Covered by lines and conic connected varieties.

*(English)*Zbl 1256.14052Let \(X\) 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.

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.

Reviewer: Yoshiaki Fukuma (Kochi)

##### MSC:

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 |