Iterated blowups of two-dimensional regular local rings
For an algebraically closed field \(k\) and \(X_0=P_k^2\), one may consider a sequence \(X_0\xleftarrow{\pi_1}X_1\xleftarrow{\pi_2}X_2\xleftarrow{\pi_3}\dots\) where each \(\pi_i\) is a quadratic transform (blowup) at an \(x_i\in X_i\), with \(\pi_i(x_i)=x_{i-1}\), for every \(i\geq 1\). The local domains \(R_i=\mathcal O_{X_i,x_i}\) and their upward directed union \(V=\bigcup R_i\) have \(k(x,y)\) as their quotient field, and by an old result of \textit{S. Abhyankar} [Am. J. Math. 78, 321--348 (1956; Zbl 0074.26301)] (see also [\textit{R. Hartshorne}, Algebraic geometry. New York-Heidelberg-Berlin: Springer-Verlag (1977; Zbl 0367.14001)]), \(V\) is a valuation domain. The authors are focusing on the value group of \(V\). Two main results are as follows: Theorem 3.3. Let \(\mathbf a =[a_0, a_1, a_2,\dots\)] be a sequence of positive integers. Then the value group \(\Gamma_{\mathbf a}\) is order isomorphic to \(\mathbb{Z}\oplus\mathbb{Z}_\gamma\subseteq\mathbb{R}\), where the irrational number \(\gamma\) is the value of the infinite continued fraction \(a_0+1/(a_1+1/(a_2+1/(a_3\dots\). Theorem 3.4. For every positive irrational number \(\gamma\), there is a sequence \(\mathbf a\) of positive integers such that \(\Gamma(\mathbf a)\) is order-isomorphic to \(\mathbb{Z}\oplus\mathbb{Z}_\gamma\).
scientific article; zbMATH DE number 7367422
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