Higher Initial Ideals of Homogeneous Ideals (Memoirs of the American Mathematical Society) by Gunnar Floystad (PDF)

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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 $sgeq0$. 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.

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