Lines on projective varieties and applications
This paper gives a review on the Hilbert scheme \(\mathcal{L}_{x,X}\) of lines contained in a smooth projective variety \(X\subset \mathbb{P}^N\) and passing through a general point \(x\in X\). In particular the author focuses on the relation between the properties of \(\mathcal{L}_{x,X}\) and those of the variety \(X\). In the case of varieties defined by quadratic equations, he proves that \(\mathcal{L}_{x,X}\) coincides with the base locus of the second fundamental form of \(X\) at \(x\). Moreover the behaviour of the Hilbert scheme of lines through a general point is studied with respect to projective extensions. Given a projective variety \(X\subset \mathbb{P}^N\), an extension \(Y\subset\mathbb{P}^{N+1}\) of \(X\) is a variety such that \(X\) can be obtained as hyperplane section of \(Y\). In this case also the Hilbert scheme \(\mathcal{L}_{x,Y}\) is an extension of \(\mathcal{L}_{x,X}\), for \(x\in X\) general. This fact allows the author to describe a sufficient condition for the non-existence of non-trivial extensions via Hilbert schemes and to deduce some applications.
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