Higher Initial Ideals of Homogeneous Ideals

Given a homogeneous ideal $I$ and a monomial order, one may form the initial ideal $\textnormal{in}(I)$. The initial ideal gives information about $I$, for instance $I$ and $\textnormal{in}(I)$ have the same Hilbert function. However, if $\mathcal I$ is the sheafification of $I$ one cannot read the higher cohomological dimensions $h^i({\mathbf P}^n, \mathcal I(\nu)$ from $\textnormal{in}(I)$. This work remedies this by defining a series of higher initial ideals $\textnormal{in}_s(I)$ for $s\geq0$. Each cohomological dimension $h^i({\mathbf P}^n, \mathcal I(\nu))$ may be read from the $\textnormal{in}_s(I)$. The $\textnormal{in}_s(I)$ are however more refined invariants and contain considerably more information about the ideal $I$. This work considers in particular the case where $I$ is the homogeneous ideal of a curve in ${\mathbf P}^3$ and the monomial order is reverse lexicographic.Then the ordinary initial ideal $\textnormal{in}_0(I)$ and the higher initial ideal $\textnormal{in}_1(I)$ have very simple representations in the form of plane diagrams. It enables one to visualize cohomology of projective schemes in ${\mathbf P}^n$. It provides an algebraic approach to studying projective schemes. It gives structures which are generalizations of initial ideals.