Lines on Brieskorn-Pham surfaces
Motivated by the problem posed by \textit{J. F. Nash jun.} in 1968 [see Duke Math. J. 81, 31-38 (1995; Zbl 0880.14010)] in order to study the space of arcs in an algebraic or analytic variety over \(\mathbb C\), and since the problem had already been solved for minimal surface singularities by \textit{A. J. Reguera} [Manuscr. Math. 88, 321-333 (1995; Zbl 0867.14012)], \textit{G. Gonzalez-Springberg} and \textit{M. Lejeune-Jalabert} proposed a paper [Ann. Pol. Math. 67, 179-190 (1997; Zbl 0894.14017)] trying to understand how the property of the existence of smooth curves in the minimal surface singularities helps to solve the problem. They characterized the existence of smooth curves in a surface singularity \((X,0)\) by the fact that the maximal cycle \({\mathcal Z}_X\) for its minimal desingularization \(\pi: {\widetilde X} \rightarrow X\) (i.e. the underlying cycle of the ideal sheaf \({\mathfrak m} {\mathcal O}_{\widetilde X}\), where \(\mathfrak m\) is the maximal ideal of \({\mathcal O}_{X,0}\)) has at least one reduced component. They also gave a natural partition of the set \(\mathcal L\) of smooth arcs in \((X,0)\) in families \({\mathcal L}_E\), each \({\mathcal L}_E\) defined by a reduced component of \({\mathcal Z}_E\). And they showed that there is no wedge centered at smooth curves of two different families. In the paper under review, the authors study smooth curves on the Brieskorn-Pham type surfaces \(G(p,q,r)\), i.e. defined in \(\mathbb C^3\) by \(x^p+y^q+z^r=0\), for some positive integers \(p \leq q \leq r\). The minimal desingularization of such a surface can be obtained by making a regular subdivision of the Newton fan defined by its equation. Using this, the authors characterize the existence of smooth curves in \(G(p,q,r)\) as follows: If \(s=\text{gcd}(p,q,r)\) then, either \(r> \frac {pq} {\text{gcd}(p,q)}\), or else two of the three integers \(\frac ps\), \(\frac qs\), \(\frac rs\) are coprime and the other is divisible by at least one of the coprime numbers. The authors also compute the number of reduced components in \({\mathcal Z}_X\), i.e. the number of families of smooth curves, and the equations of the curves in one family.
scientific article; zbMATH DE number 1547125