Newton-Okounkov bodies of algebraic varieties and their applications

• Applicant: Dr. David Schmitz
• Subject Area: Mathematics
• Term: from 2015 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 274229163

It is planned to investigate geometric properties and applications of Newton-Okounkov bodies of algebraic varieties. These convex bodies, which are associated to linear series, carry rich information on the associated linear series but are notoriously hard to determine. Only in certain special cases the determination can be carried out without great difficulty. However, for certain classes of algebraic varieties, e.g., smooth projective surfaces, the last couple of years have seen great improvement both in the description of geometric properties of Okounkov bodies as well as their determination. Moreover, important applications of Okounkov bodies have been introduced, e.g., in constructing integrable systems or in reformulating classical conjectures of Nagata type. In this project, both questions shall be investigated. More concretely, we turn to global Okounkov bodies in order to investigate under which conditions on the variety they satisfy certain geometric properties, in particular rational polyhedrality. It is known that a variety which admits a so-called Minkowski basis has a polyhedral global Okounkov body. The planned strategy to show rational polyhedrality therefore consists in an attempt to prove the existence of a Minkowski basis. It is a main goal of this project to carry out this proof for the case of Mori dream spaces. Among the applications of Okounkov bodies we concentrate in particular on the observation made by D. Anderson that varieties with finitely generated valuation semi-group admit a toric degeneration. Our goal is to improve this result by fixing the prima facie drawback that the special fiber in the degeneration need not be a normal variety.

• DFG Programme: Research Fellowships
• International Connection: USA
• Participating Institution: State University of New York at Stony Brook ; Department of Mathematics
• Host: Professor Robert Lazarsfeld, Ph.D.