A classical Reye congruence
X
X
is an Enriques surface of rational equivalence class
(
3
,
7
)
(3,7)
in the grassmannian
G
(
1
,
3
)
G(1,3)
of lines of
P
3
{{\mathbf {P}}^3}
.
X
X
is the locus of lines of
P
3
{{\mathbf {P}}^3}
which are included in two quadrics of
W
=
W=
web of quadrics. A generalization to
G
(
1
,
t
)
G(1,t)
is given (1) for each
t
>
2
t > 2
there exist Enriques surfaces
X
X
of class
(
t
,
3
t
−
2
)
(t,3t - 2)
in
G
(
1
,
t
)
G(1,t)
, (2) the determinant of the dual of the universal bundle on
X
X
is
O
X
(
2
E
+
R
+
K
X
)
{\mathcal {O}_X}(2E + R + {K_X})
, with
E
=
E=
isolated elliptic curve,
R
2
=
−
2
{R^2} = - 2
,
E
⋅
R
=
t
E \cdot R = t
, (3)
X
X
parameterizes lines of
P
t
{{\mathbf {P}}^t}
which are included in a codimension
2
2
subsystem of
W
W
,
W
=
W=
linear system of quadrics of dimension
(
t
2
)
\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
10
in
P
5
{{\mathbf {P}}^5}
.