In this research project, the investigator studies the uniform nature of constructions from Algebraic Geometry and Commutative Algebra. Uniformity has to be understood here in the following sense. Suppose an algebraic-geometric object, such as a variety or a coherent sheaf, is presented to us by means of some polynomials over an algebraically closed field. From this object other objects or invariants can be derived by applying some algebraic-geometric process. The question posed is now when can this construction be carried out by using only polynomials of degree bounded by the degree of the initial polynomials? In particular, when is an invariant associated to this object uniformly bounded by the initial degrees only? Provided such uniform bounds exist, the following transfer principle can be applied. Encode, allowing negations and quantification, the construction or some of its properties in the field by means of the coefficients of the polynomials involved--the existence of uniform bounds is certainly a necessary, but often even a sufficient condition for this to hold. Then use the Lefschetz Principle to carry over results from positive characteristic to zero characteristic, or, conversely, from zero characteristic to almost all positive characteristics. The investigator has already successfully applied this method to the following non-trivial facts: the Bass Conjecture, the Zariski-Lipman Conjecture and the New Intersection Theorem. In this proposal, he is especially interested in applications to tight closure in characteristic zero. Moreover, he proposes to infer from the existence of uniform bounds, the constructible nature of certain algebraic-geometric constructions. For instance, he seeks to continue the following program originally initiated by Nagata and Grothendieck: for a given geometric property of a point on a variety, when is the set of all points on the variety for which this property holds, constructible?
This program provides an alternative approach to some open problems in the field, as well as a simplified treatment of previous results. The author takes a special interest in the following old problem, simple to state but yet so far resisting all known proof methods: is every curve in three dimensional space obtained by intersecting two surfaces? On a deeper level, the current proposal can be seen as an attempt to analyze and better understand the constructive nature of Algebraic Geometry, by looking at it from a logicians point of view. In this way, the Principal Investigator intends to present the domain in a more coherent and unified way and obtain an improved link between Geometry and Algebra.
