Simultaneous algorithmic resolution of singularities (Q357834)

The problem of resolution of singularities in characteristic 0 was solved by Hironata at the beginning of the sixties of the last century. Much later algorithmic methods have been developed for solving this problem. The study of algorithmic equiresolution started about 15 years ago. Some basic results in this direction can be found in the article of \textit{S. Encinas, A. Nobile} and \textit{O. E. Villamayor} [Proc. Lond. Math. Soc., III. Ser. 86, No. 3, 607--648 (2003; Zbl 1076.14020)]. Here families of ideals or of embedded schemes, parametrized by smooth varieties are studied. The equiresolution proposed required that the centers for the transformations are smooth over the parameter variety (condition AE) or required the local constancy of a certain invariant associated to each fibre (condition \(\tau\)). In this paper a definition of equiresolution is proposed for families parametrized by not necessarily reduced schemes, called condition E. Other approaches are proposed, condition A, C and F. Condition A corresponds to AE mentioned above. The main objective of the paper is to prove that when the parameter space is regular all these conditions are quivalent. Assuming the properness of certain projections it is proved that they are also equivalent to \(\tau\) mentioned above.