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Reye constructions for nodal Enriques surfaces


Authors: A. Conte and A. Verra
Journal: Trans. Amer. Math. Soc. 336 (1993), 79-100
MSC: Primary 14J28; Secondary 14J60
DOI: https://doi.org/10.1090/S0002-9947-1993-1079052-5
MathSciNet review: 1079052
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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}$ .


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1079052-5
Article copyright: © Copyright 1993 American Mathematical Society

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