Jet schemes of complex plane branches and equisingularity
The \(m^{th}\)-jet scheme \(X_m\) of an algebraic variety \(X\) over an algebraically closed field \(k\) is a \(k\)-scheme of finite type which parametrizes morphisms \(\mathrm{Spec}(k[t]/t^{m+1})\to X\). Jet schemes have attracted, from various viewpoints, the attention of many authors such as \textit{J. F. Nash, jun.} [Duke Math. J. 81, No.1, 31--38 (1995; Zbl 0880.14010)] and, more recently, \textit{M. Mustată} [Invent. Math. 145, No. 3, 397--424 (2001; Zbl 1091.14004)], with \textit{L. Ein} and \textit{R. Lazarsfeld} [Compos. Math. 140, No. 5, 1229--1244 (2004; Zbl 1060.14004)].NEWLINENEWLINEIn the present paper the author considers a curve \(C\) in the complex plane, with a singularity at \(O\) at which it is analytically irreducible (i.e. the formal neighborhood \((C,O)\) of \(C\) at \(O\) is a branch). He determines the irreducible components of the \(m\)-Jet scheme of \((C,O)\) and shows that their number is not bounded as \(m\) grows. He also gives formulas for their number and for their codimension, in terms of \(m\) and of the generators of the semigroup of \((C,O)\).NEWLINENEWLINEAs it is well known, the semigroup of the branch \((C,O)\) and the topological type of \(C\) near \(O\) are equivalent data and characterize the equisingularity class of \((C,O)\), as defined by Zariski. The author shows in particular that the structure of the Jet Scheme determines the topological type of \(C\) and conversely.
scientific article; zbMATH DE number 6043080