TY - JOUR

T1 - A Series of Smooth Irregular Varieties in Projective Spaci

AU - Ciliberto, Ciro

AU - Hulek, Klaus

N1 - Publisher Copyright: © 1999 Scuola Normale Superiore. All rights reserved.

PY - 1999

Y1 - 1999

N2 - One of the simplest examples of a smooth, non degenerate surface in 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 (linearly 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 (Equation Presented) and (Equation Presented), 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 F is a smooth threefold in 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.

AB - One of the simplest examples of a smooth, non degenerate surface in 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 (linearly 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 (Equation Presented) and (Equation Presented), 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 F is a smooth threefold in 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.

UR - http://www.scopus.com/inward/record.url?scp=84877718039&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84877718039

VL - 28

SP - 357

EP - 380

JO - Annali della Scuola normale superiore di Pisa - Classe di scienze

JF - Annali della Scuola normale superiore di Pisa - Classe di scienze

SN - 0391-173X

IS - 2

ER -