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We show that a nodal hypersurface X in P 3 of degree d with a sufficiently large number l of nodes, l > 8 3 (d 2 − 5 2 d), is algebraically quasi-hyperbolic, i.e. X can only have finitely many rational and elliptic curves. Our results use the theory of symmetric differentials and algebraic foliations and give a very striking example of the jumping of the number of symmetric differentials in families.doi:10.1515/crelle.2006.053 fatcat:qh6ft73fdzbhdewarjyhsfl7pq