Introduction to Combinatorial Birational Geometry

Meetings

Content

Algebraic geometry studies solution sets to systems of algebraic equations in many variables and considers them as geometric objects called algebraic varieties. It turns out that the best understanding of algebraic varieties arises only after they are reduced by means of certain transformations to the most convenient geometric models for their study. Our goal is to familiarize students with the most accessible transformations of algebraic varieties which are called birational. Using birational transformations one comes to definition of birational equivalence classes of algebraic varieties, and one of the main tasks of birational geometry is to identify "the best" representatives which are called minimal models. In general, finding minimal birational models is a rather difficult task. But in these lectures we will restrict ourselves to the case of non-degenerate toric hypersurfaces. For these it turns out to be possible to completely solve this problem in arbitrary dimension using methods of combinatorial birational geometry based on the elementary convex geometry of Newton polyhedra of toric hypersurfaces. To maintain maximum accessibility of the presented methods of combinatorial birational geometry, we will devote the bulk of the lectures to the classical case of algebraic curves on algebraic surfaces.

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Literature

• Igor R. Schafarewitsch: Grundzüge der algebraischen Geometrie. Springer 1972.
• Klaus Hulek: Elementare Algebraische Geometrie. Springer 2012.
• David Mumford: Lectures on curves on an algebraic surface. Princeton 1966.
• Robin Hartshorne: Algebraic Geometry. Springer 1977.
• Tadao Oda: Convex Bodies and Algebraic Geometry. Springer 1988.