Let
A
A
be an
n
n
-dimensional Abelian variety,
n
≥
2
n \geq 2
; let
CH
0
(
A
)
{\text {CH}_0}(A)
be the group of zero-cycles of
A
A
, modulo rational equivalence; by regarding an effective, degree
k
k
, zero-cycle, as a point on
S
k
(
A
)
{S^k}(A)
(the
k
k
-symmetric product of
A
A
), and by considering the associated rational equivalence class, we get a map
γ
:
S
k
(
A
)
→
CH
0
(
A
)
\gamma :{S^k}(A) \to {\text {CH}_0}(A)
, whose fibres are called
γ
\gamma
-orbits. For any
n
≥
2
n \geq 2
, in this paper we determine the maximal dimension of the
γ
\gamma
-orbits when
k
=
2
k = 2
or
3
3
(it is, respectively,
1
1
and
2
2
), and the maximal dimension of families of
γ
\gamma
-orbits; moreover, for generic
A
A
, we get some refinements and in particular we show that if
dim
(
A
)
≥
4
\dim (A) \geq 4
,
S
3
(
A
)
{S^3}(A)
does not contain any
γ
\gamma
-orbit; note that it implies that a generic Abelian four-fold does not contain any trigonal curve. We also show that our bounds are sharp by some examples. The used technique is the following: we have considered some special families of Abelian varieties:
A
t
=
E
t
×
B
{A_t} = {E_t} \times B
(
E
t
{E_t}
is an elliptic curve with varying moduli) and we have constructed suitable projections between
S
k
(
A
t
)
{S^k}({A_t})
and
S
k
(
B
)
{S^k}(B)
which preserve the dimensions of the families of
γ
\gamma
-orbits; then we have done induction on
n
n
. For
n
=
2
n = 2
the proof is based upon the papers of Mumford and Roitman on this topic.