Quadratic transforms inside their generic incarnations (Q2846836)

Let \(R\) be a two-dimensional regular local ring with maximal ideal \(M(R)=(x,y)R\), field of quotients \(L\), residue field \(K\), and let \(\kappa\subset R\) be a coefficient set. Let \(\kappa[X,Y]^{\mathrm{hm}}\) be the set of homogeneous, \(Y\)-monic polynomials in \(R[X,Y]\) having coefficients in \(\kappa\), and let \(\kappa[X,Y]^{\mathrm{hmi}}\) be the subset of those \(\lambda\in \kappa[X,Y]^{\mathrm{hm}}\) having an irreducible image in \(K[X,Y]\), the associated graded ring of \(R\). Let \(Q_1(R)\) be the set of first quadratic transforms of \(R\), and let \(\delta: \kappa[X,Y]^{\mathrm{hmi}}\to Q_1(R)\) be the well-known tangential bijection (cf.\ e.g., [\textit{S. S. Abhyankar}, Proc. Am. Math. Soc. 139, No. 9, 3067--3082 (2011; Zbl 1227.14004)]). For every \(T\in Q_1(R)\) and non-zero ideal \(I\) in \(R\) let \((R,T)(I)\) be the transform of \(I\) in \(T\), and let \(\mathfrak Q_1(R,I)\) resp.\ \(\mathfrak P_1(R,I)\) be the set of \(T\in Q_1(R)\) such that \((R,T)(I)\) is not a principal ideal of \(T\) resp.\ \((R,T)(I)\neq T\).NEWLINENEWLINELet \(L(t)\) be the rational function field in an indeterminate \(t\) over \(L\); for every subring \(S\) of \(L\) let \(S^t=S(t)\) be the \(t\)-extension of \(S\): it is the localization of \(S[t]\) with respect to the multiplicatively closed set consisting of all polynomials in \(S[t]\) whose coefficients generate the unit ideal of \(S\). The \(t\)-extension \(R^t\) of \(R\) is a two-dimensional regular local ring with field of quotients \(L(t)\), maximal ideal \(M(R^t)=M(R)R^t\) and residue field \(K(\tau)\) where \(\tau=t\bmod {M(R^t)}\) is transcendental over \(K\). The set \(\kappa(t)\), consisting of all members of \(L(t)\) which, when written in reduced form with monic denominator, has for its numerator and denominator polynomial expressions in \(t\) with coefficients in \(\kappa\), is a coefficient set for \(R^t\), and \(\kappa[X,Y]^{\mathrm{hmi}}\subset\kappa(t)[X,Y]^{\mathrm{hmi}}\). Let \(\delta^t: \kappa(t)[X,Y]^{\mathrm{hmi}}\to Q_1(R^t)\) be the tangential bijection. Let \(T\in Q_1(R)\) and \(\lambda\in \kappa[X,Y]^{\mathrm{hmi}}\) with \(\delta(\lambda)=T\); then \(\delta^t(\lambda)=T^t\in Q_1(R^t)\).NEWLINENEWLINELet \(F,G\in M(R)\setminus\{0\}\) generate an \(M(R)\)-primary ideal \(J\) in \(R\), and set \(\Phi=F+tG\in M(R^t)\). The aim of this paper is to compare \(\mathfrak Q_1(R,J)\) and \(\mathfrak P_1(R^t,\Phi)\). The main result is Theorem (4.6); we state some of the results. Let \(R^*_1,\ldots,R^*_{h^*}\) be the distinct members of \(\mathfrak Q_1(R,J)\), let \(\lambda^*_1,\ldots\lambda^*_{h^*}\in \kappa[X,Y]^{\mathrm{hmi}}\) with \(\delta(\lambda^*_i)=R^*_i\) for \(i\in\{1,\ldots,h^*\}\), and set \(R'_i =(R^*_i)^t=\delta^t(\lambda_i^*)\). Then \(R'_1,\ldots,R'_{h^*}\) are exactly all those distinct members of \(\mathfrak P_1(R^t,\Phi)\) which are of the form \(\delta^t(\lambda)\) for some \(\lambda\in \kappa[X,Y]^{\mathrm{hmi}}\), and \(\lambda^*_1,\ldots,\lambda^*_{h^*}\) are exactly all the distinct tangential directions of \(\Phi\) in \(\kappa[X,Y]^{\mathrm{hmi}}\).NEWLINENEWLINERemarks: (1) In the paper [\textit{S. S. Abhyankar}, Rev. Mat. Complut. 26, No. 2, 735--752 (2013; Zbl 1334.14004)], Abhyankar gives in Lemma 2.1 a detailed proof of (4.1) of the paper under review, in Theorem 2.9 the relation to the dicritical divisors of the pencil generated by \(F\) and \(G\) mentioned at the end of p.\ 4111, and in Theorem 3.2 the result of squarefreeness mentioned in (4.7).NEWLINENEWLINE(2) Typos: On p.\ 4122, formula \((1\Phi)\), replace \(R\) with \(R^t\). On p.\ 4124, second line of Calculation (4.3), replace [Ab7] with [Ab11].