Finite Type and the Effective Nullstellensatz

Abstract

Improved local and global versions of the effective Nullstellensatz for ideal sheaves on nonsingular complex varieties are obtained, based on a new invariant motivated by the notion of finite type from the theory of several complex variables. Two closely related curve selection theorems for curves with maximal adjusted tangency orders to a given ideal sheaf are established along the way, using normalized blow-ups and integral closures.

Key Words:

• Effective Nullstellensatz
• Finite type domain
• Integral closure
• Normalized blow-up

2000 Mathematics Subject Classification:

• 14Q20
• 14C17
• 32S10
• 32T25

ACKNOWLEDGMENTS

The author wishes to thank Brian Conrad, Jeffery McNeal, Mircea Mustaţǎ, and, most of all, Rob Lazarsfeld for valuable discussions. Bernard Teissier kindly alerted us to, and provided a copy of Lejeune-Jalabert and Teissier (1974).

This research was supported by the DFG Schwerpunkt Global methods in complex geometry.

Notes

Communicated by L. Ein.