Multiplicity Structures on Conics

Abstract

Let C ? P 3 be a conic. A multiplicity structure on C is a closed subscheme Z ? P 3 such that Supp Z = Supp C . The multiplicity of Z is defined by the ratio deg Z /deg C , which we prove to be an integer. In this dissertation we give complete classification of double conics on C . This classification includes descriptions of their of total ideals, minimal free resolutions of their total ideals, their Rao modules, descriptions of general surfaces containing such structures and the criterion for two double conics on C to be linked by a complete intersection, which extends a well-known theorem of Migliore on self-linkage of double lines to double conics. We also give a partial classification of triple conics on C , which is complicated by new behaviors such as the jumping of cohomology groups and the non-splitting nature of the restriction of total ideals of the second Cohen-Macaulay filtrant of odd genera.