On conjectures of Chudnovsky and Demailly and

motivation
homogeneous ideals
On conjectures of Chudnovsky and Demailly
and Waldschmidt constants
Tomasz Szemberg
Pedagogical University of Cracow
MEGA 2017
Nice, June 16, 2017
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Schneider-Lang Theorem in one variable
Theorem
Let f1 , . . . , fk be meromorphic functions in C with f1 , f2
algebraically independent. Let K be a number field. Assume that
for all j = 1, . . . , k
fj0 ∈ K[f1 , . . . , fk ].
Then the set
S = {z ∈ C : z is not a pole of fj , fj (z) ∈ K, j = 1, . . . , k}
is finite.
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Schneider-Lang Theorem in one variable
Theorem
Let f1 , . . . , fk be meromorphic functions in C with f1 , f2
algebraically independent. Let K be a number field. Assume that
for all j = 1, . . . , k
fj0 ∈ K[f1 , . . . , fk ].
Then the set
S = {z ∈ C : z is not a pole of fj , fj (z) ∈ K, j = 1, . . . , k}
is finite.
Corollary (Hermite-Lindemann)
For ω ∈ C∗ at least one of the numbers ω, exp(ω) is
transcendental.
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Schwarz Lemma in one variable
Theorem
Let f be an analytic function in a disc {|z| ≤ R} ⊂ C with at least
N zeroes in a disc {|z| ≤ r } with r < R. Then
|f |r ≤
3r
R
N
|f |R ,
where
|f |γ = sup |f (z)|.
|z|≤γ
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Schneider-Lang Theorem in several variables
Theorem (Bombieri 1970)
Let f1 , . . . , fk be meromorphic functions in Cn with f1 , . . . , fn+1
algebraically independent. Let K be a number field. Assume that
for all i = 1, . . . , n, j = 1, . . . , k
∂
fj ∈ K[f1 , . . . , fk ].
∂zi
Then the set
S = {z ∈ Cn : z is not a pole of fj , fj (z) ∈ K, j = 1, . . . , k}
is contained in an algebraic hypersurface.
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Hörmander version of Schwarz lemma in several variables
Theorem
Let S ⊂ Cn be a finite set. Let m be a positive integer. There
exists M(m) > 0 such that there exists r > 0 such that for R > r
and a function f analytic in the ball {|z| ≤ R} ⊂ Cn vanishing
with multiplicity ≥ m at each point of S
|f |r ≤
c(n) · r
R
M(m)
|f |R ,
where c(n) is a constant depending only on n.
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Hörmander version of Schwarz lemma in several variables
Theorem
Let S ⊂ Cn be a finite set. Let m be a positive integer. There
exists M(m) > 0 such that there exists r > 0 such that for R > r
and a function f analytic in the ball {|z| ≤ R} ⊂ Cn vanishing
with multiplicity ≥ m at each point of S
|f |r ≤
c(n) · r
R
M(m)
|f |R ,
where c(n) is a constant depending only on n.
Problem
Make the statement effective. In particular: what is the maximal
value of M(m)?
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Waldschmidt constant should be Moreau constant
Problem
Make the statement effective. In particular: what is the maximal
value of M?
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Waldschmidt constant should be Moreau constant
Problem
Make the statement effective. In particular: what is the maximal
value of M?
Theorem (Moreau)
Let S ⊂ Cn be a finite set. Let m be a positive integer. There
exists r > 0 such that for R > r and a function f analytic in the
ball {|z| ≤ R} ⊂ Cn vanishing with multiplicity ≥ m at each point
of S
exp(n) · r α(mS)
|f |r ≤
|f |R ,
R
(m)
where α(mS) is the initial degree of IS .
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Waldschmidt constant should be Moreau constant
Problem
Make the statement effective. In particular: what is the maximal
value of M?
Theorem (Moreau)
Let S ⊂ Cn be a finite set. Let m be a positive integer. There
exists r > 0 such that for R > r and a function f analytic in the
ball {|z| ≤ R} ⊂ Cn vanishing with multiplicity ≥ m at each point
of S
exp(n) · r α(mS)
|f |r ≤
|f |R ,
R
(m)
where α(mS) is the initial degree of IS .
Remark
The constant α(mS) is optimal.
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Symbolic powers
Definition
