Log canonical thresholds of quasi-ordinary hypersurface singularities (Q2846832)

Given a smooth complex variety \(X\), and a nonzero sheaf of ideals \(\mathcal{I}\) on it with zero locus \(Z\), the log canonical threshold of the pair \((X, Z)\) is an important local invariant of the singularities of \(Z\). In many cases one could assume \(X = \mathbb{C}^{d+1}\), and \(\mathcal{I}\) generated by a non-zero polynomial \(f \in \mathbb{C}[x_1, \dots x_{d+1}]\) with \(f(0) = 0\). Take a log resolution \(\pi: \widetilde{U} \rightarrow U\) of \(Z \cap U\) in a neighborhood \(U\) of 0, with \(E_i\) the irreducible components of \(\pi^{-1}(Z \cap U) = {\bigcup}_{i \in J}E_i\). If \(a_i = \mathrm{ord}_{E_i}(f \circ \pi)\) and \(b_i = \mathrm{ord}_{E_i} \det(\mathrm{Jac}(\pi))\) for all \(i \in J\), the log canonical threshold of \(f\) at 0 is defined as \(\mathrm{lct}_{0}f = \min_{i}\{\frac{b_{i}+1}{a_i}\}\). It is independent of the choice of the log resolution, and the same definition holds for a germ of complex analytic function. It is the reciprocal of Arnold multiplicity, and it evaluates how bad the singularity is. Intuitively, the worse the singularity, the higher the multiplicities \(a_i\) are, so the smaller the log canonical threshold is. It is related with other notions, for example, it could be computed in terms of the jet spaces of \(Z\) [\textit{M. Mustaţă}, J. Am. Math. Soc. 15, No. 3, 599--615 (2002; Zbl 0998.14009)] or, as the negative of the biggest root of Bernstein-Sato polynomial for \(f\). Also, it is the smallest \(\alpha > 0\) such that \(|f|^{-2\alpha}\) is not locally integrable.NEWLINENEWLINEIn the article under review the log canonical threshold is computed for irreducible quasi ordinary hypersurface singularities. A germ \((Z, 0) \subset (\mathbb{C}^{d+1}, 0)\) of complex analytic hypersurface is quasi-ordinary if there is a finite morphism \(h: (Z, 0) \rightarrow (\mathcal{C}^d, 0)\) whose discriminant locus is contained in a simple normal crossings divisor. For such \((Z, 0)\) there exists an embedding in \((\mathcal{C}^{d+1}, 0)\), defined by the zero set \(Z(f)\) of a quasi-ordinary polynomial \(f \in \mathbb{C}\{x_1, \dots x_d \}[y]\). This means that \(f\) is an irreducible Weierstrass polynomial in \(y\) with discriminant \(\Delta_{f} = x^{a}.u\), where \(a \in \mathbb{Z}^d_{\geq 0}\), and \(u \in \mathbb{C}\{x_1, \dots, x_d\}\) is a unit in the ring of convergent power series. Moreover, in terms of coordinates \((x_1, \dots x_d, y)\) the morphisms \(h\) is the projection on the first \(d\) coordinates.NEWLINENEWLINEThe geometry of such hypersurface germ could be described in terms of its characteristic exponents. These are uniquely defined vectors \(\lambda_1 \leq \dots \leq \lambda_s\) in \(\mathbb{Q}^d\), and they determine a nested sequence of characteristic lattices [\textit{Y.-N. Gau}, Mem. Am. Math. Soc. 388, 109--129 (1988; Zbl 0658.14004)]. The main theorem of the article computes \(\mathrm{lct}_{0}(f)\) in terms of the combinatorics of associated characteristic exponents for \(f\). The proof uses earlier result of the last two authors, based on the fact that the biggest pole of the local motivic zeta function \(Z_{\mathrm{mot}, f}(\mathbb{L}^{-s})_0\) is equal to \(-\mathrm{lct}_{0}(f)\) (see, for example, [\textit{L. H. Halle} and \textit{J. Nicaise}, Adv. Math. 227, No. 1, 610--653 (2011; Zbl 1230.11076)]). This result describes explicitly the possible poles of \(Z_{\mathrm{mot}, f}\), and from it and some combinatorial lemmas \(\mathrm{lct}_{0}(f)\) is obtained by discarding some of the candidates. As a corrolary is given a necessary and sufficient condition for the polynomial \(f\) to be log canocical in terms of its (this time unique) characteristic exponent. The calculations are demonstrated on particular examples.