ON TWO SIMPLE CRITERIA FOR RECOGNIZING COMPLETE INTERSECTIONS IN CODIMENSION 2

Abstract

Developing a previous idea of Faltings, we characterize the complete intersections of codimension 2 in P ⁿ , n ≥ 3, over an algebraically closed field of any characteristic, among l.c.i. X, as those that are subcanonical and scheme-theoretically defined by p ≤ n − 1 equations. Moreover, we give some other results assuming that the normal bundle of X extends to a numerically split bundle on P ⁿ and p ≤ n. Finally, using our characterization, we give a (partial) answer to a question posed recently by Franco, Kleiman and Lascu ([5]) on self-linking and complete intersections in positive characteristic.

ACKNOWLEDGMENTS

It is a pleasure to thank Philippe Ellia and Alexandru Lascu for useful discussions, and Steve Kleiman for valuable remarks and corrections; I would like also to thank the members of the Department of Mathematics of Ferrara University for the kind hospitality and the warm and stimulating atmosphere.