Asymptotic constructions and invariants of graded linear series
Authors: Chang Chih-Wei , Jow Shin-Yao 
Published: December 2021

Let

X
X

be a complete variety of dimension 

n
n

over an algebraically closed field

K

\mathbb {K}

. Let

V
∙

V_\bullet

be a graded linear series associated to a line bundle

L
L

on

X
X

, that is, a collection

{

V
m

}

m
∈

N

\{V_m\}_{m\in \mathbb {N}}

of vector subspaces

V
m

⊆

H
0

(
X
,

L

⊗
m

)

V_m\subseteq H^0(X,L^{\otimes m})

such that

V
0

=

K

V_0=\mathbb {K}

and

V
k

⋅

V
ℓ

⊆

V

k
+
ℓ

V_k\cdot V_\ell \subseteq V_{k+\ell }

for all

k
,
ℓ
∈

N

k,\ell \in \mathbb {N}

. For each

m
m

in the semigroup
\[

N

(

V
∙

)
=
{
m
∈

N

∣

V
m

≠
0
}
,

\mathbf {N}(V_\bullet )=\{m\in \mathbb {N}\mid V_m\ne 0\},

\]
the linear series

V
m

V_m

defines a rational map
\[

ϕ
m

:
X
⇢

Y
m

⊆

P

(

V
m

)
,

\phi _m\colon X\dashrightarrow Y_m\subseteq \mathbf {P}(V_m),

\]
where

Y
m

Y_m

denotes the closure of the image

ϕ
m

(
X
)

\phi _m(X)

. We show that for all sufficiently large

m
∈

N

(

V
∙

)

m\in \mathbf {N}(V_\bullet )

, these rational maps

ϕ
m

:
X
⇢

Y
m

\phi _m\colon X\dashrightarrow Y_m

are birationally equivalent, so in particular

Y
m

Y_m

are of the same dimension 

κ
\kappa

, and if

κ
=
n

\kappa =n

then

ϕ
m

:
X
⇢

Y
m

\phi _m\colon X\dashrightarrow Y_m

are generically finite of the same degree. If

N

(

V
∙

)
≠
{
0
}

\mathbf {N}(V_\bullet )\ne \{0\}

, we show that the limit
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v o l Subscript kappa Baseline left-parenthesis upper V Subscript bullet Baseline right-parenthesis equals limit Underscript m element-of bold upper N left-parenthesis upper V Subscript bullet Baseline right-parenthesis Endscripts StartFraction dimension Subscript double-struck upper K Baseline upper V Subscript m Baseline Over m Superscript kappa Baseline slash kappa factorial EndFraction">

v
o

l
κ

(

V
∙

)
=

lim

m
∈

N

(

V
∙

)

dim

K

⁡

V
m

m
κ

/

κ
!

vol_\kappa (V_\bullet )=\lim _{m\in \mathbf {N}(V_\bullet )}\frac {\dim _\mathbb {K} V_m}{m^\kappa /\kappa !}

\]
exists, and

0
>
v
o

l
κ

(

V
∙

)
>
∞

0>vol_\kappa (V_\bullet )>\infty

. Moreover, if

Z
⊆
X

Z\subseteq X

is a general closed subvariety of dimension 

κ
\kappa

, then the limit
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper V Subscript bullet Superscript kappa Baseline dot upper Z right-parenthesis Subscript normal m normal o normal v Baseline equals limit Underscript m element-of bold upper N left-parenthesis upper V Subscript bullet Baseline right-parenthesis Endscripts StartFraction number-sign left-parenthesis left-parenthesis upper D Subscript m comma 1 Baseline intersection midline-horizontal-ellipsis intersection upper D Subscript m comma kappa Baseline intersection upper Z right-parenthesis minus upper B s left-parenthesis upper V Subscript m Baseline right-parenthesis right-parenthesis Over m Superscript kappa Baseline EndFraction">

(

V
∙
κ

⋅
Z

)

m
o
v

=

lim

m
∈

N

(

V
∙

)

#

(

(

D

m
,
1

∩
⋯
∩

D

m
,
κ

∩
Z
)
∖
B
s
(

V
m

)

)

m
κ

(V_\bullet ^\kappa \cdot Z)_{\mathrm {mov}}=\lim _{m\in \mathbf {N}(V_\bullet )}\frac {\#\bigl ((D_{m,1}\cap \cdots \cap D_{m,\kappa }\cap Z)\setminus Bs(V_m)\bigr )}{m^\kappa }

\]
exists, where

D

m
,
1

,
…
,

D

m
,
κ

∈

|

V
m

|

D_{m,1},\ldots ,D_{m,\kappa }\in |V_m|

are general divisors, and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper V Subscript bullet Superscript kappa Baseline dot upper Z right-parenthesis Subscript normal m normal o normal v Baseline equals degree left-parenthesis phi Subscript m Baseline vertical-bar Subscript upper Z Baseline colon upper Z right dasheD arrow phi Subscript m Baseline left-parenthesis upper Z right-parenthesis right-parenthesis v o l Subscript kappa Baseline left-parenthesis upper V Subscript bullet Baseline right-parenthesis">

(

V
∙
κ

⋅
Z

)

m
o
v

=
deg
⁡

(

ϕ
m

|

Z

:
Z
⇢

ϕ
m

(
Z
)

)

v
o

l
κ

(

V
∙

)

(V_\bullet ^\kappa \cdot Z)_{\mathrm {mov}}=\deg \bigl (\phi _m|_Z\colon Z\dashrightarrow \phi _m(Z)\bigr )vol_\kappa (V_\bullet )

\]
for all sufficiently large

m
∈

N

(

V
∙

)

m\in \mathbf {N}(V_\bullet )

.