Prym varieties of etale double covers produce canonical projective structures whose variation is governed by Thetanullwert maps, and this variation is nonzero, so the base structure depends on the cover.
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Prym varieties and projective structures on Riemann surfaces
Prym varieties of etale double covers produce canonical projective structures whose variation is governed by Thetanullwert maps, and this variation is nonzero, so the base structure depends on the cover.