1998/05/26 by Ciro Ciliberto, C. Ciliberto, Klaus Hulek +3
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #FOS: Mathematics #Nonlinear Waves and Solitons #math.AG
paper · pdf · doi:10.48550/arxiv.math/9805115
23 pages, Plain Tex. Some corrections made. To appear: Annali Sc. Norm. Sup. di Pisa
openalex publication_date 1998/05/26 · arxiv created 1999/03/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
One of the simplest examples of a smooth, non degenerate surface in P4 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 (d1,d2) and r=d1d2, 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 P6 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.