Reye constructions for nodal Enriques surfaces

Abstract:

A classical Reye congruence $X$ is an Enriques surface of rational equivalence class $(3,7)$ in the grassmannian $G(1,3)$ of lines of ${{\mathbf {P}}^3}$. $X$ is the locus of lines of ${{\mathbf {P}}^3}$ which are included in two quadrics of $W=$ web of quadrics. A generalization to $G(1,t)$ is given (1) for each $t > 2$ there exist Enriques surfaces $X$ of class $(t,3t - 2)$ in $G(1,t)$, (2) the determinant of the dual of the universal bundle on $X$ is ${\mathcal {O}_X}(2E + R + {K_X})$, with $E=$ isolated elliptic curve, ${R^2} = - 2$, $E \cdot R = t$, (3) $X$ parameterizes lines of ${{\mathbf {P}}^t}$ which are included in a codimension $2$ subsystem of $W$, $W=$ linear system of quadrics of dimension $\left ( \begin {array}{*{20}{c}} t \\ 2 \\ \end {array} \right )$. The paper includes a description of the variety of trisecant lines to a smooth Enriques surface of degree $10$ in ${{\mathbf {P}}^5}$ .