## Helmke, Stefan

**Helmke, Stefan**

In my last report I stated the view that with the help of
a certain effective version of the local uniformization theorem
which I could prove in~[6], I could quickly finish my proof
of Fujita's Conjecture. As it turned out, this view was slightly
too optimistic. In fact, while my method indeed proved the
kind of result I mentioned in that report, for the applications
I had in mind, this was not quite enough. One also needs good
upper bounds for the number of blow-ups the algorithm uses
to obtain such an effective local uniformization, and that
was rather hopeless with my method, so that I decided to
postpone the publication of those results. However, recently I was
able to considerably improve the method, so that it seems to
work now. In what follows, I like to briefly outline the crucial
ideas of this new method.

We start, quite generally, with a regular local ring~$R$ together
with a regular system of parameters $x_1,\dots,x_d$. Suppose~$v$
is a valuation of the quotient field of~$R$, dominating~$R$. We
want to find an infinite sequence of blow-us of~$R$, such that
eventually every element of~$R$ would become a monomial at some stage
of that process, and in a somewhat effective way. For simplicity,
we may assume here that the residue field of~$v$ coincides with
that of~$R$. In that case, the most elementary singularities we
need to uniformize are binomials of the form $m_1-u\cdot m_2$,
where~$m_1$ and~$m_2$ are two different monomials, i.e.~products
of powers of the coordinate functions~$x_i$, and~$u$ is a unit
in~$R$. Without the unit, such binomials are easy to uniformize
with a purely combinatorial method and if~$R$ contains the field
of rational numbers, the unit can be essentially absorbed in one
of the coordinate functions, so that it is not a cause of
trouble here. Aside, over a field of positive characteristics,
or in the case of mixed characteristics, this unit is indeed the
sole cause of trouble, but unfortunately there seems to be no
easy way to dispense of this unit from the expression. But in
characteristic zero the only problem now is that, when iterating
this process, some other binomials transform into functions
of the form $m_1-u\cdot m_2$ with $m_1=m_2$, which apparently
looks like a rather arbitrary function! But a careful analysis
shows that the extra singularities introduced by this process
can be surprisingly easily dealt with, and in an effective way.
In this way then, one uniformizes one binomial after another
in order of their value with respect to the valuation~$v$,
each time producing new binomials, and shows that this
value tends to infinity, which in turn shows that any element
of~$R$ with value in the minimal isolated subgroup of the
valuation group is uniformized by this process. This argument
contrasts to all previously known arguments by the fact that
it totally avoids the use of any kind of multiplicity in order
to show that it uniformizes a given function, which then makes it
much easier to prove its effectiveness.

- S. Helmke, On Fujita's conjecture, Duke Math. J. 88 (1997), 201--216.
- S. Helmke, On global generation of adjoint linear systems, Math. Ann. 313 (1999), 635--652.
- S. Helmke, The base point free theorem and the Fujita conjecture, Vanishing theorems and effective results in algebraic geometry, ICTP Lecture Notes 6, Trieste, 2001, 215--248.
- S. Helmke, Multiplier ideals and basepoint freeness, Oberwolfach reports 1, 2004, 1137--1139.
- S. Helmke, New Combinatorial Methods in Algebraic Geometry, in preparation.
- S. Helmke, On local uniformizations, in preparation.