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

Readership

Research mathematicians in commutative algebra, computer algebra and algebraic geometry.