A mathematical instanton
bundle on P3 (over an algebraically closed field) is a rank two vector bundle ℰ on P3
with c1= 0 and with H0(ℰ) = H1(ℰ(−2)) = 0. Let c2(ℰ) = n. Then n > 0. A
jumping line of ℰ of order a, (a > 0), is a line ℓ in P3 on which ℰ splits as
𝒪ℓ(−a) ⊕𝒪ℓ(a). It is easy to see that the jumping lines of ℰ all have order ≤ n. We
will say that ℰ has a maximal order jumping line if it has a jumping line of order n.
Our goal is to show that such an ℰ is unobstructed in the moduli space of stable rank
two bundles, i.e., H2(ℰ⊗ℰ) = 0. The technique can be slightly extended. We show
that when c2= 5, any ℰ with a jumping line of order 4 is unobstructed. We
describe at the end how mathematical instantons with maximal order jumping
lines arise and estimate the dimension of this particular smooth locus of
bundles.
