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Severi varieties parametrize irreducible curves of fixed geometric genus in a given linear
system.
They are classical objects that have been studied extensively. My talk will survey results
concerning
the irreducibility of such varieties for linear systems defined on toric surfaces. The case of
the
projective plane in characteristic zero was settled in the landmark paper of Harris in 1986.
After recalling
his approach, I will sketch further developments since then. Finally, I will report on ongoing
work with Xiang He and Ilya Tyomkin, in which we use tropical geometry to study the question of
irreducibility. In particular, we can prove irreducibility for the projective plane in positive
characteristic.
Time permitting, I will indicate some further directions we plan to investigate.
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