Abstract
LetY be an irreducible nodal hyperelliptic curve of arithmetic genusg such that its nodes are also ramification points (char ≠2). To the curveY, we associate a family of quadratic forms which is dual to a singular pencil of quadrics in\(\mathbb{P}^{2g + 1} \) with Segre symbol [2...21...1], where the number of 2's is equal to the number of nodes. We show that the compactified Jacobian ofY is isomorphic to the spaceR of (g−1) dimensional linear subspaces of\(\mathbb{P}^{2g + 1} \) which are contained in the intersectionQ of quadrics of the pencil. We also prove that (under this isomorphism) the generalized Jacobian ofY is isomorphic to the open subset ofR consisting of the (g−1) dimensional subspaces not passing through any singular point ofQ.
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Bhosle, U.N. Singular pencils of quadrics and compactified Jacobians of curves. Proc. Indian Acad. Sci. (Math. Sci.) 100, 95–102 (1990). https://doi.org/10.1007/BF02880954
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DOI: https://doi.org/10.1007/BF02880954