A subvariety $V$ of an abelian variety $X$ is *geometrically nondegenerate* if it meets any subvariety of $X$ of dimension bigger than or equal $codim(V)$. 


My question is about the Prym varieties as subvarieties of the Jacobians of curves. Are they geometrically nondegenerate? 



Are the Prym varieties geometrcally nondegenerate subvarieties of the Jacobians?
Pryms are geometrically degenerate. Let $f:Y\to X$ be an étale double cover of a genus $g>1$ curve $X$. Up to taking isogenous quotients, which will not affect "geometric degeneracy", the Prym of $f$ is the kernel of $\text{Alb}\_f : \text{Alb}\_Y\to \text{Alb}\_X$, where the domain is an Abelian variety of dimension $2g-1$, and where the target is an Abelian variety of dimension $g$. The Prym has dimension $g-1$, so codimension $g$ in $\text{Alb}\_Y$. So now let $C\subset \text{Alb}\_X$ be any curve that does not contain the origin. Then the inverse image $\text{Alb}\_f^{-1}(C)$ is a subvariety of $\text{Alb}\_Y$ of dimension $g$ that is disjoint from the Prym. 


