The Hilbert Scheme of Lines Contained in a Variety and Passing Through a General Point

Abstract

We recall without proofs the basic definitions and results, leading to the construction of various Hilbert Schemes and describing the infinitesimal properties of these parameter spaces. Then we introduce and study in great detail the Hilbert scheme of lines passing through a (general) point x ∈ X of a projective variety \(X \subset \mathbb{P}^{N}\) and contained in it, indicated by \(\mathcal{L}_{x}\). We included all the details and proofs concerning the geometrical properties of \(\mathcal{L}_{x}\) inherited from X. In particular, we analyze the singularities of \(\mathcal{L}_{x}\) and the relations between the equations defining scheme-theoretically \(X \subset \mathbb{P}^{N}\) and those defining \(\mathcal{L}_{x} \subset \mathbb{P}^{n-1}\). As an application we treat from a new perspective the classical problem of the existence of projective extensions \(X \subset \mathbb{P}^{N+1}\) of a manifold \(Y \subset \mathbb{P}^{N} \subset \mathbb{P}^{N+1}\). In particular, we show in an elementary way that a lot of homogeneous varieties admit only trivial extensions, that is, those obtained by taking the cone S(p, Y ) with \(p \in \mathbb{P}^{N+1}\setminus\), by constructing explicitly the cone via the study of the singularities of the corresponding Hilbert schemes of lines through a general point of X.