Deformation and Unobstructedness of Determinantal Schemes
Authors: Kleppe Jan , Miró-Roig Rosa 
Published: May 2023

A closed subscheme

X
⊂

P

n

X\subset \mathbb {P}^n

is said to be determinantal if its homogeneous saturated ideal can be generated by the

s
×
s

s\times s

minors of a homogeneous

p
×
q

p\times q

matrix satisfying

(
p
−
s
+
1
)
(
q
−
s
+
1
)
=
n
−
dim
⁡
X

(p-s+1)(q-s+1)=n - \dim X

and it is said to be standard determinantal if, in addition,

s
=
min
(
p
,
q
)

s=\min (p,q)

. Given integers

a
1

≤

a
2

≤
⋯
≤

a

t
+
c
−
1

a_1\le a_2\le \cdots \le a_{t+c-1}

and

b
1

≤

b
2

≤
⋯
≤

b
t

b_1\le b_2 \le \cdots \le b_t

we consider

t
×
(
t
+
c
−
1
)

t\times (t+c-1)

matrices

A

=
(

f

i
j

)

\mathcal {A}=(f_{ij})

with entries homogeneous forms of degree

a
j

−

b
i

a_j-b_i

and we denote by

W
(

b
_

;

a
_

;
r
)

¯

\overline {W(\underline {b};\underline {a};r)}

the closure of the locus
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W left-parenthesis b underbar semicolon a underbar semicolon r right-parenthesis subset-of upper H i l b Superscript p left-parenthesis t right-parenthesis Baseline left-parenthesis double-struck upper P Superscript n Baseline right-parenthesis">

W
(

b
_

;

a
_

;
r
)
⊂
H
i
l

b

p
(
t
)

(

P

n

)

W(\underline {b};\underline {a};r)\subset Hilb^{p(t)}(\mathbb {P}^{n})

of determinantal schemes defined by the vanishing of the

(
t
−
r
+
1
)
×
(
t
−
r
+
1
)

(t-r+1)\times (t- r+1)

minors of such

A

\mathcal {A}

for

max
{
1
,
2
−
c
}
≤
r
>
t

\max \{1,2-c\} \le r > t

.

W
(

b
_

;

a
_

;
r
)

W(\underline {b};\underline {a};r)

is an irreducible algebraic set.

First of all, we compute an upper

r
r

-independent bound for the dimension of

W
(

b
_

;

a
_

;
r
)

W(\underline {b};\underline {a};r)

in terms of

a
j

a_j

and

b
i

b_i

which is sharp for

r
=
1

r=1

. In the linear case (

a
j

=
1
,

b
i

=
0

a_j = 1, b_i=0

) and cases sufficiently close, we conjecture and to a certain degree prove that this bound is achieved for all

r
r

. Then, we study to what extent the family

W
(

b
_

;

a
_

;
r
)

W(\underline {b};\underline {a};r)

fills in a generically smooth open subset of the corresponding component of the Hilbert scheme

H
i
l

b

p
(
t
)

(

P

n

)

Hilb^{p(t)}(\mathbb {P}^{n})

of closed subschemes of

P

n

\mathbb {P}^n

with Hilbert polynomial

p
(
t
)
∈

Q

[
t
]

p(t)\in \mathbb {Q}[t]

. Under some weak numerical assumptions on the integers

a
j

a_j

and

b
i

b_i

(or under some depth conditions) we conjecture and often prove that

W
(

b
_

;

a
_

;
r
)

¯

\overline {W(\underline {b};\underline {a};r)}

is a generically smooth component. Moreover, we also study the depth of the normal module of the homogeneous coordinate ring of

(
X
)
∈
W
(

b
_

;

a
_

;
r
)

(X)\in W(\underline {b};\underline {a};r)

and of a closely related module. We conjecture, and in some cases prove, that their codepth is often 1 (resp.

r
r

). These results extend previous results on standard determinantal schemes to determinantal schemes; i.e. previous results of the authors on

W
(

b
_

;

a
_

;
1
)

W(\underline {b};\underline {a};1)

to

W
(

b
_

;

a
_

;
r
)

W(\underline {b};\underline {a};r)

with

1
≤
r
>
t

1\le r > t

and

c
≥
2
−
r

c\ge 2-r

. Finally, deformations of exterior powers of the cokernel of the map determined by

A

\mathcal {A}

are studied and proven to be given as deformations of

X
⊂

P

n

X \subset \mathbb {P}^n

if

dim
⁡
X
≥
3

\dim X \ge 3

.

The work contains many examples which illustrate the results obtained and a considerable number of open problems; some of them are collected as conjectures in the final section.