ag.algebraic geometry – embedding abelian varieties in projective spaces of small dimension

Given a (complex) Abelian variety $ A $ a fixed dimension $ g $, To let $ d (A) $ Be the dimension of the smallest complex projective space in which it is embedded.

is $ d (A) $ uniform over all abelian varieties of a solid $ g $? Or are there special ones that can be embedded in even smaller projective rooms?

Can $ d (A) $ be calculated explicitly? I am especially interested in the case $ g = 2 $,