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Remarks on \(r\)-planes in complete intersections. (English) Zbl 1372.14003
Consider smooth complete intersections of dimension \(n\) and multidegree \(d = (d_1 , \dots , d_{N-n} )\) in \(\mathbb P^{N}\). The author is interested in the scheme \(H_r\) of such complete intersections which contain a linear space of dimension \(r\). He notes that if \( n \leq 2r -1 \) then Lefschetz theorem implies that \(H_r = \emptyset \). He then proves for \( n \geq 2r \) that \(H_r \neq \emptyset \) and he computes its dimension. Under the same assumption, he further shows that if a second inequality holds, which is a function of the multidegree \(d\) and of \(r\), then every complete intersection of type \(d\) is covered by \(r\)-linear spaces. The method of proof is based on the study of the tangent spaces to the incidence correspondences which naturally appear in the description of the geometry of the problem.

14C05 Parametrization (Chow and Hilbert schemes)
14M10 Complete intersections
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