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Varieties connected by chains of lines. (English) Zbl 1285.14054
The main result of the paper is the following:
Theorem: Let \(X \subset \mathbb{P}^n\) be a variety set theoretically defined by homogeneous polynomials \(G_i\) of degree \(d_i, i=1, \dots, m\), and let \(l\) be an integer. If \[ \sum_i d_i \leq \frac{N(l-1)+m}{l}, \] then \(X\) is rationally chain connected by chains of lines of length at most \(l\).
In particular if \(X\) is smooth and the above inequality is satisfied then \(X\) is rationally connected by rational curves of degree at most \(l\).

MSC:
14M22 Rationally connected varieties
14N05 Projective techniques in algebraic geometry
14J45 Fano varieties
14M07 Low codimension problems in algebraic geometry
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