Stable Pairs and Log Flips

the blow-up of |KC +D| along the image of C (which we will also denote by C). In contrast, the Serre correspondence is a morphism only when d ≤ 2, as it is undefined at points of C (and certain secant varieties) for larger values of d. The structure of the Serre correspondence for large d (particularly d > 2g − 2, in which case it is dominant) is, however, well understood now, due largely to recent work of the author and Michael Thaddeus. The first goal of this paper (§2) is to explain Thaddeus’ interpretation of the Serre correspondence as a sequence of simple birational maps of moduli spaces of stable pairs followed by a contraction. We will discuss the birational maps at length, but let’s start with an example of a contraction. This is another ancient result: