Many shapes that occur in nature are given by polynomial equations, and computers are especially fast in computing polynomials. Understanding the solutions of systems of polynomial equations is a key to advances in many areas, including computer aided geometric design, robotics and physics. In most cases a small change of the coefficients results in a small change of the geometric objects. For example, increasing the radius of a sphere a little results in a slightly larger sphere. The aim of this project is to give a general theoretical framework for analyzing those cases when a small change of the coefficients results in a dramatic change of the geometric objects. These are the cases when a small error in measurement or manufacturing may have large consequences.
The main aim of the project is to understand what a good family of algebraic varieties is. The moduli of curves and its compactification constructed by Deligne and Mumford are among the most important objects in mathematics and in string theory. The goal of higher dimensional moduli theory is to construct more general versions and to use this in understanding the geometry of higher dimensional algebraic varieties. An approach to compactifying the moduli space of varieties of general type was proposed by Kollar and Shepherd-Barron in 1988. The limiting objects are called stable varieties. These satisfy a local condition (having only log-canonical singularities) and a global condition (having ample canonical class). More generally, following Alexeev, one should consider moduli spaces of pairs of a projective, reduced scheme and a non-negative linear combination (with rational or real coefficients) of divisors. In order to get a reasonable moduli space, these pairs should satisfy a local condition (the pair has only semi-log-canonical singularities) and a global condition (having ample log canonical class). The aim of the project is to apply the study of semi-log-canonical singularities to complete the moduli theory of stable pairs. The theory is complete as far as the underlying varieties are concerned, but the divisor parts of the pairs exhibit non-flat behavior that is harder to approach with the usual techniques. For stable varieties without divisors, the existence of the moduli space are known, but it needs to be written down in a systematic way. When a divisor is added, we run into the problem (first observed by Hassett) that a deformation of the pair need not induce a flat deformation of the divisor part. For seminormal base spaces the theory of Chow varieties is ideal to understand non-flat deformations over. The first main aim is to generalize this to reduced base spaces. In general we may expect a better theory when the coefficients in the divisorial part are bigger than a half. The second main aim is to work out the moduli theory by treating the variety and the divisor as essentially independent objects varying flatly. A third aim is to understand what happens when the coefficients in the divisorial part are allowed to be a half, since this case comes up frequently in applications. Besides being important in its own right, this is a test case for the various techniques that one can use in general. The fourth, most speculative part is to develop a general moduli theory that works optimally in all cases. A key problem is that there are several competing definitions that give slightly different answers for some non-reduced schemes. We need to understand the precise relationships and to find the optimal theory.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
