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Intermediate Jacobians of intersections of two quadrics

This is not an answer, but this is too long for a comment. The hyperelliptic curve (let me call its $C$) associated to an intersection of quadrics $X$ has an intrinsic meaning as the moduli space of …
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Degenerations of hyperelliptic coverings

If $p_1 = p_2 \ne p_3 = p_4 \ne p_5 \ne p_6$ then the normalization of the double cover branched at the divisor $D = \sum_{i=1}^6 p_i$ is a smooth irreducible rational curve. If also $p_5$ and $p_6$ c …
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