Given a homogeneous ideal \(I\) and a monomial order, one may form the initial ideal \(\mathrm{in}(I)\). The initial ideal gives information about \(I\), for instance \(I\) and \(\mathrm{in}(I)\) 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 \(\mathrm{in}(I)\). This work remedies this by defining a series of higher initial ideals \(\mathrm{ in}_s(I)\) for \(s\geq0\). Each cohomological dimension \(h^i({\mathbf P}^n, \mathcal I(\nu))\) may be read from the \(\mathrm{in}_s(I)\). The \(\mathrm{in}_s(I)\) 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 \(\mathrm{in}_0(I)\) and the higher initial ideal \(\mathrm{in}_1(I)\) have very simple representations in the form of plane diagrams.

Features:

• enables one to visualize cohomology of projective schemes in \({\mathbf P}^n\)
• provides an algebraic approach to studying projective schemes
• gives structures which are generalizations of initial ideals