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Varieties swept out by grassmannians of lines

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 Publication date 2009
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and research's language is English




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We classify complex projective varieties of dimension $2r geq 8$ swept out by a family of codimension two grassmannians of lines $mathbb{G}(1,r)$. They are either fibrations onto normal surfaces such that the general fibers are isomorphic to $G(1,r)$ or the grassmannian $mathbb{G}(1,r+1)$. The cases $r=2$ and $r=3$ are also considered in the more general context of varieties swept out by codimension two linear spaces or quadrics.



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In this article, we present a massively parallel framework for computing tropicalizations of algebraic varieties which can make use of finite symmetries. We compute the tropical Grassmannian TGr$_0(3,8)$, and show that it refines the $15$-dimensional skeleton of the Dressian Dr$(3,8)$ with the exception of $23$ special cones for which we construct explicit obstructions to the realizability of their tropical linear spaces. Moreover, we propose algorithms for identifying maximal-dimensional tropical cones which belong to the positive tropicalization. These algorithms exploit symmetries of the tropical variety even though the positive tropicalization need not be symmetric. We compute the maximal-dimensional cones of the positive Grassmannian TGr$^+(3,8)$ and compare them to the cluster complex of the classical Grassmannian Gr$(3,8)$.
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