Geometry of the genus 9 Fano 4-folds
The symplectic Grassmannian \(G_w = G_w(3,6)\) is the variety of all 3-spaces in \({\mathbb{C}}^6\) where a non-degenerate 3-form \(w\) vanishes. \(G_w\) is a smooth Fano 6-fold with \(\mathrm{Pic}(G_w) = {\mathbb{Z}}H\), and canonical class \(K_{G_w} = -4H\). The ample generator \(H\) embeds \(G_w\) in \({\mathbb{P}}^{13}\) as a smooth 6-fold of degree 16. In this paper the author studies the geometry of the general linear section \(B\) of \(G_w\) with a codimension 2 subspace \({\mathbb{P}}^{11}\) of \({\mathbb{P}}^{13}\). In Section 2 is shown that the family of lines \(F_B\) of \(B\) is a linear section of the Segre 4-fold \({\mathbb{P}}^1 \times {\mathbb{P}}^1 \times {\mathbb{P}}^1 \times {\mathbb{P}}^1\). In Section 3 is computed the Chow ring of \(B\). In Sections 1 and 4 are constructed four rank two vector bundles \(E_i\) on \(B\) such that \(E_i(1)\) give embeddings \(f_i\) of \(B\) in the Grassmannian \(G(2,6)\). One interesting property of these embeddings is that the isomorphic images of \(B\) in \(G(2,6)\) under \(f_i\), \(i = 1,\dots,4\) can be realized as congruences of lines described by \textit{E. Mezzetti} and \textit{P. De Poi} [Geom. Dedicata 131, 213--230 (2008; Zbl 1185.14042)].
scientific article; zbMATH DE number 5793937
scientific article