Recent zbMATH articles in MSC 14H20https://www.zbmath.org/atom/cc/14H202021-04-16T16:22:00+00:00WerkzeugIntroduction to Lipschitz geometry of singularities. Lecture notes of the international school on singularity theory and Lipschitz geometry, Cuernavaca, Mexico, June 11--22, 2018.https://www.zbmath.org/1456.580022021-04-16T16:22:00+00:00"Neumann, Walter (ed.)"https://www.zbmath.org/authors/?q=ai:neumann.walter-d"Pichon, Anne (ed.)"https://www.zbmath.org/authors/?q=ai:pichon.annePublisher's description: This book presents a broad overview of the important recent progress which led to the emergence of new ideas in Lipschitz geometry and singularities, and started to build bridges to several major areas of singularity theory. Providing all the necessary background in a series of introductory lectures, it also contains Pham and Teissier's previously unpublished pioneering work on the Lipschitz classification of germs of plane complex algebraic curves.
While a real or complex algebraic variety is topologically locally conical, it is in general not metrically conical; there are parts of its link with non-trivial topology which shrink faster than linearly when approaching the special point. The essence of the Lipschitz geometry of singularities is captured by the problem of building classifications of the germs up to local bi-Lipschitz homeomorphism. The Lipschitz geometry of a singular space germ is then its equivalence class in this category.
The book is aimed at graduate students and researchers from other fields of geometry who are interested in studying the multiple open questions offered by this new subject.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Aguilar-Cabrera, Haydée; Cisneros-Molina, José Luis}, Geometric viewpoint of Milnor's fibration theorem, 1-43 [Zbl 1457.32081]
\textit{Snoussi, Jawad}, A quick trip into local singularities of complex curves and surfaces, 45-71 [Zbl 1457.32074]
\textit{Neumann, Walter D.}, 3-manifolds and links of singularities, 73-86 [Zbl 1457.32071]
\textit{Trotman, David}, Stratifications, equisingularity and triangulation, 87-110 [Zbl 1457.32017]
\textit{Ruas, Maria Aparecida Soares}, Basics on Lipschitz geometry, 111-155 [Zbl 1457.32016]
\textit{Birbrair, Lev; Gabrielov, Andrei}, Surface singularities in \(\mathbb{R}^4\) : first steps towards Lipschitz knot theory, 157-166 [Zbl 07303684]
\textit{Pichon, Anne}, An introduction to Lipschitz geometry of complex singularities, 167-216 [Zbl 1457.32073]
\textit{Giles Flores, Arturo; da Silva, Otoniel Nogueira; Teissier, Bernard}, The biLipschitz geometry of complex curves: an algebraic approach, 217-271 [Zbl 1457.32069]
\textit{Popescu-Pampu, Patrick}, Ultrametrics and surface singularities, 273-308 [Zbl 1457.32079]
\textit{Pham, Frédéric; Teissier, Bernard}, Lipschitz fractions of a complex analytic algebra and Zariski saturation, 309-337 [Zbl 1457.32072]Enumerative geometry of plane curves.https://www.zbmath.org/1456.140692021-04-16T16:22:00+00:00"Caporaso, Lucia"https://www.zbmath.org/authors/?q=ai:caporaso.luciaThe author gives a nice account of some results in the enumerative geometry of plane curves. After explaining how curves \(C \subset \mathbb P^2\) of degree \(d\) are parametrized by projective space \(P_d = \mathbb P^{c_d}\) with \(c_d = d(d+3)/2\) via their coefficients,
she gives a very readable proof of the fact that the locus \(S_d \subset P_d\) corresponding to singular curves is a hypersurface of degree \(3(d-1)^2\) in the second section. Then she moves on to consider the more refined \textit{Severi variety} \(S_{d,\delta} \subset P_d\) parametrizing degree \(d\) curves with at least \(\delta\) nodes, explaining why \(S_{d, \delta}\) is empty for \(\delta > \binom{d-1}{2}\). For \(\delta \leq \binom{d-1}{2}\), \textit{J. Harris} [Invent. Math. 84, 445--461 (1986; Zbl 0596.14017)] showed that \(S_{d, \delta}\) is irreducible of dimension \(c_d - \delta\) and the general curve in this family has geometric genus \(g=\binom{d-1}{2} - \delta\). Next the author considers the general problem of computing the number of irreducible plane curves of degree \(d\) and genus \(g\) passing through \(3d+g-1\) general points, explaining the complicated recursive formula of \textit{M. Kontsevich} in the case \(g=0\) [Prog. Math. 129, 335--368 (1995; Zbl 0885.14028)], which originally came from physical considerations as a condition characterizing the associativity of the quantum product on quantum cohomology. For curves of genus \(g>0\), she gives a few examples, but the full story is explained in work of \textit{L. Caporaso} and \textit{J. Harris} [Invent. Math. 131, No. 2, 345--392 (1998; Zbl 0934.14040)].
Reviewer: Scott Nollet (Fort Worth)