A. NOBILEIntuitively, in the Nash blowing-up process each singular point of an algebraic (or analytic) variety is replaced by the limiting positions of tangent spaces (at non-singular points). The following properties of this process are shown: 1) It is, locally, a monoidal transform; 2) in characteristic zero, the process is trivial if and only if the variety is nonsingular. Examples show that this is not true in characteristic p >0; that, in general, the transform of a hypersurface is not locally a hypersurface; and that this process does not give, in general, minimal resolutions.

IntroductionBy now, the theory of desingularization of varieties in characteristic zero seems to be pretty well understood. The ®rst general result on resolution, valid for any dimension, is the famous theorem of Hironaka, published in 1964 [15]. This is a precise result, showing not only that desingularization is possible, but also that given a variety, or more generally an excellent scheme X over a ®eld of characteristic zero, it is possible, by means of a ®nite sequence of blowing-ups such that each one has as center a regular subscheme, to obtain a birational, projective morphism f : X H 3 X, with X H regular. Moreover, one obtains`embedded desingularization'. This means the following: if X is a subscheme of a regular scheme W, a situation can be reached where X H is a subscheme of a regular W H , and f is induced by a projective morphism g: W H 3 W, inducing an isomorphism from W H À g À1 S onto W À S, where S is the singular set of X, in such a way that X H has normal crossings with the exceptional divisor of g (see (1.1)(b) of this article for the precise de®nition). A possiblè shortcoming' of this important result is the fact that, although the morphism f (or g, in the embedded situation) is a composition of`nice' blowing-ups, the method of choosing each center is not speci®ed. More recent work shows how to obtain more constructive proofs. We refer here to [6], [10] and [28] for different algorithms of desingularization. A more self-contained presentation of the latter appears in [11] and [29]. These algorithms tell us, given a subvariety X of a regular variety W (over a ®eld of characteristic zero), how to choose the different centers in order to get embedded desingularization, by repeatedly blowing-up along the centers provided by the algorithm.We mention that the algorithm treated in [11] and [29] can certainly bè implemented'. In fact, in [7] a number of explicit examples (involving surfaces in three-space) are resolved by using that algorithm with the aid of computers (see also [8]).The development in [11] also led to a short and simpli®ed proof of desingularization avoiding Hironaka's notion of normal¯atness (see [12] and the addendum in [11]).We should mention other recent proofs (short, but non-constructive) obtained using very different techniques, namely the use of the theory of moduli of curves. See [17,5,9,4,3].Once the problem of resolving, in an explicit (or constructive) way, a single algebraic variety has been settled, it is natural to consider the question of `classi®cation' of varieties according to the resolution of their singularities. A closely related question is the study of criteria saying when a family of varieties can be simultaneously resolved, in an appropriate sense. The present article is concerned with this type of question.Simple examples indicate that some restrictions have to be imposed on the family. It seems that the ®rst systematic, rigorous analysis of the problem, in the case of families of curves, is due to Zariski, as part of his program of studying equi-singulari...

A theory of simultaneous resolution of singularities for families of embedded varieties (over a field of characteristic zero) parametrized by the spectrum of a suitable artinian ring, and compatible with a given algorithm of resolution, is presented. As usually, this a simple consequence of a similar theory for analogous families of basic objects, to which the main portion of this article is devoted.

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Abstract.The following is proved: Given a family of projective reduced curves X -T ( T irreducible), if X, (the general curve) is integral and X0 is a special curve (having irreducible components Y,.Xr), then Z'=-, g,(Xf) « g(Y,), where g(Z) = geometric genus of Z. Conversely, if A is a reduced plane projective curve, of degree n with irreducible components Y,.Xr, and g satisfies 2'= | g,(X,) *£ g 'S .¡(n -1)(« -2), then a family of plane curves Y -7" (with 7" integral) exists, where for some tn G 7", Y, = Z and for r generic, X, is integral and has only nodes as singularities. Results of this type appear in an old paper by G. Albanese, but the exposition is rather obscure.