The Classification Of Local Equations Of Branch Locus Of Rational Triple Points

The main purpose of this paper is to give all local equations of local locus of rational triple point in the meaning of topo-logical equivalence under the triple covers of surfaces. It's well known that there are only nine types of rational triple singu-larities isomorphically [1,26,27] and every rational triple point can be defined by a cubic equation in C3. Moreover, S.-L. Tan gave explicitly a canonical method to resolve a singularity de-fined by a cubic equation in [23]. Z.-J. Chen and R. Du etc. [7] gave an local equation of branch locus for each type of rational triple point and then construnted an explicit cubic equation re-spectively, from which we can equate rational triple points with their local equations of local locus under the triple covers of surfaces. Recently, J. Lu [16] gave some criterions for rational points generated by triple cover and he also gave all possible local equations of branch locus of rational double points and ra-tional triple points of type E6,0, E7,0 and E0,7 in the meaning of topological equivalence. In this paper, we will give all possible local equations of branch locus for the remaining six types.