Some Questions in Algebraic Geometry

Frans Oort \Een zot kan meer vragen \One fool can ask more questions dan tien wijzen kunnen beantwoorden." than ten wise men can answer." Een oud Nederlands spreekwoord. An old Dutch saying. Introduction In June 1995 several mathematicians will gather in Utrecht for a conference on \Arithmetic and geometry of abelian varieties," and this seems a good occasion to share with them some of the questions that have occupied my mind over the years. These pages contain some of these problems. Their choice, approach and presentation do, I am afraid, bear a distinctly personal stamp-my own. Very likely they will bring a smile to the lips of some of the experts present at the conference, and there may even be some raisings of eyebrows. Yet I hope that among you there may be some that are willing to spend some time with these musings, even though there are, as we all realize, more important questions to be studied in the mathematics of today. Most of these ideas presented here I have over the past thirty years shared with colleagues, but the part which each of them has played in shaping my thoughts is in most cases no longer traceable. Only such ideas that I believe to have originated in my own mind and which, moreover, I consider likely to be true, are here labelled as \conjectures". Very likely some of these questions might have been formulated more precisely. I may also not be aware of the fact that some of them have already been solved, or were found to be connected to other problems: references to that eeect I may easily have missed. If you have any comments on the problems as they are presented here, please let me know. NB This is the 1995 version, NOT UPDATED.