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Let $C\rightarrow C'$ be a double cover of curves, the restriction of the polarisation of $J_C$ to the Prym varieties $P$ attached to this double cover, gives a polarization on $P$,

Does this polarization have a square root in general?

I khow that, in some special cases, this is true an a square root gives a principal polarization. thanks

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    $\begingroup$ If and only if the double cover is étale or has 2 branch points. See Mumford, Prym varieties I. $\endgroup$
    – abx
    Jan 26, 2017 at 6:37
  • $\begingroup$ @abx Your statement holds true if $g\geq 3$. See Proposition 6.4. of math.ucsd.edu/~eizadi/papers/JAG2012.pdf $\endgroup$
    – user21574
    Jan 26, 2017 at 6:59
  • $\begingroup$ The type of the polarisation is in general (1,...,1,2,...,2), and hence it has'nt square root ! but a translation of it may. $\endgroup$
    – Z.A.Z.Z
    Jan 26, 2017 at 7:05
  • $\begingroup$ @Z.h I do not understand your comment. It seems to me that the existence of a square root of a line bundle on an abelian variety $A$ only depends on its algebraic class (because $\mathrm{Pic}^0(A)$ is a divisible group). So it should not depend on the translate chosen. $\endgroup$
    – Francesco Polizzi
    Jan 26, 2017 at 7:14
  • $\begingroup$ @FrancescoPolizzi, you r right, I was thinking in general problem: given a family of $SO_r-$bundles over a curve, after translating this family with atheta characteristic, we get a family whose determinant bundle has a square root, see for example Sorger And Laszlo "line bundle on the moduli of parapolic bundles" section 7. $\endgroup$
    – Z.A.Z.Z
    Jan 26, 2017 at 7:24

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Let $R$ be the ramification of $C\rightarrow C'$, and let $\eta$ be a square root of $\mathcal O(R)$. Let $\mathcal L$ be a family of line bundles of $P$ parametrized by some noetherian scheme $T$, and finally consider $$\mathcal L'=\mathcal L\otimes p_1^*\eta$$ where $p_1:X\otimes T\rightarrow X$. Now consider the direct image of $\mathcal L'$by $\pi\times id:C\times T\rightarrow C'\times T$. This direct image has a non degenerate symmetric bilinear form with value in $\mathcal O_{C'}$ (by the relative duality), hence take the tensor product with some theta characteristic, we get a family of line bundles with quadratic form in $K_{'C'}$, by Sorger and Laszlo "line bundle on the moduli of parapolic bundles" section 7; the determinant bundle of this family has a square root. Hope this may help!

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    $\begingroup$ This has not much to do with the question, which is about a very standard fact on Prym varieties (and would be more appropriate on MSE). Once again, look at Mumford's paper (or Birkenhake-Lange). $\endgroup$
    – abx
    Jan 26, 2017 at 7:48

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