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
 In my long term research project, a proof of the Fujita Conjecture, I made overall progress but there where also some setbacks. In fact, it turned out that in order to apply the local theory of graded rings associated to valuations of Abhyankar type, their filtrations and deformations I have developed in recent years, to the problem of constructing global sections of algebraic line bundles, some small but important modifications where needed. Unfortunately, this consumed most of my time, so that the global theory is still not quite completed, as I had expected last year. On the other hand, the local theory is now much more coherent and it is certainly quite interesting on its own. In particular, it includes a very general version of the Restriction Theorem for multiplier ideals and dual versions of Teissier's inequality on the multiplicity of ideals and Demailly-Ein-Lazarsfeld's Subadditivity Theorem on the multiplier ideal of an effective divisor. The deformation theory includes generalizations of Galligo's Borel-invariance of generic initial ideals, the existence of universal Gröbner bases and many more new results. Those results are crucial to construct good filtrations of certain linear systems on projective varieties and to prove their rigidity. Earlier approaches [1-4] to the problem always relied on very special filtrations which do not have the appropriate strong properties necessary to proof the Fujita Conjecture. But as a drawback with these new filtrations, the minimal center of log-canonical singularities does not anymore always decrease in this process; it can even increase! This poses the new difficulty that the process may not terminate. But rigidity of the filtrations allows one to prove that it indeed terminates. The final result of this process will then be either a filtration with an isolated log-canonical singularity, or a filtration whose centers on the variety have very small degrees. In the first case, it is already well-known that the point under consideration is then not a base point. On the other hand, in the second case, the degrees of the subvarieties are so small, that it leads to an obvious contradiction to the assumptions of the Fujita Conjecture. These exciting new developments will appear in a hopefully forthcoming preprint [5].
  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.