Singularities in Arbitrary Characteristic and Positivity

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The project concerns questions in algebraic geometry and commutative algebra. Algebraic geometry is the study of algebraic varieties, which are solution sets for systems of polynomial equations. For example, in the xy-plane, the solutions for y=0 consist of all points along the x-axis, while the solutions for xy=0 consist of all points along both coordinate axes. Since the tangent line at the origin (0,0) is not defined for the algebraic variety defined by xy=0, we say that this variety has a singularity at the origin (0,0). The first goal of the present project is to further develop the theory of singularities of algebraic varieties and more general objects. The second goal is to build the tools and techniques necessary for this research program, which would have applications to other fundamental open questions in algebraic geometry and commutative algebra.The research will investigate singularities of algebraic varieties over arbitrary fields, or more generally of arbitrary Noetherian rings and schemes. Even when the primary interest is in non-singular complex algebraic varieties, working in this much more general context is often unavoidable. A major focus in the present project is to build the foundations necessary for birational geometry and the minimal model program for schemes of arbitrary characteristic. In previous work, the PI proved that Kodaira-type vanishing theorems hold for schemes in equal characteristic zero of arbitrary dimension and developed new methods to replace Kodaira-type vanishing theorems in positive characteristic. In the present project, the PI will apply the techniques from this previous work to study the behavior of singularities under flat morphisms. The PI will also investigate inversion of adjunction-type results for various classes of singularities. Finally, the PI plans to extend techniques in positive characteristic to study positivity of line bundles on algebraic varieties over arbitrary fields.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.