One of the simplest examples of a smooth, non degenerate surface in P^4 is
the quintic elliptic scroll. It can be constructed from an elliptic normal
curve E by joining every point on E with the translation of this point by a
non-zero 2-torsion point. The same construction can be applied when E is
replaced by a (lineaerly normally embedded) abelian variety A. In this paper we
ask the question when the resulting scroll Y is smooth. If A is an abelian
surface embedded by a line bundle L of type (d_1,d_2) and r=d_1d_2, then we
prove that for general A the scroll Y is smooth if r is at least 7 with the one
exception where r=8 and the 2-torsion point is in the kernel K(L) of L. In this
case Y is singular.The case r=7 is particularly interesting, since then Y is a
smooth threefold in P^6 with irregularity 2. The existence of this variety
seems not to have been noticed before. One can also show that the case of the
quintic elliptic scroll and the above case are the only possibilities where Y
is smooth and the codimension of Y is at most half the dimension of the
surrounding projective space.Comment: 23 pages, Plain Tex. Some corrections made. To appear: Annali Sc.
Norm. Sup. di Pis
