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Tropical geometry is a powerful tool that allows one to compute enumerative algebraic
invariants through the use of some correspondence theorem, transforming an algebraic
problem into a combinatorial problem. Moreover, the tropical approach also allows one
to twist definitions to introduce mysterious refined invariants, obtained by counting
curves with polynomial multiplicities. So far, this correspondence has mainly been
implemented in toric varieties. In this talk we will study enumeration of curves in
abelian surfaces and line bundles over an elliptic curve.
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