The investigator plans to adapt his simplified algorithm of Hironaka's resolution of singularities to different types of resolution problems. The Hironaka theorem on resolution of singularities plays a central role in Algebraic Geometry. It constitutes a basis of proofs of many theorems in Algebraic and Analytic Geometry. The theorem on desingularization is proven in characteristic zero and is known only in some special cases in positive characteristic. This is a major drawback for many theorems which rely on desingularization and which can be proven in characteristic zero only. The project deals with studying some new ideas and developing a new relatively simple approach to the resolution problem in positive characteristic. By suitable modification of the basic notions (like order, marked ideals and etc) introduced in the algorithm in characteristic zero the investigator plans to approach the problem of resolution of singularities in positive characteristic. The new notions open some new possibilities in related areas. In particular, the new invariant proposed in the project, so called p-order, may have interesting applications in problems related to studying singularities in positive characteristic.

The goal splits into two parts. First, the investigator plans to show how to reduce in a canonical way the resolution of general singularities to the singularities of the special form (singularities of Giraud hypersurfaces of maximal contact). Understanding these singularities is considered to be the key for solving the problem. Second, the investigator plans to approach the resolution of singularities by using the introduced language and the inductive scheme developed in the first part. Although the problem seems very difficult, using new tools significantly simplifies it and allows us to better understand the problem in many particular situations. Moreover the investigator plans to analyze and generalize the simplified algorithms of Hironaka desingularization and the Weak factorization from the point of view of further potential applications.

Project Report

Project summary The project deals mainly with three major topics in Algebraic Geometry: -factorization of birational maps - the invariants of singularities - resolution of singularities In one of the papers written jointly with Donu Arapura and Parsa Bakhtary the investigator strengthens his Weak factorization Theorem. The generalization says that given two birationally equivalent compact nonsingular algebraic varieties together with simple normal crossing divisors i.e. subsets locally analytically isomorphic to unions of coordinate hyperplanes in the affine space, there exits a sequence of , so called, monoidal transformations which transform one of the sets ( a variety with the divisor ) into the other. The main difference is that we do not require the complements of the simple normal crossing subvarieties to be isomorphic ( they are just birational). This powerful tool allows to compare two seemingly very different objects by reducing the comparison to very simple monoidal transformations. The latter are easy to describe and control. This allows, for instance, to establish some new birational invariants. As an application of the strengthening of the Weak factorization theorem we show that the dual complexes associated with a variety and a simple normal crossings subvariety are very similar ( homotopically equivalent) for biartionaly equivalent varieties. In particular their cohomology and homotopy groups are identical. 2. The above allows us to construct the invariant of singular points i.e. the points where the variety is not locally isomorphic to an affine space. A possible invariant is a homotopy type of the dual complex of the fiber of a sufficiently good resolution of singularities or the cohomology and homotopy group of this dual complex. We also study some of these invariants from a different perspective of, so called, Deligne weight filtration. As was shown by Deligne one can find inside the cohomology groups of algebraic varieties some uniquely defined sequence of nested subgroups forming a Weight filtration. In particular, this can be related to the dual complexes associated to the fibers of a sufficiently good resolution of singularities. The investigator shows jointly with the coauthors that the higher cohomoloqy groups of the dual complex associated to a good resolution of rational hypersurface singularity vanish. This is motivated by a classical results of Artin for rational surfaces. 3. In the previous papers the investigator gave a simplified proof of the celebrated Hironaka resolution theorem. It says that any algebraic variety X over a field of characteristic zero (or an analytic space) can be desingularize i.e. there exists a nonsingular variety Y and an proper birational map from Y to X, Building upon the above the simplification the investigator jointly with Bierstone, Grigoriev and Milman gives in one of the recent papers an effective algorithmic description of the Hironaka resolution algorithm. They found the estimates for the vital functions: degree of polynomials, number of monoidal transformations, and the number of operations needed to complete the algorithm. This is a fundamental question when dealing with algorithms and its computational and practical aspects. In particular, it has already found some recent applications over the fields of positive characteristic. One can show that if the degree of polynomials defining a variety and their number is relatively small comparing to the characteristic then the variety can be desingularize. Intellectual merit: Good understanding and description of singular points is critical for many problems in Algebraic geometry and far beyond, as these points contain vital information for studying algebraic and analytic varieties, and more general spaces. The Hironaka theorem on resolution of singularities plays a central role in Algebraic Geometry. It constitutes a basis of proofs of many theorems not only in Algebraic and Analytic Geometry. The projects deals with more efficient approach to the algorithm. It is vital from the point of view of the potential applications. It allows to extend the desingularization techniques to the other unknown cases (for instance to some special cases in positive characteristic). On the other hand the investigator in his project studies new interesting topological invariants describing the resolution of singularities. One of the important tools applied here is the previously proven by the investigator the Weak factorization theorem. It is frequently used for comparing birational varieties.In the project the investigator generalizes his Weak factorization theorem and shows how to construct the new invariants of singularities. Some of these invariants can be computed for certain types of singularities. Broader impacts: Hironaka's theorem on resolution of singularities plays an important role in Mathematics far beyond Algebraic and Analytic Geometry. Also the Weak factorization theorem found its important applications beyond birational algebraic geometry. Simple and accessible resolution algorithm has impact on developing research tools and enhancing scientific cooperation of people from different areas of Mathematics an Applied Mathematics and from different regions of the world. The results were disseminated through the world wide web as well as through publications or direct communication.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Tie Luo
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Purdue University
West Lafayette
United States
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