Let K be a field and let R = K[x0 , . . . , xn ] be the ring of
polynomials. For a homogeneous ideal 0 6= I ( R its m-th
symbolic power is
\
I (m) =
(I m RP ∩ R) .
P∈Ass(I )
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Symbolic powers
Definition
Let K be a field and let R = K[x0 , . . . , xn ] be the ring of
polynomials. For a homogeneous ideal 0 6= I ( R its m-th
symbolic power is
\
I (m) =
(I m RP ∩ R) .
P∈Ass(I )
Theorem (Zariski-Nagata)
Let X ⊂ Pn (K) be a projective variety (in particular reduced).
Then I (X )(m) is generated by all forms which vanish along X to
order at least m.
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Symbolic powers of ideals of points
Let Z = {P1 , . . . , Ps } be a finite set of points in Pn (K). Then
I (Z ) = I (P1 ) ∩ . . . ∩ I (Ps )
and
I (Z )(m) = I (P1 )m ∩ . . . ∩ I (Ps )m
for all m ≥ 1.
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
The initial degree and the Waldschmidt constant
Definition
For a graded ideal I its initial degree α(I ) is the least number t
such that It 6= 0.
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
The initial degree and the Waldschmidt constant
Definition
For a graded ideal I its initial degree α(I ) is the least number t
such that It 6= 0.
The Waldschmidt constant of I is the real number
α(I (m) )
.
m≥1
m
α
b(I ) = inf
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Waldschmidt constants are hard to compute
Conjecture (Nagata)
Let I be a saturated ideal of s ≥ 10 very general points in P2 (C).
Then
√
α(I (m) ) > m s
for all m ≥ 1.
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Waldschmidt constants are hard to compute
Conjecture (Nagata)
Let I be a saturated ideal of s ≥ 10 very general points in P2 (C).
Then
√
α(I (m) ) > m s
for all m ≥ 1.
Equivalently
α
b(I ) =
Tomasz Szemberg Pedagogical University of Cracow
√
s.
Waldschmidt constants
motivation
homogeneous ideals
Chudnovsky and Demailly Conjectures
Conjecture (Chudnovsky)
Let I be a saturated ideal of points in Pn (K). Then
α
b(I ) ≥
α(I ) + n − 1
.
n
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Chudnovsky and Demailly Conjectures
Conjecture (Chudnovsky)
Let I be a saturated ideal of points in Pn (K). Then
α
b(I ) ≥
α(I ) + n − 1
.
n
Conjecture (Demailly)
Let I be a saturated ideal of points in Pn (K). Then
α
b(I ) ≥
α(I (m) ) + n − 1
.
m+n−1
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Chudnovsky and Demailly Conjectures
Conjecture (Chudnovsky)
Let I be a saturated ideal of points in Pn (K). Then
α
b(I ) ≥
α(I ) + n − 1
.
1+n−1
Conjecture (Demailly)
Let I be a saturated ideal of points in Pn (K). Then
α
b(I ) ≥
α(I (m) ) + n − 1
.
m+n−1
Remark
The Chudnovsky Conjecture is the m = 1 case of the Demailly
Conjecture.
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
The containment problem
Problem
Compare ordinary and symbolic powers of homogeneous ideals.
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
The containment problem
Problem
Compare ordinary and symbolic powers of homogeneous ideals.
More precisely, given I determine all pairs (m, r ) such that
a) I r ⊂ I (m) ;
b) I (m) ⊂ I r .
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
The containment problem
Problem
Compare ordinary and symbolic powers of homogeneous ideals.
More precisely, given I determine all pairs (m, r ) such that
a) I r ⊂ I (m) ;
b) I (m) ⊂ I r .
Proposition
I r ⊂ I (m) ⇔ r ≥ m.
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
The containment problem
Problem
Compare ordinary and symbolic powers of homogeneous ideals.
More precisely, given I determine all pairs (m, r ) such that
a) I r ⊂ I (m) ;
b) I (m) ⊂ I r .
Proposition
I r ⊂ I (m) ⇔ r ≥ m.
Theorem (Ein-Lazarsfeld-Smith, Hochster-Huneke)
If m ≥ bight(I )r , then I (m) ⊂ I r .
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Containment and Chudnovsky Conjecture
Theorem (Ein, Lazarsfeld, Smith; Hochster, Huneke)
Let I be a saturated ideal in K[x0 , . . . , xn ]. Then for all m ≥ nr
I (m) ⊂ I r .
