Portal de Periódicos da CAPES

Resolutions of singularities in prime characteristic for almost all primes

1969; American Mathematical Society; Volume: 146; Linguagem: Inglês

10.1090/s0002-9947-1969-0263818-1

1088-6850

Paul C. Eklof,

Polynomial and algebraic computation

0. Introduction.The existence of resolutions of singularities of algebraic varieties defined over fields of characteristic zero has been proved by Hironaka [2] ; for algebraic varieties defined over fields of characteristic />#0, the existence of resolutions is known only for varieties of dimension 2 (for all p) and dimension 3 (for /?#2, 3, 5).In this paper we use Hironaka's Theorem and techniques of ultraproducts to obtain a partial answer to this question, namely a result which (roughly) asserts the existence for all but a finite number of primes of resolutions of singularities of varieties defined over fields of prime characteristic, where the exceptional set of primes depends on certain numerical parameters of the varieties.In particular, we prove the following (where "nonsingular" means "smooth" in Grothendieck's terminology) Theorem A. For any pair (n, d) of positive integers there exists a finite set P0(n, d) of primes, and positive integers t', n', d' such that if X is any projective F-variety satisfying:(1) characteristic of F is not in P0(n, d); and(2) X can be embedded in projective n-space, as a subvariety of degree d; then there exists a finite sequence of monoidal transformations of F-varieties rr, : X¡ + y -> Xh 0 SI < t, such that t S t', X0 = X, Xt is nonsingular, andsatisfying for all I :(3) the center o/w,, say Y¡, is nonsingular; (4) Yi contains no nonsingular points of X¡ ; and (5) Xi can be embedded in projective ri-space as a subvariety of degree S d' (1=0,..., t).Condition (1) is satisfied if Char f=0.In that case, the existence of the resolution is Hironaka's Theorem-which is used in the proof of Theorem A-but we also assert the existence of the bounds t', n', and a".(We thank Abraham Robinson for calling to our attention this application of our methods to characteristic zero.)