Existence of dicritical divisors
Let \(R\) be a two-dimensional regular local ring with field of quotients \(L\), and let \(D(R)^\Delta\) be the set of all prime divisors of \(R\). Let \(z\in L\); the subset \(\mathfrak D(R,z)\) of \(D(R)^\Delta\) consisting of all \(V\in D(R)^\Delta\) such that \(z\bmod{ M(V)}\in V/M(V)\) is transcendental over \(R/M(R)\) is called the set of dicritical divisors of \(z\) in \(R\). (The analytical (topological) concept of dicritical divisors was developed by several authors (see the first line of the introduction of the paper under review); in [\textit{S. S. Abhyankar}, Inversion and Invariance of Characteristic Terms: Part I. The Legacy of Alladi Ramakrishnan in the Mathematical Sciences. New York, NY: Springer. (2010; Zbl 1200.00039) pp. 93--168] Abhyankar algebracized this concept). For a finite subset \(U\) of \(D(R)^\Delta\), let \(\mathfrak D^*(R,U)\) be the set of all \(z\in L\) such that \(\mathfrak D(R,z)=U\). The main results of this paper are the existence theorem ET and the existence question EQ:NEWLINENEWLINEET: Given a finite subset \(U\) of \(D(R)^\Delta\), there exists \(z\in L^\times\) such that \(U=\mathfrak D(R,z)\). In case the residue field \(R/M(R)\) is infinite, the element \(z\) can be chosen in such a way that it satisfies additional properties.NEWLINENEWLINEEQ: Can you describe the set \(\mathfrak D^*(R,U)\)?NEWLINENEWLINEThe authors give a sketch of the proof of ET in section 3.1; a complete proof of ET is given in section 9 where also EQ is answered.NEWLINENEWLINEThe authors use the theory of complete ideals for two-dimensional regular local rings as developed by Zariski in Appendix 5 of [\textit{O. Zariski} and \textit{P. Samuel}, Commutative algebra. Vol. II. (The University Series in Higher Mathematics.) Princeton, N.J.-Toronto- London-New York: D. Van Nostrand Company, Inc. (1960; Zbl 0121.27801)]; another source for this theory is the book [\textit{I. Swanson} and \textit{C. Huneke}, Integral closure of ideals, rings, and modules. London Mathematical Society Lecture Note Series 336. Cambridge: Cambridge University Press. (2006; Zbl 1117.13001)] of Swanson and Huneke. In reading this article, one should have at hand the book [\textit{S. S. Abhyankar}, Lectures on algebra. Vol. 1. Hackensack, NJ: World Scientific. (2006; Zbl 1121.13001)].
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