Maximal contact in positive characteristic and small multiplicity. (Q1814227)

The author proves the following theorem: Fix an infinite base field \(k\) of characteristic \(p >0\), let \(X\) be a subscheme of the regular \(k\)- scheme \(Z\), and let \(x\in X\) be a closed point such that \(X\) has multiplicity \(e(x)<p\) at \(x\). Then there exists, in a neighborhood of \(x\), a regular subscheme \(W\) of \(Z\) which has maximal contact with \(X\) in \(x\) and which has dimension \(\leq\dim X\). \textit{J. Giraud} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 8, 201--234 (1975; Zbl 0306.14004)] had proved a similar result, showing the existence of a subscheme \(W'\) as above, but which is not in general regular. In the present paper it is shown that Giraud's \(W'\) is contained in a regular \(W\) with the same property. Using a result of \textit{S. S. Abhyankar} [``Resolution of singularities of embedded algebraic surfaces'', Pure Appl. Math. 24. New York etc.: Academic Press (1966; Zbl 0147.20504)] it follows that every projective 3-dimensional variety (over a field of positive characteristic \(p\)) is birationally equivalent to one which can be desingularized.