It is classical that ample line bundles on an algebraic variety satisfy many beautiful geometric and cohomological properties, but it was long believed that arbitrary effective divisors are mired in pathology. However in recent years it has become clear that many features of the classical picture do have analogues in the general case, provided that one works asymptotically. I will discuss a construction, introduced in passing by Okounkov, whereby one associates a convex body in Euclidean space to a linear series on a projective variety. This construction renders transparent most of the asymptotic properties of line bundles that have been studied in recent years.
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