As we know, the albanese map $Alb$ assoicates a smooth proper variety $X$ of dimension $n$ to an abelian variety $Alb(X)$ of dimension $g=H^0(X,\Omega^1_X)$. Another well known fact is the moduli spaces $A_g$, $g\leq 3$ have dense subsets consisting of $Alb(X)$ for those $X$ satisfying $n=1$, which is no longer true for $g>3$. My questions are

1. for $g>3$, could one always find , s.t. $A_g$ contains a dense subset consisting of $Alb(X)$, where $X$ satisfies dim $X\leq n$ and $g=H^0(X,\Omega^1_X)$.

2. could anyong give explicitly of a lower bound of such $n$ for some $g>3$

As we know, the albanese map $Alb$ assoicates a smooth proper variety $X$ of dimension $n$ to an abelian variety $Alb(X)$ of dimension $g=H^0(X,\Omega^1_X)$. Another well known fact is the moduli spaces $A_g$, $g\leq 3$ have dense subsets consisting of $Alb(X)$ for those $X$ satisfying $n=1$, which is no longer true for $g>3$. My questions are

1. for $g>3$, could one always find $n$ < $g$, s.t. $A_g$ contains a dense subset consisting of $Alb(X)$, where $X$ satisfies dim $X\leq n$ and $g=H^0(X,\Omega^1_X)$.

2. could anyong give explicitly a lower bound of such $n$ for some $g>3$