On the Hilbert scheme of degeneracy loci of twisted differential forms (Q2790631)

In the interesting and well written paper under review, the author studies degeneracy loci of morphisms of the form \(\phi: \mathcal{O}_{\mathbb{P}^{n-1}}^m \to \Omega_{\mathbb{P}^{n-1}}(2)\) in the case \(2 < m < n-1\). The interest in these degeneracy loci goes back a long time ago. Indeed, many classical varieties arise as degeneracy locus of such morphisms, for instance the projected Veronese surface in \(\mathbb{P}^4\), the elliptic scroll surface of degree six, the Palatini scroll and the Segre cubic primal.NEWLINENEWLINEConsider an algebraically closed field \(\mathbf{k}\) of characteristic 0, and let \(U\) and \(V\) be two \(\mathbf{k}\)-vector spaces of dimension \(m\) and \(n\) with bases \(\{y_0,\ldots,y_{m-1}\}\) and \(\{x_0,\ldots,x_{n-1}\}\). Every morphism \(\phi\) can be naturally associated to a \(n \times n\) skew-symmetric matrix \(N\) whose entries \(N_{ij} = \sum_{k=0}^{m-1} a_{i,j}^k y_k\) (\(a_{i,j}^k = - a_{j,i}^k\) for all \(i,j,k\)) are linear forms in the coordinates of the projective space \(\mathbb{P}^{m-1} = \mathbb{P}(U)\) or, equivalently, to \(m\) elements of the vector space \(\bigwedge^2 V\). The linear group \(\text{GL}(U)\) acts projectively on the \(m\) elements not affecting the subspace they span, so the \(\text{GL}(U)\)-orbit of a general morphism \(\phi\) corresponds to an element of the Grassmannian \(\mathbf{Gr}(m,\bigwedge^2 V)\).NEWLINENEWLINEMoreover, the degeneracy locus of a general morphism \(\phi\) corresponds to the subscheme \(X_{\phi} \subset \mathbb{P}^{n-1} = \mathbb{P}(V)\) cut out by the maximal minors of the \(n\times m\) matrix NEWLINE\[NEWLINE M = \left(\begin{alignedat}{2} \sum_{i=0}^{n-1} a_{i,0}^0 x_i & \dots & \sum_{i=0}^{n-1} a_{i,0}^{m-1} x_i\\ \vdots && \vdots \\ \sum_{i=0}^{n-1} a_{i,n-1}^0 x_i & \dots & \sum_{i=0}^{n-1} a_{i,n-1}^{m-1} x_i \end{alignedat}\right). NEWLINE\]NEWLINE As the Hilbert polynomial of \(X_{\phi}\) is generically fixed, and the action of \(\text{GL}(U)\) induced on \(M\) does not change the ideal generated by the maximal minors defining \(X_{\phi}\), we have rational maps NEWLINE\[NEWLINE \text{Hom}\big(\mathcal{O}_{\mathbb{P}(V)}^m,\Omega_{\mathbb{P}(V)}(2)\big) \dashrightarrow \mathbf{Gr}(m,\bigwedge^2 V) \overset{\rho}{\dashrightarrow} \mathcal{H}, NEWLINE\]NEWLINE where \(\mathcal{H}\) is the union of the components of the Hilbert scheme containing the degeneracy locus of a general morphism \(\phi\).NEWLINENEWLINEIn the paper under review, the author gives a complete description of the behavior of the map \(\rho\) in the case \(2 < m < n-1\). Precisely, {\parindent=0.6cm\begin{itemize}\item[--] for \(m \geq 4\) or \((m,n)=(3,5)\), the map \(\rho\) is birational; \item[--] for \(m=3\) and \(n\neq 6\), the map \(\rho\) is generically injective and dominant on a closed subscheme \(\mathcal{H}'\subset \mathcal{H}\) of codimension \(\frac{1}{8}n(n-3)(n-5)\) if \(n\) odd or \(\frac{3}{8}(n-4)(n-6)\) if \(n\) even. NEWLINENEWLINE\end{itemize}} In the case \((m,n) = (3,6)\), the map \(\rho\) is not generically injective, as proved in [\textit{D. Bazan} and \textit{E. Mezzetti}, Geom. Dedicata 86, No. 1--3, 191--204 (2001; Zbl 1042.14022)]. The case \(m=2\) was treated by the same author in [Arch. Math. 105, No. 2, 109--118 (2015; Zbl 1349.14015)].