The Limiting Shape of the Generic Initial System of a Complete Intersection

Abstract

Consider a complete intersection I of type (d ₁,…, d _r ) in a polynomial ring over a field of characteristic 0. We study the graded system of ideals {gin(I ⁿ )} _n obtained by taking the reverse lexicographic generic initial ideals of the powers of I and describe its asymptotic behavior. This behavior is nicely captured by the limiting shape which is shown to depend only on the type of the complete intersection.

Key Words:

• Asymptotic
• Complete intersection
• Generic initial ideal
• Graded system
• Multiplier ideal

2010 Mathematics Subject Classification:

• Primary: 13P10
• 13C40
• Secondary: 13A02
• 14F18

ACKNOWLEDGMENTS

I would like to thank Karen Smith for introducing this problem to me, for guiding my work, and for her continued encouragement. I would also like to thank Daniel Erman for suggestions on an earlier draft. Finally, I would like to thank Mel Hochster, Robert Lazarsfeld, Irena Swanson, and Michael Von Korff for useful discussions. Calculations leading to the statement of the main theorem were performed using the computer software Macaulay2 [13].

The author is partially supported by the Natural Sciences and Engineering Research Council of Canada.

Notes

This follows from the fact that multiplication of monomial ideals corresponds to adding Newton polytopes: if q = ps, then implies that sP _{a p}  ⊆ P _{a q} so, scaling by , (also see [22]).

Communicated by I. Swanson.