Gross, Mark The distribution of bidegrees of smooth surfaces in \(Gr(1,\mathbb{P}^ 3)\). (English) Zbl 0741.14017 Math. Ann. 292, No. 1, 127-147 (1992). 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. Reviewer: M.Gross (Ann Arbor) Cited in 1 ReviewCited in 6 Documents MSC: 14J25 Special surfaces 14M15 Grassmannians, Schubert varieties, flag manifolds 14J99 Surfaces and higher-dimensional varieties Keywords:bidegree of a surface; bound on the hyperplane section genus PDFBibTeX XMLCite \textit{M. Gross}, Math. Ann. 292, No. 1, 127--147 (1992; Zbl 0741.14017) Full Text: DOI EuDML References: [1] Arbarello, E., Cornalba, M., Griffiths, P., Harris, J.: Geometry of algebraic curves. Berlin Heidelberg New York: Springer 1985 · Zbl 0559.14017 [2] Arrondo, E.: On congruences of lines in the projective space. Ph.D. Thesis, Universidad Complutense de Madrid, 1990 [3] Arrondo, E., Sols, I.: Classification of smooth congruences of low degree. J. Reine Angew. 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