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Monomial resolutions of morphisms of algebraic surfaces. (English) Zbl 1003.14004

From the introduction: Let \(k\) be a perfect field and \(L/K\) be a finite separable field extension of one-dimensional function fields over \(k\). A classical result states that \(K\) (respectively \(L)\) has a unique proper and smooth model \(C\) (respectively \(D)\), and that there is a unique morphism of curves \(f:D\to C\) inducing the field inclusion \(K\subset L\) at the generic points of \(C\) and \(D\). It has the following properties:
(i) \(f\) is a finite morphism.
(ii) \(f\) is monomial on its tamely ramified locus; let \(\beta\in D\) be any point, with \(\alpha:=f(\beta)\in C\), such that the extension of discrete valuation rings \({\mathcal O}_{Y,\beta} /{\mathcal O}_{X,\alpha}\) is tamely ramified. There exists a local-étale ring extension \(R\) of \({\mathcal O}_{Y, \beta}\) and regular parameters \(u\) of \({\mathcal O}_{X,\alpha}\) and \(\overline x\) of \(R\) such that \(u=\overline x^a\) for some \(a\) prime to the characteristic of \(k\).
The authors investigate a two-dimensional version of this statement, that is \(L/K\) is a finite separable field extension of two-dimensional function fields over \(k\). By birational resolution of singularities and elimination of indeterminacies, there exists a proper and smooth model \(X\) (respectively \(Y)\) of \(K\) (respectively \(L)\), together with a morphism \(f:Y\to X\) inducing the field inclusion \(K\subset L\) at the generic points of \(X\) and \(Y\). Such a morphism is in general neither finite nor monomial.
S. Abhyankar [Am. J. Math. 78, 761-790 (1956; Zbl 0073.37902)] raised the question of whether this can be arranged by blowing-up: Does there exist compositions of point blow-ups \(Y'\to Y\) and \(X'\to X\), together with a map \(f':Y'\to X'\) such that \(f'\) is finite and/or monomial? It is actually shown (loc. cit.) that \(f'\) finite cannot in general be achieved. The obstruction is local for the Riemann-Zariski manifold of \(L/k\).
This leaves open the question of whether \(f'\) can be taken to be monomial. For complex surfaces, a positive answer has been given by S. Akbulut and H. King [“Topology of real algebraic sets”, Math. Sci. Res. Inst. Publ. 25 (1992; Zbl 0808.14045)]. Their method however does not generalize to positive characteristic. In general, it is not possible to monomialize an arbitrary morphism in characteristic \(p>0\), even for a morphism of curves. The obstruction to monomialization is the appearence of wild ramification. In the presence of wild ramification, monomialization is possible only in some very special cases.
The authors present a quite general solution to this problem: Any proper, tamely ramified morphism \(f:Y\to X\) of surfaces (which are separated but not necessarily proper over \(k)\),inducing the field inclusion \(K\subset L\) at the generic points of \(X\) and \(Y\), can, after performing suitable compositions of point blow-ups \(Y'\to Y\) and \(X'\to X\), be arranged to a monomial morphism \(f':Y'\to X'\). Moreover, there is a unique minimal such \(f'\).
The authors’ method is constructive. That is, they give an algorithm, which, starting from an arbitrary proper \(f\) as above, produces its associated minimal \(f'\).

MSC:

14E05 Rational and birational maps
12F10 Separable extensions, Galois theory
14H05 Algebraic functions and function fields in algebraic geometry
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References:

[1] Abhyankar S., Amer. J. of Math 78 pp 321– (1956) · Zbl 0074.26301
[2] Abhyankar S., Amer. J. of Math 78 pp 761– (1956) · Zbl 0073.37902
[3] Abhyankar S., Annals of Math. Studies 43 (1959)
[4] Abhyankar S., Annals of Math 63 pp 491– (1956) · Zbl 0108.16803
[5] Abhyankar S., Amer. J. Math 81 (1959) · Zbl 0100.16401
[6] Abramovich D., Weak semistable reduction in characteristic 0 · Zbl 0958.14006
[7] Akbulut S., MSRI publications 25 (1992)
[8] Cutkosky S.D., Astérisque 25 (1992)
[9] Grothendieck A., The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme 208 (1971) · Zbl 0216.33001
[10] Hartshorne R., Graduate Texts in Math 52 (1971)
[11] Orbanz U., Resolution of singularities 1101 (1980)
[12] Kempf G., Toroidal embeddings I 339 (1973) · Zbl 0271.14017
[13] Lipman J., Publ. Math. IHES 36 pp 195– (1969) · Zbl 0181.48903
[14] Lipman J., Annals of Math 107 pp 115– (1978) · Zbl 0349.14004
[15] Milne J.S., Princeton Series in Math 33 (1980)
[16] Grothendieck A., Revêtements étales et groupe fondemental 224 (1971)
[17] Spivakovsky M., Amer. J. of Math 112 pp 107– (1990) · Zbl 0716.13003
[18] Zariski O., Annals of Math 41 pp 852– (1940) · Zbl 0025.21601
[19] Zariski O., Graduate Texts in Math 28 (1958)
[20] Zariski O., The Univ. Series in Higher Math (1960)
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