Poncelet's theorem states that if there exists an n-sided polygon which is inscribed in a given conic C and circumscribed about another conic D, then there are infinitely many such n-gons. Proofs of ...this theorem that we are aware of, including Poncelet's original proof and the celebrated modern proof by Griffiths and Harris, assume the two conics to be in general position (that is, not tangent or at least not osculating), or be defined over
, or both. Here we show that Poncelet's theorem holds for any two conics C and D in the projective plane
over an algebraically closed field k of any characteristic other than two. If C and D are osculating and
, our result shows that there always exist infinitely many polygons of
sides that are inscribed in C and circumscribed about D. We also describe the situation in characteristic two in the appendix.
Let X be a complete variety of dimension n over an algebraically closed field \mathbb {K}. Let V_\bullet be a graded linear series associated to a line bundle L on X, that is, a collection ...\{V_m\}_{m\in \mathbb {N}} of vector subspaces V_m\subseteq H^0(X,L^{\otimes m}) such that V_0=\mathbb {K} and V_k\cdot V_\ell \subseteq V_{k+\ell } for all k,\ell \in \mathbb {N}. For each m in the semigroup \ \mathbf {N}(V_\bullet )=\{m\in \mathbb {N}\mid V_m\ne 0\},\ the linear series V_m defines a rational map \ \phi _m\colon X\dashrightarrow Y_m\subseteq \mathbf {P}(V_m), \ where Y_m denotes the closure of the image \phi _m(X). We show that for all sufficiently large m\in \mathbf {N}(V_\bullet ), these rational maps \phi _m\colon X\dashrightarrow Y_m are birationally equivalent, so in particular Y_m are of the same dimension \kappa, and if \kappa =n then \phi _m\colon X\dashrightarrow Y_m are generically finite of the same degree. If \mathbf {N}(V_\bullet )\ne \{0\}, we show that the limit \ vol_\kappa (V_\bullet )=\lim _{m\in \mathbf {N}(V_\bullet )}\frac {\dim _\mathbb {K} V_m}{m^\kappa /\kappa !}\ exists, and 0<vol_\kappa (V_\bullet )<\infty. Moreover, if Z\subseteq X is a general closed subvariety of dimension \kappa, then the limit \ (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},\ldots ,D_{m,\kappa }\in |V_m| are general divisors, and \ (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\in \mathbf {N}(V_\bullet ).
Let
X
be a projective variety. If
L
is a line bundle on
X
, for each positive integer
m
in
N
(
L
)
=
{
m
∈
N
∣
H
0
(
X
,
L
⊗
m
)
≠
0
}
, the global sections of
L
⊗
m
define a rational map
ϕ
m
:
X
⤏
Y
...m
⊆
P
(
H
0
(
X
,
L
⊗
m
)
)
,
where
Y
m
is the closure of
ϕ
m
(
X
)
. It is well-known that for all sufficiently large
m
∈
N
(
L
)
, the rational maps
ϕ
m
:
X
⤏
Y
m
are birationally equivalent to a fixed fibration (the Iitaka fibration), and
κ
(
L
)
:
=
dim
Y
m
is called the Iitaka dimension of
L
. In a recent paper titled “Iitaka fibrations for vector bundles”, Mistretta and Urbinati generalized this to a vector bundle
E
on
X
. Let
N
(
E
)
be the set of positive integers
m
such that the evaluation map
H
0
(
X
,
S
m
E
)
→
S
m
E
x
is surjective for all points
x
in some nonempty open subset of
X
. For each
m
∈
N
(
E
)
, the global sections of
S
m
E
define a rational map
φ
m
:
X
⤏
Y
m
⊆
G
(
H
0
(
X
,
S
m
E
)
,
rank
S
m
E
)
,
where
G
(
H
0
(
X
,
S
m
E
)
,
rank
S
m
E
)
is the Grassmannian of
rank
S
m
E
-dimensional quotients of
H
0
(
X
,
S
m
E
)
. Mistretta and Urbinati showed that for every
m
∈
N
(
E
)
, the rational maps
φ
km
are birationally equivalent for sufficiently large
k
, and called
κ
(
E
)
:
=
dim
Y
km
the Iitaka dimension of
E
. Here we first slightly improve Mistretta and Urbinati’s result to show that the rational maps
φ
m
are birationally equivalent for all sufficiently large
m
∈
N
(
E
)
. Then we show that
κ
(
E
)
≥
κ
(
O
P
(
E
)
(
1
)
)
-
rank
E
+
1
.
An immediate corollary of this inequality is that if
E
is big then
κ
(
E
)
=
dim
X
, which answers a question of Mistretta and Urbinati. Another corollary is that if
E
is big then
det
E
is big, provided that
N
(
E
)
≠
∅
.
Given a big divisor
D on a normal complex projective variety
X, we show that the restricted volume of
D along a very general complete-intersection curve
C
⊂
X
can be read off from the Okounkov body ...of
D with respect to an admissible flag containing
C. From this we deduce that if two big divisors
D
1
and
D
2
on
X have the same Okounkov body with respect to every admissible flag, then
D
1
and
D
2
are numerically equivalent.
We determine the effective cone of the Quot scheme parametrizing all rank
r
, degree
d
quotient sheaves of the trivial bundle of rank
n
on
. More specifically, we explicitly construct two effective ...divisors which span the effective cone, and we also express their classes in the Picard group in terms of a known basis.
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