On the second section of a rank 2 reflexive sheaf on \(\mathbb{P}^ 3\) (Q1911634)

Let \(F\) be a normalized rank-2 reflexive sheaf on the three-dimensional projective space \(P\) (over an algebraically closed field of characteristic 0). Denote by \(\alpha\) the first integer such that \(H^0 F(\alpha)\neq 0\). Any non-zero section of \(H^0 F(\alpha)\) defines a locally Cohen-Macaulay curve \(C\), which is locally a complete intersection except at finitely many closed points. On the other hand, a general section \(s\) of \(H^0 F(m)\), where \(\alpha<m\), may not define a curve since the locus of zeros of such a section may include a two-dimensional component. The authors study the question of how far we twist, beyond \(\alpha\), to guarantee a one-dimensional locus. They define \(\beta\) to be the first integer \(n\) such that \(h^0 F(n)> h^0 {\mathcal O}_P(n-\alpha)\). This \(\beta\) is precisely the first integer such that the general section of \(H^0 F(n)\) defines a curve, for every \(n\geq\beta\). Denoting by \(a\) the first integer such that the restriction of \(F\) to a general plane in \(P\) has a non-zero section, they formulate the main results: (i) if \(F\) is a semistable sheaf, with \(a=\alpha\), then we have \[ \beta\leq \sqrt{6(c_2+\alpha c_1+\alpha^2)+1}- 3\alpha/2-1-c_1/4; \] (ii) if \(F\) is semistable, with \(a<\alpha\), then \(\beta\leq \sqrt{6(c_2+\alpha c_1+\alpha^2)+1}-1\). In addition, the case when \(F\) is nonstable is also considered leading to some cubic equations.