Time: Saturday, January 7, from 2:30pm to 4:30pm (Hanoi time)

Abstract: Classical Nash blowups of algebraic varieties are parameter spaces of tangent spaces of smooth points and their limits, and it is natural to ask whether the iteration of Nash blowups leads to smooth varieties. Although this is not true in general, at least it holds in dimension 1, namely, any curve in characteristic zero can be desingularized by its n-th Nash blowup with n large enough. Based on Oneto and Zatini’s work, Yasuda showed that the n-th Nash blowup of a reduced Noetherian scheme can be recovered by the Nash blowup associated to a certain coherent sheaf. Inspired by this, as well as by an approach of Duarte for hypersurfaces, we define the n-th Nash blowup of a regular function, and we prove that the n-th Nash blowup of a regular function f can be read off by the n-th order Jacobian matrix of f. Furthermore, we prove that the second order Jacobian matrix of f completely determines the motivic zeta function of f. In fact, we are also developing these topics toward formal schemes (higher-order Jacobian matrices, higher Nash blowups of formal schemes) and obtaining their interesting applications, but this talk will discuss only on regular functions.