The space of lines in cyclic covers of projective space (Q2001440)

Let \(W\) be a smooth complex projective variety of dimension \(n\), endowed with an ample line bundle \(h\). A line of \((W, h)\) is a rational curve in \(W\) of \(h\)-degree \(1\). Let \(F(W)\) be the space of lines of \((W, h)\). If non-empty, its dimension is \(\dim(F(W)) \leq\exp\dim (F(W))= (-K_W) \cdot \ell + n - 3\), where \(\ell\) is a line of \((W, h)\), and it is of interest to know when this is an equality. This happens e.g., for a general hypersurface \(W \subset \mathbb P^{n+1}\) of degree \(d > 2n-1\), with \(h\) being the hyperplane bundle. In the paper under review, the authors consider the space of lines in cyclic covers of projective spaces and prove the following result. Let \(m, n, d\) be positive integers such that \(md > 2n-3\) and \(k := 2(n-1)-d(m-1) \geq 0\), let \(w : Y \to \mathbb P^n\) be a cyclic cover of degree \(m\), branched along a general hypersurface \(X \subset \mathbb P^n\) of degree \(md\) and \(h = w^*\mathcal O_{\mathbb P^n}(1)\). Then \(F(Y)\) is smooth of dimension \(k\) and irreducible if \(k \geq 1\); in particular, \(F(Y)\) has the expected dimension. For \(n = 3, m = d = 2\), this was already proven by [\textit{A. S. Tikhomirov} [Izv. Akad. Nauk SSSR, Ser. Mat. 44, 415--442 (1980; Zbl 0434.14023)]. The result is proven by connecting the lines of \((Y, h)\) to the \(m\)-contact order lines in \(X\), i.e., lines \(\ell \subset \mathbb P^n\) whose local intersection number with \(X\) at each point of \(\ell \cap X\) is a multiple of \(m\). Actually, the authors study the subvariety \(S_m(X)\) of the Grassmannian \(G\) of lines of \(\mathbb P^n\) parameterizing such lines, proving that it is smooth of dimension \(k\) and show that \(F(Y)\) is an unramified cover of \(S_m(X)\) of degree \(m\). Furthermore, when \(k = 0\), they obtain an explicit enumerative formula expressing the number of \(m\)-contact order lines of \(X\) in terms of intersection numbers of Schubert cycles on \(G\).