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Let $X\subset \mathbf{P}^n$ be a hypersurface such that the singular locus of $X$ consists of a single ordinary double point. I'm trying to find a reference to the "generalized" intermediate Jacobian of $X$.

When $n=2$, i.e. $X$ is a nodal curve, it is standard that the is a notion of generalized Jacobian of $X$, which is a $\mathbf{G}_m$-extension of the usual Jacobian of its normalization. See Ex II.6.9 in Hartshorne's book.

For higher dimensional cases, I have seen in several papers that experts use a notion of the generalized intermediate Jacobian of $X$, which is also a $\mathbf{G}_m$-extension of an abelian variety, without citing a reference.

So I'm wondering if there is a standard reference and definition for such generalized Jacobians. Thanks in advance.

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  • $\begingroup$ maybe try a mathscinet search for articles referencing ams.org/journals/tran/1979-253-00/S0002-9947-1979-0536936-4/… ? i would do this myself, but i cannot at the moment $\endgroup$
    – Joseph Harrison
    Commented Sep 23 at 19:29
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    $\begingroup$ See the top of page 216 in S. Zucker, Generalized intermediate Jacobians and the theorem on normal functions, Invent. Math. 33 (1976), no.3, 185–222. Note that Zucker points to Theorem 16.16 in P. Griffiths, On the periods of certain rational integrals. II, Ann. of Math. (2) 90 (1969), 496–541. Griffiths in turn says this is due to M. Rosenlicht, Generalized Jacobian varieties, Ann. of Math. (2) 59 (1954), 505–530. $\endgroup$
    – Oli Gregory
    Commented Sep 23 at 19:46
  • $\begingroup$ One doesn't get an extension of an abelian variety by $\mathbb{G}_m$ without some extra condition on $X$: the intermediate Jacobian of even a smooth hypersurface is not in general an abelian variety. $\endgroup$
    – naf
    Commented Sep 24 at 2:29
  • $\begingroup$ @naf Sorry for confusing, I'm mainly interested in cubic or quartic threefolds, so their intermediate Jacobians are ppav. $\endgroup$
    – Jooh
    Commented Sep 24 at 5:49
  • $\begingroup$ @OliGregory Thanks, this is exactly what I'm searching for! $\endgroup$
    – Jooh
    Commented Sep 24 at 6:17

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Posting my comment as an answer:

See the top of page 216 in S. Zucker, Generalized intermediate Jacobians and the theorem on normal functions, Invent. Math. 33 (1976), no.3, 185–222. Note that Zucker points to Theorem 16.16 in P. Griffiths, On the periods of certain rational integrals. II, Ann. of Math. (2) 90 (1969), 496–541. Griffiths in turn says this is due to M. Rosenlicht, Generalized Jacobian varieties, Ann. of Math. (2) 59 (1954), 505–530.

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