In this thesis we discuss a moduli space of projective curves with a map to a given projective space P. The functor CM parametrizes curves, that is, Cohen-Macaulay schemes of pure dimension 1, together with a finite map to P that is an isomorphism onto its image away from a finite set of closed points. We proof that CM is an algebraic space by contructing a scheme W and a representable, surject...

We prove a version of the Halphen Speciality Theorem for locally Cohen-Macaulay curves in P3. To prove the theorem, we strengthen some results of Okonek and Spindler on the spectrum of the ideal sheaf of a curve. As an application, we classify curves C having index of speciality as large as possible once we fix the degree of C and the minimum degree of a surface

This thesis presents an expository account of the use of Frobenius splitting techniques in the study of Schubert varieties. After developing the basic theory of Frobenius splitting, we show that the Schubert and Bott-Samelson varieties are split and use this to derive geometric consequences in arbitrary characteristic. The main result highlighted is that Schubert varieties are normal, Cohen-Mac...

We prove that any rank two arithmetically CohenMacaulay vector bundle on a general hypersurface of degree at least three in P must be split.

In this article we present a fast recursive orthogonalization scheme for two important subspaces of the Macaulay matrix: its row space and null space. It requires a graded monomial ordering and exploits the resulting structure of the Macaulay matrix induced by this graded ordering. The resulting orthogonal basis for the row space will retain a similar structure as the Macaulay matrix and is as ...

Let C ⊂ N be an affine semigroup, and R = K[C] its semigroup ring. This paper is a collection of various results on “C-graded” R-modules M = ⊕ c∈C Mc, especially, monomial ideals of R. For example, we show the following: If R is normal and I ⊂ R is a radicalmonomial ideal (i.e., R/I is a generalization of Stanley-Reisner rings), then the sequentially Cohen-Macaulay property of R/I is a topologi...

Here we study rank 2 arithmetically Cohen-Macaulay vector bundles on a a ruled surface over a smooth genus q curve, essentially proving their non-existence if q ≥ 2 and the ruled surface is rather balanced.