instituto de matemáticas universidad de sevilla
Antonio de Castro Brzezicki
Teissier's conjecture on extending a valuation centered in a local domain to its formal completion
Seminario de Álgebra

Let $(R,m,k)$ be a local noetherian domain with field of fractions $K$, let $$ \nu:K^*\twoheadrightarrow\Gamma $$ be a valuation centered at $R$ and let $R_\nu$ be the corresponding valuation ring of $K$, dominating $R$. Denote by $\widehat R$ the $m$-adic completion of $R$. In the applications of valuation theory to commutative algebra and the study of singularities, one is often induced to replace $R$ by its $m$-adic completion $\widehat R$ and $\nu$ by a suitable extension $\widehat\nu_-$ to $\frac{\widehat R}P$ for a suitably chosen prime ideal $P$, such that $P\cap R=(0)$.

In [1] we gave a systematic description of all such extensions$\widehat\nu_-$ and defined the notion of {\bf tight} extensions, which are of particular interest for applications. One of the key properties of a tight extension $\widehat\nu_-$ is that its value group equals $\Gamma$ and its graded algebra is birational to that of $\nu$. The existence of such extensions (known as Teissier's conjecture and dating to the early nineteen ninetiies) is a crucial step in at least two recent approaches to local uniformization in positive characteristic. The subject of this talk is our (very recent) progress on Teissier's conjecture.

[1]: Extending a valuation centered in a local domain to its formal completion. J. Herrera, M. A. Olalla, M. Spivakovsky, B. Teissier. Proceedings of the London Mathematical Society. Vol (3) 105, 2012, pp. 571--621