Lecture Announcement
Date: Tuesday, 5 August 2025
Time: 3:30 PM
Venue: Seminar Hall
Local monomialization and stable forms of mappings in characteristic zero and p
Dale Cutkosky
University of Missouri, United States.
05-08-25

---

Abstract

Let $\psi:X\rightarrow Y$ be a dominant morphism of nonsingular varieties over a field $l$ and let $v$ be a valuation of the function field of $X$. The problem of local monomialization is to find sequences of monoidal transforms (blowups of nonsingular subvarieties) $X_1\rightarrow X$ and $Y_1\rightarrow Y$ so that there is an induced morphism $\psi_1:X_1\rightarrow Y_1$ such that if $p_1$ is the center of $v$ on $X_1$ and $q_1$ is the center of $v$ on $Y_1$, then $\psi_1$ has a monomial form at $p_1$. To state this more explicitly, if $\dim X=m$ and $\dim Y=n$, there exist regular parameters $w_1,\ldots,w_n$ in $\mathcal O_{Y_1,q_1}$ and $z_1,\ldots,z_m$ in $\mathcal O_{X_1,p_1}$ and units $u_i$ in $\mathcal O_{X,p_1}$ such that $w_i=u_i\prod_jz_j^{a_{ij}}$ for $1\le i\le n$, where $(a_{ij})$ has rank $n$.

We discuss some ideas from our proof of local monomialization over fields of characteristic zero. We also discuss our counterexample to local monomialization in positive characteristic, which appears already in dimension 2. We discuss stable forms which can be in positive dimension on surfaces, and the role of defect in local mononialization and the stable forms.