AbstractLet $$\mathcal {H}_{a,b}^n$$
H
a
,
b
n
denote the component of the Hilbert scheme whose general point parameterizes an a-plane union a b-plane meeting transversely in $${\mathbf {P}}^n$$
P
n
. We show that $$\mathcal {H}_{a,b}^n$$
H
a
,
b
n
is smooth and isomorphic to successive blow ups of $$\mathbf {Gr}(a,n) \times \mathbf {Gr}(b,n)$$
Gr
(
a
,
n
)
×
Gr
(
b
,
n
)
or $$\text {Sym}^2 \mathbf {Gr}(a,n)$$
Sym
2
Gr
(
a
,
n
)
along certain incidence correspondences. We classify the subschemes parameterized by $$\mathcal {H}_{a,b}^n$$
H
a
,
b
n
and show that this component has a unique Borel fixed point. We also study the birational geometry of this component. In particular, we describe the effective and nef cones of $$\mathcal {H}_{a,b}^n$$
H
a
,
b
n
and determine when the component is Fano. Moreover, we show that $$\mathcal {H}_{a,b}^n$$
H
a
,
b
n
is a Mori dream space for all values of a, b, n.
Birational geometry of varieties, fibred into complete intersections of codimension two
