Lines on the Gushel' threefold (Q1344161)

Let \(Y\) be a smooth intersection of the Grassmannian \(G(2,5)\subset \mathbb{P}^9\) with a \(\mathbb{P}^6\subset \mathbb{P}^9\), then the Gushel' threefold \(X\) is a double covering of \(Y\) ramified along a smooth intersection of \(Y\) with a smooth quadric. In this paper the author studies the curve \(\Gamma (X)\) parametrizing the lines contained in \(X\) and their incidence correspondence \(\Sigma\). He proves that the intermediate Jacobian of \(X\), \((J(X), \Theta_X)\), is isomorphic to the Prym variety \((P, \Xi)\) of the pair \((\Gamma (X), \Sigma)\).