Tropical convexity, linear systems on metric graphs, and a generalized notion of reduced divisors

Series: 
Algebra Seminar
Monday, November 21, 2011 - 16:05
1 hour (actually 50 minutes)
Location: 
Skiles 006
,  
Georgia Tech
,  
Organizer: 
Metric graphs arise naturally in tropical tropical geometry and Berkovich geometry. Recent efforts have extend conventional notion of divisors and linear systems on algebraic curves to finite graphs and metric graphs (tropical curves). Reduced divisors are introduced as an essential tool in proving graph-theoretic Riemann-Roch. In short, a q-reduced divisor is the unique divisor in a linear system with respect to a point q in the graph. In this talk, I will show how tropical convexity is related to linear systems on metric graphs, and define a canonical metric on the linear systems. In addition, I will introduce a generalized notion of reduced divisors, which are defined with respect to any effective divisor as in comparison a single point (effective divisor of degree one) in the conventional case.