Birational Geometry of Moduli Spaces and Applications

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The principal investigator is interested in the study of the geometry of moduli spaces, especially those that are birational to modular varieties of orthogonal or unitary type (examples include the moduli space of K3 surfaces or low genus curves). Laza's work exploits the existence of multiple birational models for a moduli space (e.g. obtained by using different compactification methods, such as Geometric Invariant Theory (GIT) or Hodge theory) to extract useful geometric information about a given moduli space. The principal investigator plans to apply this type of ideas to various projects involving moduli spaces. A first question that the PI proposes to investigate is the problem of finding a geometric compactification for the moduli of polarized K3 surfaces. The methods of the variation of GIT quotients and ideas coming from the minimal model program offer a promising approach to the low degree cases. A second project addresses various questions about the moduli of compact hyperkaehler manifolds, higher dimensional analogues of the K3 surfaces. In particular, the PI plans to investigate from an arithmetic and geometric point of view the case of moduli space of double EPW sextics (introduced by O'Grady). A third project is concerned with some concrete questions about the birational geometry of moduli spaces of genus 4 curves and cubic threefolds. The general area of the proposal is algebraic geometry, the branch of mathematics that is concerned with the geometric properties of algebraic varieties (geometric objects defined by polynomial equations). A simple example of algebraic variety is the complex torus, that has the shape of a doughnut. While in other branches of mathematics (such as topology) all doughnuts have the same shape, in algebraic geometry the precise shape (in this case the ratio between the diameter and width) is very important. In fact, the precise quantification of the shape of the geometric objects within a given topological class is the subject of the moduli theory. Moduli theory is a central field of study in algebraic geometry and has numerous applications in mathematics and modern physics.