The relative Hilbert scheme of projection morphisms (Q2796983)

Let \(Y \subset \mathbb P^n\) be a smooth hypersurface of degree \(d\). For a point \(y \not \in Y\) the projection \(\pi: Y \to \mathbb P^{n-1}\) gives a finite morphism of degree \(d\) and one can consider the relative Hilbert scheme \(\text{Hilb}_y^{[m]} (Y/\mathbb P^{n-1})\) parametrizing zero-dimensional subschemes of length \(m\) contained in the fibers of the projection for each \(1 \leq m \leq d\). It is easy to see that \(\text{Hilb}_y^{[m]} (Y/\mathbb P^{n-1})\) is isomorphic to \(\mathbb P^{n-1}\) for \(m=d\) and to \(Y\) for \(m=1\) or \(m=d\), so the author focuses on the intermediate values \(1 < m < d\). Here it is known from work of \textit{L. Gruson} and \textit{C. Peskine} that \(\text{Hilb}_y^{[m]} (Y/\mathbb P^{n-1})\) is a smooth connected projective variety of dimension \(n-1\) for general \(y \not \in Y\) [Duke Math. J. 162, No. 3, 553--578 (2013; Zbl 1262.14058)]. In the paper under review the author gives an explicit embedding \(j:\text{Hilb}_y^{[m]} (Y/\mathbb P^{n-1}) \hookrightarrow \mathbb P\) into a weighted projective space for \(n > 1\) and gives gives equations defining the image \(X \subset \mathbb P\). The equations show that \(X \subset \mathbb P\) is a smooth weighted complete intersection, and hence results of \textit{I. Dolgachev} [Lect. Notes Math. 956, 34--71 (1982; Zbl 0516.14014)] yield the generating function for the dimensions of the graded pieces of \(\oplus_{k=0}^\infty H^0(X, {\mathcal O}_X (k))\), the dualizing sheaf \(\omega_X\), and the Picard group for \(n \geq 4\).