A Depth Formula for Generic Singularities and Their Weak Normality

Abstract

Let X be a smooth projective variety of dimension r and π:X → ℙ^m a generic projection with r + 1 ≤ m ≤ 2r. It is shown that, at any point on X′ = π(X) of multiplicity μ, off a closed subset of the triple locus of codimension four, the depth of the local ring is equal to r − (μ − 1)(m − r − 1). This leads to some improvements on the affirmation of a conjecture of Andreotti–Bombieri–Holm on the weak normality of X′ and a conjecture of Piene on the weak normality of Sing(X′).

Key Words:

• Fitting ideals
• Generic projections
• Generic singularities
• Seminormality
• Weak normality

2000 Mathematics Subject Classification:

• 14B05
• 14M05
• 14B25
• 13F45

ACKNOWLEDGMENTS

The author would like to thank Joel Roberts for valuable communications and encouragements. He thanks Paolo Salmon for a long period of cooperation and his influence on this article. He is grateful to Bernard Teissier for bringing attention to the importance of the work on weak normality, Joseph Oesterle for suggesting the natural form for the finite presentation treated in Section 2, Hassan Haghighi for useful discussions on tensor products of algebras, Siamk Yassemi for his help on homological arguments, and Ofer Gabber for his comments on multiplicities. Part of this work was carried out when the author visited Institut de Mathématiques de Jussieu (January–March 2006) and Institut Hautes Études Scientifiques (April–June 2006). He wishes to thank Joseph Oesterle for arranging the first visit and IHES authorities for their hospitality during the second occasion. The author thanks the Research Council of the College of Science, University of Tehran, for the financial support (Grant No. 6103011/1/01).

Notes

Communicated by R. Piene.