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Higher Initial Ideals of Homogeneous Ideals

Gunnar Fløystad University of Bergen, Bergen, Norway
Available Formats:
Electronic ISBN: 978-1-4704-0227-3
Product Code: MEMO/134/638.E
List Price: $45.00 MAA Member Price:$40.50
AMS Member Price: $27.00 Click above image for expanded view Higher Initial Ideals of Homogeneous Ideals Gunnar Fløystad University of Bergen, Bergen, Norway Available Formats:  Electronic ISBN: 978-1-4704-0227-3 Product Code: MEMO/134/638.E  List Price:$45.00 MAA Member Price: $40.50 AMS Member Price:$27.00
• Book Details

Memoirs of the American Mathematical Society
Volume: 1341998; 68 pp
MSC: Primary 13; 14;

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.

• Chapters
• Introduction
• 1. Borel-fixed ideals
• 2. Monomial orders
• 3. Some algebraic lemmas
• 4. Defining the higher initial ideals
• 5. Representing the higher initial ideals
• 6. Group action on $R^{s+1}(I)$
• 7. Describing the action on $R^{s+1}(I)$
• 8. Borel-fixedness
• 9. Higher initial ideals of hyperplane sections
• 10. Representing the higher initial ideals of general hyperplane sections
• 11. Higher initial ideals as combinatorial structures
• 13. Examples: Points and curves in $\mathbf {P}^3$
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Volume: 1341998; 68 pp
MSC: Primary 13; 14;

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.

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