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The distribution of bidegrees of smooth surfaces in $$Gr(1,\mathbb{P}^ 3)$$. (English) Zbl 0741.14017
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.

##### MSC:
 14J25 Special surfaces 14M15 Grassmannians, Schubert varieties, flag manifolds 14J99 Surfaces and higher-dimensional varieties
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##### References:
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