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Covered by lines and conic connected varieties. (English) Zbl 1256.14052
Let \(X\) be a smooth irreducible variety of dimension \(n\) in \(\mathbb{P}^{N}\) defined over the field of complex numbers. We set \(c:=N-n\). Let \(L\) be an irreducible component of the Hilbert scheme of lines of \(X\) and we denote by \(L_{x}\) the variety of lines from \(L\) passing through \(x\). It is said that \(X\) is covered by the lines in \(L\) if \(L_{x}\not=\emptyset\) for general \(x\in X\). Set \(a=\dim L_{x}\). Then \(a=\deg N_{l/X}\), where \(l\) is a line from \(L\) and \(N_{l/X}\) is its normal bundle.
First, the authors prove the following: Let \(X\subset \mathbb{P}^{N}\) be a variety covered by an irreducible family of lines \(L\). If \(a\geq n-c\), then \(a\leq \frac{n+c-3}{2}\). Furthermore if this equality holds, then \(X\) is dual defective and \(\dim X=\dim X^{*}\), where \(X^{*}\) is the dual variety of \(X\). As an application of this result, the authors show that prime Fano varieties of high index are quite special (see Proposition 3.6).
Next the authors study conics on \(X\) and obtained the following result (see Theorem 4.3): Let \(X\subset \mathbb{P}^{N}\) be a variety set theoretically defined by homogeneous polynomials \(G_{i}\) of degree \(d_{i}\) for \(i=1, \dots , m\). If \(\sum_{i=1}^{m}d_{i}\leq \frac{N+m}{2}\), then \(X\) is connected by singular conics. Assume that \(X\) is smooth and the equations \(G_{i}\) are scheme theoretical equations for \(X\) and in decreasing order of degrees. If \(\sum_{i=1}^{c}d_{i}\leq \frac{N+c}{2}\), then \(X\) is connected by smooth conics.

14M22 Rationally connected varieties
14N05 Projective techniques in algebraic geometry
14J45 Fano varieties
14M07 Low codimension problems in algebraic geometry
14C05 Parametrization (Chow and Hilbert schemes)
14M10 Complete intersections
14J10 Families, moduli, classification: algebraic theory
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