所員 -Helmke, Stefan-

名前 Helmke, Stefan
助教
E-Mail helmke (emailアドレスには@kurims.kyoto-u.ac.jp をつけてください)
U R L
研究内容 Algebraic Geometry
紹 介
 Following my plan to take care of the final steps of my proof of Fujita's Conjecture, I tried to find an effective bound for the number of blow-ups needed to locally uniformize certain log-canonical singularities. This is crucial in order to show that my previously found algorithm to construct special flags of subvarieties with very low degree of a given variety (in case the linear system fails to separate a given set of points) terminates. Unfortunately, Hironaka's approach to resolutions of singularities turned out to be inappropriate for this purpose. Instead, I developed a kind of multi-dimensional Newton-Puiseux expansion. That worked! Already in 1940, O. Zariski proved the existence of local uniformizations of singularities over a field of characteristic zero. His proof also generalizes Newton's ideas to higher dimensions. However, the decisive estimate in Zariski's proof is quite ineffective. On the other hand, especially for plane curve singularities, the algorithm is known to be equivalent to a sequence of euclidean algorithms, which are actually very effective, even though this is not clear from Zariski's argument. In my new approach, this estimate is now replaced by a more combinatorial method, very much in the spirit of the whole proof of point separation, which makes it indeed effective and brings the long search for a proof of Fujita's Conjecture started in [1-3] and even the much more general conjecture stated in [4] to its conclusion, but it is still in the process of writing [5]. Incidentally, there is some hope that my new approach to local uniformizations of singularities could also work over a field of positive characteristic, since it was exactly this particular argument from Zariski's proof which prevented this so far and which is now replaced by a method which is apparently independent of the characteristic of the field.
  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.