This project addresses three central problems in algebraic geometry: Can one compute the ample cone of a polarized holomorphic-symplectic variety from its Hodge structure? What is the right functorial definition for compact moduli spaces of higher-dimensional varieties (and what might they be good for)? To what extent does the birational geometry of moduli spaces govern the behavior of related Geometric Invariant Theory problems, and vice versa? These questions are intertwined in intricate and beautiful ways: Intersection-theoretic constructions govern curve classes on both the moduli space of stable curves and holomorphic-symplectic varieties. The Torelli Theorem for K3 surfaces is the starting point for their moduli theory; the lack of such a result for higher dimensional holomorphic-symplectic manifolds is a major impetus for analyzing their ample cones. The elusive dream of a geometric compactification for the moduli space of K3 surfaces animates work on the interplay between Geometric Invariant Theory and moduli spaces.
Algebraic geometry is the study of geometric objects defined by polynomial equations, which are called varieties. Examples of varieties include circles, ellipses, parabolas, spheres, etc. A fundamental problem is to classify all the varieties of a given type. One approach is to analyze all the varieties defined by polynomials of given degree, e.g., the conic sections studied in high school analytic geometry. Here the type of the variety is expressed in algebraic terms. Alternately, one can study all the varieties sharing common geometric characteristics, e.g., those with given numerical invariants. This project addresses the interplay between the algebraic and geometric quantities, and how these govern the behavior of families of varieties.
Algebraic geometry is the study of geometric objects defined by polynomial equations. For example, the line in the Cartesian plane passing through (0,1) and (1,0) can be expressed as the locus of points (x,y) satisfying x+y=1. What makes the field vibrant and resilient is the interplay between the different techniques: Deep algebraic results about polynomials have concise geometric interpretations; geometric intuitions become precise and rigorous once they are cast in algebraic terms. This interplay gives rise to broad applications: Computers can process geometric figures only after they are translated into numerical or algebraic terms, so the techniques of algebraic geometry are useful for geometric modeling and digital rendering of images. Many complicated algorithms from cryptography and error-correcting codes are more transparent once they are analyzed geometrically. Moduli spaces: Classification is fundamental throughout the sciences. Geometers seek to enumerate all the objects sharing common properties. For example, conic sections were classified by the ancient Greeks; in the language of algebraic geometry, each is the locus in the (x,y)-plane satisfying an equation Ax2+Bxy+Cy2+Dx+Ey+F=0, for some parameters A,B,C,D,E,F. A smooth conic section is a parabola if B2=4AC. Thus the objects with prescribed geometric properties are themselves given as solutions to equations in the parameters. The solution of a classification problem in algebraic geometry is called a moduli space. The description above gives the moduli space of conic sections. Moduli spaces are themselves solutions of polynomial equations and thus may be analyzed using the techniques of algebraic geometry. Fundamental problems studied include: Characterize possible moduli spaces for curves with given combinatorial data: how are they related geometrically as the data varies? Construct moduli spaces for smooth surfaces and their singular limits as the solution to an explicit system of polynomial equations. Find an approach to moduli spaces of surfaces allowing intersection-theoretic computations. This project yielded results on these questions, especially on the log minimal model program for the moduli space of curves. What are possible applications? History suggests these may take time to develop; the classification of conic sections (known since 200BC) found its most compelling application in Kepler's work on planetary motion. Today, moduli spaces of curves are used in string theory, and are essential in precise formulations of mirror symmetric properties of the hidden dimensions of the theory. Rational curves: Rational curves are the images of non-constant maps given by polynomials or rational functions in one variable. For example, the image of t -> (t,1-t) in the (x,y)-plane is the line with equation x+y=1; the image of t -> (t,-t,1) in (x,y,z)-space satisfies equations x+y=z-1=0 and is contained in the surface V={x4-y4+z4=1}. Lines and conic sections are both instances of rational curves. Let V denote the solutions to a system of polynomial equations. Does V admit any rational curves? What are their numerical invariants? Do these govern the structure of the rational curves? A fundamental principle in algebraic geometry is that the deformation of rational curves as V varies reflect where it sits in the classification. This is the theoretical impetus for the study of rational curves. A main result of this project is a numerical characterization of rational curves for irreducible holomorphic symplectic varieties, using the language of convex geometry. There are practical reasons to understand rational curves: Computers can plot them very nicely, because it is easy to render their points numerically. Most curve-fitting algorithms (like spline interpolation) use pieces that are rational curves; these are widely used in data-fitting, engineering, and modeling. Broader impacts: The PI leads a research group involving several postdoctoral fellows, graduate students, and undergraduate research students. Group members have gone on to make contributions to education, national security, finance and risk management, and engineering.
