## ½ê°÷ -Helmke, Stefan-

Ì¾Á° Helmke, Stefan
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¡¡As I had planned for the preceding academic year and explained in my last report, I worked on the final details of a possible proof of Fujita's Conjecture on global generation of adjoint linear systems [1--4]. I could solve all the remaining problems except the following. In order to construct a global section of an algebraic line bundle $\cal L$ on a projective variety X which is not vanishing at a closed point $P\in X$, one only needs to find a global section of the line bundle ${\cal L}^n\otimes\omega_X^{-n}$ which is singular enough at P, but not too singular in $X\setminus\{P\}$, where $\omega_X$ denotes the canonical sheaf and n is a positive integer. This idea goes back to the Italian school of algebraic geometry around 1940, where it was used in the case when X is an algebraic surface of general type, ${\cal L}=\omega_X$ and n is a very specific small integer. Then, 25 years later the techniques where considerably improved with the help of cohomology theory and in particular Kodaira's Vanishing Theorem. In the 1990's there were strong efforts to generalize these results to higher dimensions. But in contrast to the two dimensional case, the integer n could now be arbitrary large. For the procedures used at that time, this caused no essential problem since one could always increase n as desired, but the results obtained in this way were not satisfying. However, with my new techniques increasing n is impossible since then the procedure may never come to an end. Therefore, one needs an a priori estimate for n. But unfortunately, the technique I designated to this problem and which was successful for the corresponding local problem failed in the global situation. However, during this rather time consuming course or research, I developed another technique, which is more promising. In fact, at least in low dimensions the argument already works and there is good experimental evidence that this will also work in all dimensions. So the new plan is now to spend some more time on these new techniques and in case this would fail again, then to publish the already obtained results[5] independently, without 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.