A scheme
X
⊂
P
n
+
c
X\subset \mathbb {P}^{n+c}
of codimension
c
c
is called standard determinantal if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous
t
×
(
t
+
c
−
1
)
t \times (t+c-1)
matrix and
X
X
is said to be good determinantal if it is standard determinantal and a generic complete intersection. Given integers
a
0
,
a
1
,
.
.
.
,
a
t
+
c
−
2
a_0,a_1,...,a_{t+c-2}
and
b
1
,
.
.
.
,
b
t
b_1,...,b_t
we denote by
W
(
b
_
;
a
_
)
⊂
Hilb
p
(
P
n
+
c
)
W(\underline {b};\underline {a})\subset \operatorname {Hilb} ^p(\mathbb {P}^{n+c})
(resp.
W
s
(
b
_
;
a
_
)
W_s(\underline {b};\underline {a})
) the locus of good (resp. standard) determinantal schemes
X
⊂
P
n
+
c
X\subset \mathbb {P}^{n+c}
of codimension
c
c
defined by the maximal minors of a
t
×
(
t
+
c
−
1
)
t\times (t+c-1)
matrix
(
f
i
j
)
j
=
0
,
.
.
.
,
t
+
c
−
2
i
=
1
,
.
.
.
,
t
(f_{ij})^{i=1,...,t}_{j=0,...,t+c-2}
where
f
i
j
∈
k
[
x
0
,
x
1
,
.
.
.
,
x
n
+
c
]
f_{ij}\in k[x_0,x_1,...,x_{n+c}]
is a homogeneous polynomial of degree
a
j
−
b
i
a_j-b_i
. In this paper we address the following three fundamental problems: To determine (1) the dimension of
W
(
b
_
;
a
_
)
W(\underline {b};\underline {a})
(resp.
W
s
(
b
_
;
a
_
)
W_s(\underline {b};\underline {a})
) in terms of
a
j
a_j
and
b
i
b_i
, (2) whether the closure of
W
(
b
_
;
a
_
)
W(\underline {b};\underline {a})
is an irreducible component of
Hilb
p
(
P
n
+
c
)
\operatorname {Hilb} ^p(\mathbb {P}^{n+c})
, and (3) when
Hilb
p
(
P
n
+
c
)
\operatorname {Hilb} ^p(\mathbb {P}^{n+c})
is generically smooth along
W
(
b
_
;
a
_
)
W(\underline {b};\underline {a})
. Concerning question (1) we give an upper bound for the dimension of
W
(
b
_
;
a
_
)
W(\underline {b};\underline {a})
(resp.
W
s
(
b
_
;
a
_
)
W_s(\underline {b};\underline {a})
) which works for all integers
a
0
,
a
1
,
.
.
.
,
a
t
+
c
−
2
a_0,a_1,...,a_{t+c-2}
and
b
1
,
.
.
.
,
b
t
b_1,...,b_t
, and we conjecture that this bound is sharp. The conjecture is proved for
2
≤
c
≤
5
2\le c\le 5
, and for
c
≥
6
c\ge 6
under some restriction on
a
0
,
a
1
,
.
.
.
,
a
t
+
c
−
2
a_0,a_1,...,a_{t+c-2}
and
b
1
,
.
.
.
,
b
t
b_1,...,b_t
. For questions (2) and (3) we have an affirmative answer for
2
≤
c
≤
4
2\le c \le 4
and
n
≥
2
n\ge 2
, and for
c
≥
5
c\ge 5
under certain numerical assumptions.