be an especially exciting and important area of research. In the same time period, the importance of divisor class groups was realized and the connection between those of a variety and a hypersurface in the variety has been a question of interest since then. Eventually these studies lead to applications, such as the recent example of algebro-geometric codes in coding theory.