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Containment and Chudnovsky Conjecture
Theorem (Ein, Lazarsfeld, Smith; Hochster, Huneke)
Let I be a saturated ideal in K[x0 , . . . , xn ]. Then for all m ≥ nr
I (m) ⊂ I r .
Corollary
Let I be a saturated ideal of points in Pn (K). Then
α
b(I ) ≥
Tomasz Szemberg Pedagogical University of Cracow
α(I )
.
n
Waldschmidt constants
motivation
homogeneous ideals
Containment and Chudnovsky Conjecture
Theorem (Ein, Lazarsfeld, Smith; Hochster, Huneke)
Let I be a saturated ideal in K[x0 , . . . , xn ]. Then for all m ≥ nr
I (m) ⊂ I r .
Corollary
Let I be a saturated ideal of points in Pn (K). Then
α
b(I ) ≥
α(I )
.
n
Theorem (Demailly)
In the set up as above
α
b(I ) ≥
α(I )(α(I ) + 1) · . . . · (α(I ) + n − 1)
.
n! α(I )n−1
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Star configurations
Definition (Star configuration of points)
We say that Z ⊂ PN is a star configuration of degree d (or a
d-star for short) if Z consists of all intersection points of d ≥ N
general hyperplanes in PN . By intersection points we mean the
points which belong to exactly N of given d hyperplanes.
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Star configurations
Definition (Star configuration of points)
We say that Z ⊂ PN is a star configuration of degree d (or a
d-star for short) if Z consists of all intersection points of d ≥ N
general hyperplanes in PN . By intersection points we mean the
points which belong to exactly N of given d hyperplanes.
Example (Bocci, Harbourne)
For points in the star configuration in Pn , there is the equality in
the Chudnovsky Conjecture.
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Squarefree monomial ideals
Theorem (Bocci, Cooper, Guardo, Harbourne, Janssen, Nagel,
Seceleanu, Van Tuyl, Vu 2015)
Let I be a squarefree monomial ideal with bight(I ) = e. Then
α
b(I ) ≥
α(I ) + e − 1
.
e
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
More evidence for the Chudnovsky Conjecture
Theorem (Esnault – Viehweg 1983)
Let I be a radical ideal of a finite set of points in Pn with n ≥ 2.
Let k ≤ m be two integers. Then
α(I (k) ) + 1
α(I (m) )
≤
,
k +n−1
m
in particular
α(I (k) ) + 1
≤α
b(I ).
k +n−1
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
More evidence for the Chudnovsky Conjecture
Theorem (Esnault – Viehweg 1983)
Let I be a radical ideal of a finite set of points in Pn with n ≥ 2.
Let k ≤ m be two integers. Then
α(I (k) ) + 1
α(I (m) )
≤
,
k +n−1
m
in particular
α(I (k) ) + 1
≤α
b(I ).
k +n−1
Corollary
In particular
α(I ) + 1
≤α
b(I ).
n
Hence the Chudnovsky Conjecture holds in P2 .
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
General points
Theorem (Dumnicki-Tutaj-Gasińska, Fouli-Manteo-Xie 2016)
The Chudnovsky Conjecture holds for general points in Pn .
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
General points
Theorem (Dumnicki-Tutaj-Gasińska, Fouli-Manteo-Xie 2016)
The Chudnovsky Conjecture holds for general points in Pn .
Theorem (Malara, Szemberg, Szpond 2017)
The Demailly Conjecture holds for general points in Pn .
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Inductional lower bound
Theorem (Dumnicki, Szemberg, Szpond 2017)
Let H1 , . . . , Hs be s ≥ 2 mutually distinct hyperplanes in PN . Let
a1 ≥ . . . ≥ as > 1 be real numbers such that

a1 − 1 > 0





 a1 a2 − a1 − a2 > 0
..
.

Ps−1


bi . . . as−1 > 0
a1 . . . as−1 −

i=1 a1 . . . a

P

a1 . . . as − si=1 a1 . . . abi . . . as ≤ 0.
For i = 1, . . . , s let
Zi = {Pi,1 , . . . , Pi,ki } ∈ Hi \
[
Hj
j6=i
be a set of ki points such that α
b(Hi ; Zi ) ≥ ai .
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants
motivation
homogeneous ideals
Inductional lower bound, continued
Theorem (Dumnicki, Szemberg, Szpond 2017)
S
For Z = si=1 Zi and
a1 . . . as−1 −
q :=
s−1
P
i=1
a1 . . . abi . . . as−1
· as + s − 1.
a1 . . . as−1
There is
α
b(PN ; Z ) ≥ q.
Tomasz Szemberg Pedagogical University of Cracow
Waldschmidt constants

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