In this dissertation, we discuss some necessary and sufficient conditions for a curve to be arithmetically Cohen-Macaulay, in terms of its general hyperplane section. We obtain a characterization of the degree matrices that can occur for points in the plane that are the general plane section of a non arithmetically Cohen-Macaulay curve of P^3. We prove that almost all the degree matrices with positive subdiagonal that occur for the general plane section of a non arithmetically Cohen-Macaulay curve of P^3, arise also as degree matrices of some smooth, integral, non arithmetically Cohen-Macaulay curve, and we characterize the exceptions. We give a necessary condition on the graded Betti numbers of the general plane section of an arithmetically Buchsbaum (non arithmetically Cohen-Macaulay) curve in P^n. For curves in P^3, we show that any set of Betti numbers that satisfy that condition can be realized as the Betti numbers of the general plane section of an arithmetically Buchsbaum, non arithmetically Cohen-Macaulay curve. We also show that the matrices that arise as degree matrix of the general plane section of an arithmetically Buchsbaum, integral (or smooth and connected), non arithmetically Cohen-Macaulay space curve are exactly those that arise as degree matrix of the general plane section of an arithmetically Buchsbaum, non arithmetically Cohen-Macaulay space curve and have positive subdiagonal. We prove some bounds on the dimension of the deficiency module of an arithmetically Buchsbaum curve in P^n, in terms of entries of the lifting matrix of a general hyperplane section of the curve, and we show that they are sharp. We then turn to the question of whether it is possible to lift the property of being standard or good determinantal from the general hyperplane section of a scheme to the scheme itself. We produce examples of schemes that are not standard determinantal, but whose general hyperplane section is good determinantal. Using a result of Kleppe and Miro’-Roig, we show that if one hyperplane section of a scheme of codimension 3 by a hyperplane that meets it properly is good determinantal, then a general hyperplane section of the scheme is good determinantal.
Lifting Properties from the General Hyperplane Section of a Projective SchemeDoctoral Dissertation
|Contributor||Juan Migliore, Committee Member|
|Contributor||Claudia Polini, Committee Member|
|Contributor||Andrew Sommese, Committee Member|
|Contributor||Karen Chandler, Committee Member|
|Degree Level||Doctoral Dissertation|
|Departments and Units|