Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/9805115

A Series of Smooth Irregular Varieties in Projective Space

C. Ciliberto·K. Hulek
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

C. Ciliberto, K. Hulek. 1999-03-04. A Series of Smooth Irregular Varieties in Projective Space. https://arxiv.org/abs/math/9805115

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Wall divisors on irreducible symplectic orbifolds of Nikulin-type

We determine the wall divisors on irreducible symplectic orbifolds which are deformation equivalent to a special type of examples, called Nikulin orbifolds. The Nikulin orbifolds are obtained as partial resolutions in codimension 2 of a quotient by a symplectic involution of a Hilbert scheme of 2 points on a K3 surface. This builds on the previous article arXiv:2009.04873 in which the theory of wall divisors was generalized to orbifold singularities.

math.AG↗
arXiv2026

Prekosmic Grothendieck/Galois Categories

We establish a generalized version of the duality between groups and the categories of their representations on sets. Given an abstract symmetric monoidal category $K$ called Galois prekosmos, we define pre-Galois objects in $K$ and study the categories of their representations internal to $K$. The motivating example of $K$ is the cartesian monoidal category $\textit{Set}$ of sets, and pre-Galois objects in $\textit{Set}$ are groups. We present an axiomatic definition of pre-Galois $K$-categories, which is a complete abstract characterization of the categories of representations of pre-Galois objects in $K$. The category of covering spaces over a well-connected topological space is a prototype of a pre-Galois $\textit{Set}$-category. We establish a perfect correspondence between pre-Galois objects in $K$ and pre-Galois $K$-categories pointed with pre-fiber functors. We also establish a generalized version of the duality between flat affine group schemes and the categories of their linear representations. Given an abstract symmetric monoidal category $K$ called Grothendieck prekosmos, we define what are pre-Grothendieck objects in $K$ and study the categories of their representations internal to $K$. The motivating example of $K$ is the symmetric monoidal category $\textit{Vec}_k$ of vector spaces over a field $k$, and pre-Grothendieck objects in $\textit{Vec}_k$ are affine group $k$-schemes. We present an axiomatic definition of pre-Grothendieck $K$-categories, which is a complete abstract characterization of the categories of representations of pre-Grothendieck objects in $K$. The indization of a neutral Tannakian category over a field $k$ is a prototype of a pre-Grothendieck $\textit{Vec}_k$-category. We establish a perfect correspondence between pre-Grothendieck objects in $K$ and pre-Grothendieck $K$-categories pointed with pre-fiber functors.

math.AG↗