The distribution of bidegrees of smooth surfaces in \(Gr(1,\mathbb{P}^ 3)\) (Q1183058)

The bidegree of a surface \(Y\subseteq Gr(1,\mathbb{P}^ 3)\) is the class of the surface in the codimension 2 Chow ring \(A^ 2Gr(1,\mathbb{P}^ 3)\cong\mathbb{Z} \eta\oplus\mathbb{Z} \eta'\), where \(\eta\) and \(\eta'\) are classes of the two families of planes in \(Gr(1,\mathbb{P}^ 3)\). In this paper we prove that if \(Y\subseteq Gr(1,\mathbb{P}^ 3)\) is a smooth surface of bidegree \((a,b)\), then if \(Y\) is not of general type, \(a\leq 3b\) and by symmetry, \(b\leq 3a\). If \(Y\) is of general type, then we show \(a\leq O(b^{4/3})\). Our method is to study the stability of the universal rank 2 bundle \({\mathcal E}\) on \(Gr(1,\mathbb{P}^ 3)\) restricted to \(Y\). I. Dolgachev and I. Reider have conjectured this restriction is semistable if \(Y\) is non-degnerate, and this implies \(a\leq 3b\). We show that if \({\mathcal E}\mid_ Y\) is unstable, then we get a strong bound on the hyperplane section genus of \(Y\), and this enables us to conclude our theorem.