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0. Introduction This paper following a geometric approach proves new, and reproves old, vanishing and nonvanishing results on the space of twisted symmetric differentials, H 0 (X, S m Ω 1 X ⊗ O X (k)) with k ≤ m, on subvarieties X ⊂ P N. The case of k = m is special and the nonvanishing results are related to the space of quadrics containing X and lead to interesting geometrical objects associated to X, as for example the variety of all tangent trisecant lines of X. The same techniques give results on the symmetric differentials of subvarieties of abelian varieties. The paper ends with new results and examples about the jump along smooth families of projective varieties X t of the symmetric plurigenera, 1 X ⊗ αK X)). The paper is in part motivated by a previous result where we proved that while smooth hypersurfaces in P 3 do not have symmetric differentials, resolutions of nodal hypersurfaces have them if the number of nodes is sufficiently large. This is, in particular, interesting because smooth and resolutions of nodal hypersurfaces in P 3 of the same degree are deformation equivalent. So we have a case of the jumping of the symmetric plurigenera Q m (X) and a special one for that matter, as we shall see below. There is a previous example of this phenomenon in [Bo2-78], see section 2. Recall that this contrasts with the invariance of the plurigenera P m = dim H 0 (X, (∧ n Ω 1 X) m), n = dim X, [Si98]. The jumping in our example might bring back symmetric differentials to new approaches to the Kobayashi's conjecture which states that a general hypersurace in P 3 of degree d ≥ 5 is hyperbolic (the known approaches use jet differentials, i.e. higher order symmetric differentials, which exist on hypersurfaces but are quite difficult to control). The authors motivated by this unexpected appearance of the symmetric differentials realized that there are many unanswered or forgotten questions about them.