on a lower bound related to albanese map

Question

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 $g>3$

Answer 1

One can always take $n=2$ and this is the best possible:

Given a principally polarised abelian variety $(A, \Theta)$ of dimension $g$, if $S$ is the intersection of $g-2$ general elements of the linear system $|3\Theta|$, then $S$ is a smooth surface. The Lefschetz hyperplane section theorem implies that the natural map $H_1(S, \mathbb{Z}) \to H_1(A, \mathbb{Z})$ is an isomorphism which in turn implies that $Alb(S) \to A$ is an isomorphism.

(In the above I work over $\mathbb{C}$ but the conclusion holds over any infinite field; for finite fields one might have to replace $3$ by a larger integer.)