Riemann-Workshop in algebraic geometry
Venue: Leibniz Universität Hannover
Dates: September 1, 2016
Room: g117 (Main building "Welfenschloss")
11:15 - 12:00 Victor Lozovanu (Caen) - From convex geometry of certain valuations to positivity aspects in algebraic geometry
A few years ago Okounkov associated a convex set (Newton–Okounkov body) to a divisor, which encodes the asymptotic vanishing behaviour of all sections of all powers of the divisor along a fixed flag. This, on the other hand, brought to light the following guiding principle ”use convex geometry, through the theory of Newton–Okounkov bodies, to study the geometrical/algebraic/arithmetic properties of projective varieties”. The main goal of this talk is to explain some of the philosophical underpinnings of this principle with a view towards studying local positivity and syzygetic properties of algebraic varieties.
14:00 - 14:45 David Schmitz (Marburg) - On Minkowski bases for Newton-Okounkov bodies
Since their systematic introduction by Lazarsfeld and Mustata and independently by Kaveh and Khovanskii in 2008, Newton-Okounkov bodies have increasingly been studied as important invariants of effective divisors on algebraic varieties. The Newton-Okounkov body of a linear series on an n-dimensional projective variety is a compact convex body in real n-space which carries interesting information about the linear series. As we will see in the talk, however, it is in general hard to determine in practice. We show that under certain conditions there exist simple "building blocks“ for all Newton-Okounkov bodies of a given variety, a so called Minkowski basis. Additionally, we establish a consequence of the existence of a Minkowski basis for the shape of the global Okounkov body studied by Lazarsfeld and Mustata.
15:30 - 16:15 Víctor González Alonso (Hannover) - Torelli type results for irregular varieties of general type
The Torelli theorem is known for a few classes of varieties (curves, some kinds of varieties with trivial canonical bundle) but not much is known in the case of varieties of general type. In this talk I will discuss some partial results for this case concerning the local Torelli problem, which considers the variation of the Hodge structure of the fibres of a family. I will also present some application to the geography of fibred surfaces regarding the Fujita decompositions of the Hodge bundle
Riemann Center for Geometry and Physics