Rigid Subanalytic Subsets of Curves and Surfaces

Working over an algebraically closed, complete, non-Archimedean, non-trivially valued eld of characteristic zero, we show that (i) any subanalytic subset of a one-dimensional semialgebraic set is semialgebraic; (ii) any one-dimensional sub-analytic set is semianalytic; and (iii) any subanalytic subset of a two-dimensional semianalytic set is semianalytic. x1. Introduction In this paper, K denotes an algebraically closed eld of characteristic zero, complete with respect to the nontrivial ultrametric absolute value jjj: K ! R +. By R denote its valuation ring and by } the maximal ideal of R. Our main result is Theorem 1.1. Readers unfamiliar with the terminology should consult Section 2. (1.1) Theorem. Let V R m be semianalytic, dim V 2, and let S V be subanalytic. Then S is semianalytic. This has an immediate corollary. (1.2) Corollary. Let V R m be an aanoid variety, dim V 2, and let S V be subanalytic. Then S is semianalytic. This paper is a sequel to LR1], where the main result is the special case of Theorem 1.1 when V = R 2. (Also working in characteristic zero, Schoutens Sc2] obtains a result similar to the main result of LR1] for the diierent class of rigid subanalytic sets, called strongly subanalytic, that is described in Sc1]. The strongly subanalytic analogue of Theorem 1.1 is not known.) The results of this paper depend on the results of LR1]. The results from LR1] that are needed are its main result ((LR1], Theorem 1.1) and the careful desingularization given in Section 5 of LR1], in particular LR1], Lemma 5.2 and LR1], Theorem 5.4. This careful desingularization of LR1], Section 5 was carried out only in characteristic zero. Consequently the results of LR1] and the present paper are only proved in characteristic zero. We believe that the careful desingularization of LR1], Section 5 can be carried out in characteristic p. Then, with minor modiications, the arguments of LR1] and the present paper would establish Theorem 1.1 in characteristic p. *Supported in part by the NSF. The second author wishes to thank the CNR for its support and the University of Pisa for its hospitality. The authors also thank the referee for a number of suggestions which considerably improved the paper.