Degenerations of line bundles along curves

Abstract: A family of line bundles along a family of smooth curves parameterized by the punctured disk can be extended in several ways over the limit stable curve of the family. We show that the collection of all extensions can be naturally parameterized by the torus quotient of the arrangement of toric varieties associated to a certain polytope decomposition of a certain Euclidean space. We characterize all polytope decompositions arising this way in terms of combinatorial data of the stable curve. At the end I will describe how these results may be used to construct new compactifications of Jacobians of stable curves and address the problem raised by Eisenbud and Harris of constructing a useful moduli of limit linear series over the moduli of stable curves. This is an ongoing joint work with Omid Amini (École Polytechnique).

algebraic geometry

Audience: researchers in the topic

Brazilian algebraic geometry seminar

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