Resolution of Singularities

The resolution of the singularities of algebraic varieties is one of the most difficult problems in algebraic geometry, with far-reaching consequences in many other branches of mathematics. It asserts that the solution set of any system of polynomial equations in several variables can be parametrized by a smooth space. In this way, the implicit equations, which in general are hard to solve, admit at least in principle a construction of their solution set. This is a fundamental and very important result. The submitted project proposes to attack in this context two central, yet still unsolved problems: The effective construction of resolutions in characteristic 0, and the existence proof for resolutions in characteristic p. First, in characteristic zero, Hironaka`s theorem tells us that a resolution always exists -- but its construction is so complicated that already simple examples in three or four variables cannot be computed. Moreover, even in cases when the computations can be completed, Hironaka`s procedure results in such an unwieldy bulk of local data that extracting the desired information becomes intractable. In contrast, the program outlined here proposes completely new constructions of resolutions that are specifically engineered to deliver much more effective results. A key idea is to use the theory of arc spaces and techniques from infinite dimensional geometry to build more compact resolutions that provide a concrete hold on the final result. The successful achievement of this program would realize a long awaited development in the field. The second punchline is the case of positive characteristic p. Open now for more than fifty years, there has recently been a surge of new methods and insights, among them by the applicant and his ground-breaking theory of kangaroo singularities. Time seems ripe for a definite and broad attack. An advance in this problem will be a truly fantastic breakthrough, bringing about multiple applications in arithmetic geometry and number theory. The proposed approach calls for new commutative algebra tools which will be applied to quotients of polynomial rings modulo the subring of p-th powers. The applicant has recently developed a theory of local multiplicities for elements in these quotients which seems to behave much better than the residual multiplicity considered classically. The failure of upper semicontinuity of the residual multiplicity in deformations and blowups has been one of the main obstacles for transferring the characteristic zero proof to positive characteristic. The new multiplicity aims to overcome this problem and first experiments have shown promising results.