Vector bundles of rank 2 on rational ruled surfaces and on \(\mathbb{P}_k^n\times\mathbb{P}_k^m\) (Q1293326)

Let \(E\) be a topologically trivial algebraic rank 2 vector bundle on \(\mathbb{P}^2\) and \(d\geq 0\) its degree of instability, i.e. the largest integer for which \(H^0E(-d)\neq 0\). If \(d=0\) then \(E\) is (algebraically) trivial. The vector bundles of this kind are parametrized by a coarse moduli space \(M(d)\) which is irreducible, non-singular, quasi-projective, rational, of dimension \(3d^2-1\) for \(d\geq 1\). \textit{U. Schafft} [J. Reine Angew. Math. 338, 136-143 (1983; Zbl 0491.14012)] and \textit{C. Bănică} [J. Reine Angew. Math. 344, 102-119 (1983; Zbl 0512.32022)] showed, independently, that if \(d=1\) or 2 then \(E\) can be deformed to the trivial bundle. This result has been substantially generalized by \textit{S. A. Strømme} [Math. Ann. 263, 385-397 (1983; Zbl 0497.14005)] who showed that the subset of \(M(d)\) corresponding to the vector bundles which can be deformed to the trivial one is closed, of codimension \((d-1)(d-2)\). In fact, Strømme allows any Chern classes \(c_1\in \{-1,0\}\) and \(c_2\) and determines the codimension of the subset of \(M(d;c_1,c_2)\) corresponding to vector bundles which can be deformed to vector bundles with degree of instability \(d'<d\). ln this paper, the author adapts Strømme's method and results to the case of rank 2 vector bundles on a rational ruled surface. He has, of course, to overcome some technical difficulties. In particular, he shows that the topologically trivial algebraic rank 2 vector bundles on \(\mathbb{P}^1\times\mathbb{P}^1\) of bidegree of instability \((d_1,d_2)\) which can be deformed to the (algebraically) trivial vector bundle correspond to a closed subset of \(M(d_1,d_2)\) of codimension \(\max(2(d_1-1)(d_2-1)-1,0)\). Finally, I would like to mention that the problem of the existence of nontrivial rank 2 vector bundles on \(\mathbb{P}^3\) which can be deformed to the trivial one is still open in characteristic 0. In positive characteristic, there are some examples of \textit{N. Mohan Kumar} [Invent. Math. 104, 313-319 (1991; Zbl 0739.14013)].