| In this thesis we study 2-type vectors associated to Hilbert functions of complete intersections in P2 . Let Y be a finite set of points in P2 . Y is said to be a complete intersection C.I.( a,b) if the ideal of Y can be generated by two homogeneous polynomials of degrees a and b (where a ≤ b). Moreover, the Hilbert function H( Y ), and hence its associated 2-type vector THY , depends only on a and b. Thus every complete intersection C.I.(a,b) has Hilbert function HC.I.a,b and associated 2-type vector TC.I.a,b . Let T be a 2-type vector. We say that T is a permissible sub-type vector of TC.I.a,b if there exists a complete intersection Y = C.I.(a,b) and a subset X⊆Y such that the Hilbert function H( X ) is associated to T. The first original result of this thesis is the classification of such permissible sub-type vectors.; For the rest of this thesis we suppose T is a permissible sub-type vector of TC.I.a,b . Thus, there exists a complete intersection Y = C.I.(a,b) in P2 with Hilbert function H( Y ) associated to TC.I.a,b and a subset X⊆Y with Hilbert function H( X ) associated to T. Let W = Y - X have Hilbert function H( W ) with associated 2-type vector THW . In the paper titled Gorenstein Algebras and the Cayley-Bacharach Theorem, Davis, Geramita and Orecchia give the relationship (Theorem 3) between H( Y ), H( X ) and H( W ). We investigate the relationship, similar to Theorem 3, between TC.I.a,b , T, and THW . The second original result of this thesis is an algorithm (called AlgCI) which determines THW in terms of a, b, and T without first considering the Hilbert functions H( X ) and H( W ). |
