Compactifying moduli of line bundles on curves

The problem discussed in this talk is to compactify the moduli space of pairs (C,L) with C a smooth projective curve of genus g and L a line bundle of degree d on C. Such a compactification was first found by Caporaso a quarter century ago, but there is an obstruction to the existence of a universal family over it. We'll describe this obstruction, and attempts by Ran and Mainò to deal with it.

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