A skeleton key to Abhyankar's proof of embedded resolution of characteristic P surfaces (Q428171)

This paper gives a nice exposition of, and simplifies, the key aspects of Abhyankar's proof of embedded resolution of surfaces in positive characteristic. Note that Abhyankar's original proof is obtained by combining the results of 5 sources, see [\textit{S. S. Abhyankar}, Math. Ann. 153, 81--96 (1964; Zbl 0121.37901); Wiss. Abh. Arbeitsgemeinschaft Nordrhein-Westfalen 33, Festschr. Gedächtnisfeier Karl Weierstrass, 243--317 (1966; Zbl 0144.03104); Math. Ann. 170, 87--144 (1967; Zbl 0158.04101); Ann. Mat. Pura Appl., IV. Ser. 71, 25--59 (1966; Zbl 0158.04201)] and [Resolution of singularities of embedded algebraic surfaces. New York-London: Academic Press (1966; Zbl 0147.20504)]. Thus, the fact that this paper is mostly self contained and not too long, is quite valuable for researchers working in this field. The first section also gives a clean outline of the proof strategy and some discussion of the main difficulties.