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Higher Initial Ideals of Homogeneous Ideals
Gunnar Fløystad, University of Bergen, Norway
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Memoirs of the American Mathematical Society
1998; 68 pp; softcover
Volume: 134
ISBN-10: 0-8218-0853-2
ISBN-13: 978-0-8218-0853-5
List Price: US$45 Individual Members: US$27
Institutional Members: US\$36
Order Code: MEMO/134/638

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

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

• Introduction
• Borel-fixed ideals
• Monomial orders
• Some algebraic lemmas
• Defining the higher initial ideals
• Representing the higher initial ideals
• Group action on $$R^{s+1}(I)$$
• Describing the action on $$R^{s+1}(I)$$
• Borel-fixedness
• Higher initial ideals of hyperplane sections
• Representing the higher initial ideals of general hyperplane sections
• Higher initial ideals as combinatorial structures
• Examples: Points and curves in $$\mathbf P^3$$