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Oct 12, 2020 at 16:49 comment added mathphys Yes I see, thanks very much!
Oct 12, 2020 at 16:42 comment added abx Corollary 3.23 in Clemens-Griffiths tells you that there is a unique decomposition of a p.p.a.v as a product of irreducible ones. This implies immediately that $J(Y)$ is a product of some of the $J(C_i)$. Actually you don't need the polarization to be principal — see O. Debarre, C. R. Acad. Sci. Paris 323 (1996), 631-635.
Oct 12, 2020 at 16:34 history asked mathphys CC BY-SA 4.0