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I have read in the literature about quasi-trigonal curves. Such a curve C is a hyperelliptic curve X with two points p,q identified (basically a pinch). They seem to be pretty important in the theory of Prym varieties, since the prym of a double cover of a q-t curve is the jacobian of an hyperelliptic curve B.

There are a few things that are not completely clear to me: first, can the two points p,q identified be one fiber of the hyperelliptic pencil on X?

second, can the hyperelliptic curve B be reconstructed (say, find its weierstrass points) starting from the data of the weierstrass points of X plus p and q?

Third, on the other hand the canonical model of a q-t curve sits on a rational normal cone, the singular point coinciding with the vertex of the cone. Do the ruling of the cone induce the hyperelliptic map on C? Is there a model of C on a Hirzebruch surface or one just needs to blow up the cone in the vertex?

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Perhaps you could give a reference to where you've encounted quasi-trigonal curves? From your other questions and this, I seem to work in a very similar area to you, but I've not encountered the term. And are you referring to hyperelliptic Beauville covers, in the first line? – Charles Siegel Aug 19 '10 at 17:23
Err, I meant Wirtinger covers, in the previous comment. – Charles Siegel Aug 19 '10 at 17:27
Hi Charles, thank you for your answers and references, and congrats on your blog: I am a fan! Luckily enough, I managed to answer my own questions regarding the two ancient questions, whereas this one is still quite open. It is a pleasure discussing about maths and, since it is likely that we will go a bit off topic, I'll write you an email asap. – IMeasy Aug 25 '10 at 14:55

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