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## Helmke, Stefan

E-Mail helmke （emailアドレスには＠kurims.kyoto-u.ac.jp をつけてください）

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My research concentrated on an effective local uniformization theorem for singular algebraic varieties over arbitrary fields. This would be the last step in my approach [5] to the Fujita Conjecture [1--3], or more specifically, an optimal numerical criterion for separation of any set of closed points by a linear system as conjectured in [4]. Unfortunately, my expectations expressed in my previous report about this new method of local uniformizations were a little too optimistic and so far did not lead to the desired result, since one crucial estimate turned out to be more complicated than I had anticipated and only worked in dimension three. However, after recovering from this disappointment, I made some progress on the failed estimate in higher dimensions and I still believe that this will work, even over a field of positive characteristic.
The idea of {\it effective\/} local uniformizations is roughly as follows. Normally, one starts with a valuation centered at a regular local ring and an element of that ring. Then one tries to find a sequence of blow-ups, such that the total transform of the element becomes a monomial at the center of the given valuation. To make this process effective, one needs to construct the sequence of blow-ups from the valuation alone, independently of a specific element of the regular local ring and then estimate the multiplicity of the strict transform of {\it any\/} element of the original ring in terms of its value. For a two dimensional local ring (over a field of characteristic zero), this estimate is essentially achieved by the Newton-Puiseux series. This expresses one of the variables as a fractional power series in the other variable (modulo the equation of the singularity -- which defines a rank two valuation). But in higher dimensions, one only gets a fractional Laurent series in this way, which makes the estimate a lot more elaborate. But if it works, it functions as a substitute for the Zariski decomposition of a divisor on a surface and potentially has many more applications than just the Fujita Conjecture.

1. S. Helmke, On Fujita's conjecture, Duke Math. J. 88 (1997), 201--216.
2. S. Helmke, On global generation of adjoint linear systems, Math. Ann. 313 (1999), 635--652.
3. 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.
4. S. Helmke, Multiplier ideals and basepoint freeness, Oberwolfach reports 1, 2004, 1137--1139.
5. S. Helmke, New Combinatorial Methods in Algebraic Geometry, in preparation.

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