Degenerating Families of Rank Two Bundles

We construct families of rank two bundles Et on P4, in characteristic two, where for t 6= 0, Et is a sum of line bundles, and E0 is non split. We construct families of rank two bundles Et on P3, in characteristic p, where for t 6= 0, Et is a sum of line bundles, and E0 is non split. 1. A degenerating family of split bundles on P in characteristic 2 We will construct a family of rank two bundles Et on P, in characteristic two, where for t 6= 0, Et is a sum of line bundles, and E0 is non split. The construction derives from Tango’s example of a rank two bundle on P in characteristic two [5]. (For different constructions of such phenomena on P and P, see [1], [2].) Let X = P×A. Let k1, . . . , k5 be the degrees of forms a, b, c, d, e on P where the forms form a regular sequence. Impose the condition k1 = k2 + k5 = k3 + k4. Let t be a parameter for A. Consider the matrices Φ =  0 t −e d −t 0 c −b e −c 0 a −d b −a 0  ,Ψ =  0 −a −b −c a 0 −d −e b d 0 −t c e t 0  . These matrices give maps Φ : L1 → F and Ψ : F → L2, where L1 = OX(k2 − k4)⊕OX ⊕OX(−k5)⊕OX(−k4) (1.1) F = OX ⊕OX(k2 − k4)⊕OX(k3)⊕OX(k1 − k5) (1.2) L2 = OX(k2 − k4 + k1)⊕OX(k1)⊕OX(k2)⊕OX(k3). (1.3) Let Y be the hypersurface in X with equation s = at − be + cd = 0. Note that ΦΨ = ΨΦ = sI and that both Ψ and Φ have rank exactly two at each point of Y . Consider the pull backs Φ : L 1 → F (1) and Ψ : F (1) → L 2 of Φ and Ψ by the Frobenius morphism F : X → X. Form the matrix