Abstract: "Hyperelliptic Odd Coverings" are a class of odd coverings of C -> P^1, where C is a hyperelliptic curve. They are characterized by the behavior of the hyperelliptic involution of C with respect to an involution of P^1. I will talk about some ways for studying this type of coverings: by fixing an effective theta characteristic, they correspond to the solutions of a certain type of differential equations. Considering them as elements in a suitable Hurwitz space, they can be characterized using monodromy and then studied from the point of view of deformations. When C is general in H_g, the number of possible Hyperelliptic Odd Coverings C -> P^1 of minimum degree is finite. The main result will be how to compute this number. This is a work in collaboration with Gian Pietro Pirola.
In the first part of the talk, I will introduce the Hyperelliptic Odd Coverings from a geometric point of view, contextualizing them in the panorama of other enumerative works (in collaboration with Farkas, Naranjo, Pirola, Lian). In the second part of the talk I will talk about some proofs and some open problems.
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