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
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!
