In this paper we characterize the rank two vector bundles on P 2 which are invariant under action of Gp := Stabp(PGL(3)), that fixes a point in projective plane, GL StabL(PGL(3)), line, and T = ∩ GL. Moreover, prove geometrical configuration jumping locus induced by invariance does not, other hand, itself. Indeed, find infinity families almost uniform but not homogeneous.

in this paper the family of elliptic curves over q given by the equation ep :y2 = (x - p)3 + x3 + (x + p)3 where p is a prime number, is studied. itis shown that the maximal rank of the elliptic curves is at most 3 and someconditions under which we have rank(ep(q)) = 0 or rank(ep(q)) = 1 orrank(ep(q))≥2 are given.