31 research outputs found
A note on Galois embeddings of abelian varieties
In this note we show that if an abelian variety possesses a Galois embedding
into some projective space, then it must be isogenous to the self product of an
elliptic curve. We prove moreover that the self product of an elliptic curve
always has infinitely many Galois embeddings.Comment: Some typos fixed. To appear in Manuscripta Mathematic
Fixed points of endomorphisms of complex tori
We study the asymptotic behavior of the cardinality of the fixed point set of
iterates of an endomorphism of a complex torus. We show that there are
precisely three types of behavior of this function: it is either an
exponentially growing function, a periodic function, or a product of both.Comment: Some typos corrected; the introduction was also revise
Galois subspaces for projective varieties
Given an embedding of a projective variety into projective space, we study
the structure of the space of all linear projections that, when composed with
the embedding, give a Galois morphism from the variety to a projective space of
the same dimension.Comment: 14 pages, any comments welcome
The Gauss map and secants of the Kummer variety
Fay's trisecant formula shows that the Kummer variety of the Jacobian of a
smooth projective curve has a four dimensional family of trisecant lines. We
study when these lines intersect the theta divisor of the Jacobian, and prove
that the Gauss map of the theta divisor is constant on these points of
intersection, when defined. We investigate the relation between the Gauss map
and multisecant planes of the Kummer variety as well.Comment: Minor changes, to appear on the Bulletin of London Mathematical
Societ
Smooth quotients of abelian surfaces by finite groups
Let be an abelian surface and let be a finite group of automorphisms
of fixing the origin. Assume that the analytic representation of is
irreducible. We give a classification of the pairs such that the
quotient is smooth. In particular, we prove that with an
elliptic curve and that in all cases. Moreover, for
fixed , there are only finitely many pairs up to isomorphism. This
completes the classification of smooth quotients of abelian varieties by finite
groups started by the first two authors.Comment: 15 pages. Comments are welcome. arXiv admin note: text overlap with
arXiv:1801.0002
A decomposition of the Jacobian of a Humbert-Edge curve
A \textit{Humbert-Edge curve of type} is a non-degenerate smooth complete
intersection of diagonal quadrics. Such a curve has an interesting
geometry since it has a natural action of the group
. We present here a decomposition of its Jacobian
variety as a product of Prym-Tyurin varieties, and we compute the kernel of the
corresponding isogeny.Comment: 9 pages, comments welcome! To appear in Contemporary Mathematic