@JasonStarr: ..ah, and also I noticed that this result I quoted on constancy of arithmetic genus along flat family from Hartshorne, Cor. III.9.13 requires every fibre to be normal, but if special fibre is union of curves intersecting nontrivially (so connected, but not irred) then this result not applies

But does it imply the claim? Eg, why cannot be the one of the reduced components of the degeneration be an elliptic curve?

@JasonStarr: sorry for digging this old, but I'm not sure to which amount Debarre's book elaborates this statement about that a rational curve can only degenerate to union of rational curves of smaller degree. Indeed, in proof of Lemma 3.7 (p 68) (book) resp. Lemma 7.8 (linked notes) he only states this as fact, but looking through prev chapter I haven't found an elaboration. Do you maybe know if this can be immediately deduced from some of earlier results from this book? The starting point should be that the arithmetic genus not changes along flat family if general fibers are "nice enough".

Also, could you maybe elaborate the part with why $J$ neccessarily happens to be an abelian surface only if it is a product in the case above?

Thank you! Two points would like to clarify: You mentioned that for the Kodaira dimension we have always $\kappa(X) \geq \kappa(J)$ and is even an equality if $X \to B$ has no multiple fibres. Then cases $X$ rational and $X$ Enriques (esp. $X \to B$ admits multiple fibres, so we have a drop) are treated in Cossec and Dolgachev in 5.6, resp. 5.7. Could you elaborate the ideas or give a reference treating general scenario? Can this be extracted from $P_n(J) = n(2g_B-2+\chi(\mathcal{O}_X))+1-g_B$?