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Results tagged with abelian-varieties
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user 10610
Abelian varieties are projective algebraic varieties endowed with an Abelian group structure. Over the complex numbers, they can be described as quotients of a vector space by a lattice of full rank. They are analogs in higher dimensions of elliptic curves, and play an important role in algebraic geometry and number theory.
8
votes
Accepted
Quotient of an abelian surface by an antisymplectic involution
Denote by $\sigma$ the involution on the abelian surface $A$ and set $X:=A/\sigma$. The eigenvalues of the action of $\sigma$ on $H^0(\Omega^1_A)$ are $+1$ and $-1$.
So $\sigma$ has no isolated fixed …
- 6,253
13
votes
Accepted
General curves of genus 3 as plane sections of Kummer surfaces
Let $C$ be a non hyperelliptic curve of genus 3, let $f\colon C'\to C$ be an e'tale double cover and let $A$ be the Prym variety of $f$. Then $A$ is a principally polarized surface and the Abel-Prym …
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8
votes
Accepted
Can the Albanese map be anything?
Smooth hypersurfaces of fixed degree $d$ in $\mathbb P^n$ ($n\ge 3$) are simply connected, so they have trivial Albanese, and are of general type for $d>n+1$. The Hilbert polynomial is determined by …
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8
votes
Which curves can be found on Abelian varieties?
Gian Pietro Pirola in [Curves on generic Kummer varieties,
Duke Math. J. Volume 59, Number 3 (1989), 701-708] proves a rigidity theorem for curves of genus $g\le q-2$ in the Kummer variety of a $q …
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5
votes
Accepted
Properties of subvarieties of a simple abelian variety
At least over the complex numbers, $X$ is of general type by an old result of Ueno (see Damian's comment below) that says the following:
Let $E$ be the biggest abelian subvariety of $A$ such that $X …
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