Artin Approximation, Arc Spaces, Resolution of Singularities

This research project is concerned with various topics at the crossroad between algebraic geometry and commutative algebra. Its main focus can be described as the problem of solving algebraic and analytic equations via three different methods. Artin approximation looks for formal and convergent power series solutions to a given equation, for example by using an Ansatz for the Taylor expansion of the solution. Normally this approach will only yield a formal solution, given by successively calculating coefficients. Artin`s theorem now guarantees the existence of a convergent solution if a formal solution exists. In this way, one can avoid rather tedious considerations for the proof of the convergence of solutions. In our project, instead of considering just one solution, we try to construct all solutions (formal and convergent) simultaneously. Arc spaces describe all formal curves (given by power series in one variable t) on a given variety X. They were originally introduced by John Nash in order to better understand the singularities of the variety X. It turns out that there are indeed many deep connections between the arc space of X and the local geometry at singular points of X. In this project, new such relationships shall be developed and investigated. The main goal of resolution of singularities is to modify a given variety (by so-called "blowups``) to reduce the complexity of its singularities. When considering varieties over a field of characteristic 0, Hironaka`s theorem guarantees that, after finitely many steps, successive applications of these blowups will yield a smooth variety without singularities. In this project we try a more geometric approach to resolution by replacing blowups with so-called "higher Nash modifications". These are given by considering the tangent spaces and curvatures in smooth points and taking their limit as the point tends to a singularity. The resulting Gauss map yields a quasi-affine variety whose Zariski closure defines the modification of the original variety. Geometricallly, this procedure amounts to adding all limits of tangent directions and curvatures in singular points. The program has already been successfully applied to algebraic curves and should now be extended to the (much more difficult) case of singular algebraic surfaces. The three main research aspects of the project are closely related and share techniques from infinite- dimensional algebraic geometry, commutative algebra and differential geometry. All stated objectives have been formulated in a precise way and should be instrumental in understanding singularities of algebraic varieties.

In this project, substantial extensions of Artin's famous Approximation Theorems could be achieved: Firstly, the infinite-dimensional variety of all power series solutions y(x) of a system of analytic equations f(x,y)=0 could be explicitly described, yielding a geometric approach and interpretation of Artin's theorem. Further, the theory could be successfully applied to the study of arc-spaces in order to get new important insights.