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Surfaces of degree 10 in $$Gr(1,\mathbb{P}{}^ 3)$$. (English) Zbl 0761.14012
We consider smooth algebraic surfaces in $$Gr(1,\mathbb{P}^ 3)$$, the Grassmannian of lines in $$\mathbb{P}^ 3$$. Surfaces up to degree 9 via the Plücker embedding have been previously classified. We extend this classification up to degree 10, giving constructions of each possible case. Degree 10 is sufficiently complex as to require some new techniques. While many of these surfaces can be constructed via liaison or as degeneracy loci of vector bundles, several require more delicate constructions. We show how to embed one rational surface in $$Gr(1,\mathbb{P}^ 3)$$ by constructing a suitable rank 2 vector bundle on the surface, and we construct another surface by a delicate projection from a singular surface in $$\mathbb{P}^ 8$$.

##### MSC:
 14J25 Special surfaces 14M15 Grassmannians, Schubert varieties, flag manifolds
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