Normal bundles of lines on hypersurfaces (Q1980007)

Let \(X\) be a smooth hypersurface of degree \(d\) in \(\mathbb{P}^n_{\mathbb{C}}\). The Fano scheme \(F(X)\) of lines on \(X\) is the family of all lines contained in \(X\). Let \(N = \binom{n +d}{d} -1\), and \(U\) be an open subscheme in \(\mathbb{P}^N_{\mathbb{C}}\) parametrizing degree \(d\) smooth hypersurfaces \(X\) in \(\mathbb{P}^n_{\mathbb{C}}\). The Universal Fano scheme is defined by \[ \Sigma = \lbrace (L, X) \in G (1, n) \times U \vert L \subset X \rbrace. \] \noindent Since the normal bundle \(N_{L/X}\) of line \(L\) in \(X\) is a rank \(n-2\) vector bundle on \(\mathbb{P}^1_{\mathbb{C}} = L\), it can be identified with a direct of sum of line bundles \(\mathcal{O}(a_1) \oplus \cdots \oplus \mathcal{O}(a_{n-2})\), for some nondecreasing integers \(\vec{a} = (a_1, \dots , a_{n-2})\), called the splitting type. We denote \(\mathcal{O}(a_1) \oplus \cdots \oplus \mathcal{O}(a_{n-2})\) by \(\mathcal{O}(\vec{a})\), where \(\vec{a} = (a_1, \dots, a_{n-2})\) with \(a_1 \leq \cdots \leq a_{n-2}\). Let \(F_{\vec{a}}(X) := \{ L \in F(X) \vert N_{X/L} \cong \mathcal{O}(\vec{a}) \}\), and \(\sum_{\vec{a}} := \{ (L, X) \in \sum \vert N_{X/L} \cong \mathcal{O}(\vec{a}) \}.\) The main result of this article concerns the fibres of natural projection map \(\sum_{\vec{a}} \rightarrow \mathbb{P}_{\mathbb{C}}^N\), which are the strata \(F_{\vec{a}}(X)\). More precisely, it is shown that for general hypersurfaces, these strata have the expected dimension. Also in this case, the class of the closure of the strata in the Chow ring of the Grassmannian of lines in projective space is computed. Further, for certain splitting types, the upper bounds on the dimension of the strata that hold for all smooth \(X\), is provided.