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If we have a family of hyperelliptic curves of genus g, then I know that this family can be expressed as a double cover of the quadric surface branched along a certain curve. In the book [Moduli of Curves, Harris-Morrison, p.293] they say that such a family should be a covering of quadric surface branched along a curve of type (2g+2,d). But their explanation is abit vague for me. I think this covering should come from the push-forward of the relative differential forms (I have read this somewhere but I don't remember and can't give a clear argument for that). Can anyone give a better argument or a reference for this?

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Very concretely -- the double cover of P^1_{x,y} x P^1_{z,w} is obtained by adjoining a square root of f(x,y), where f is a homogenous form of degree 2g+2 in x,y. But where are the coefficients of f? They themselves are homogeneous forms in z,w, of some degree which Harris-Morrison are calling d. You should think of d as measuring the (log of the) HEIGHT of the hyperelliptic curve.

By the way, in the question I think you mean "a rationally parametrized 1-dimensional family of hyperelliptic curves of genus g.'

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