The Frobenius map, rank 2 vector bundles and Kummer's quartic surface in characteristic 2 and 3

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Let X be a smooth projective curve of genus g>1 defined over an algebraically closed field k of characteristic p>0. Let Mₓ(r) be the moduli space of semi-stable rank r vector bundles with fixed trivial determinant. The relative Frobenius map F: X → X₁ induces by pull-back a rational map V: Mₓ₁(r) → Mₓ(r). We determine the equations of V in the following two cases (1) (g,r,p) = (2,2,2) and X non-ordinary with Hasse-Witt invariant equal to 1 (see math.AG/0005044 for the case X ordinary), and (2) (g,r,p) = (2,2,3). We also show, for any triple (g,r,p), the existence of base points of V, i.e., semi-stable bundles E such that F* E is not semi-stable.