March 3, 2023

On the degree of hypersurfaces
with given singularities
Michel Waldschmidt
Professeur Émérite, Sorbonne Université,Institut de Mathématiques de Jussieu, Paris
http://www.imj-prg.fr/~michel.waldschmidt/

Abstract
Let n, t be positive integers and S be a finite set of points in
Cn . We denote by ωt (S) the least degree of a nonzero
polynomial vanishing with multiplicity at least t at each pointof S. The sequence (ωt (S)/t)t≥0 has a limit Ω(S) as t tends
to infinity. This invariant was introduced in 1975 for the proofof a Schwarz’s Lemma in several variables which occurs in thesolution by Bombieri in 1970 of a conjecture of Nagata dealing
with a generalization of a transcendence result of Schneider
and Lang. The same invariant occurs in connection withanother conjecture that Nagata introduced in 1959 in his work
on Hilbert’s 14th problem. It is closely related with Seshadri’s
constant.

Degree of hypersurfaces
Let S be a finite set of Cn and t a positive integer.
Denote by ωt (S) the smallest degree of a polynomial vanishingat each point of S with multiplicity ≥ t.
For n = 1 we have ωt (S) = t|S| :Y
P (z) =(z − s)t .
s∈SFor |S| = 1 we have ωt (S) = t :P (z 1 , . . . , z n ) = (z 1 − s1 )t .

ωt(S) : easy cases
For |S| ≤ n, ωt (S) = t :(a0 + a1 z 1 + · · · + an z n )t

For a Cartesian product (grid) S = S 1 × · · · × S n in Kn ,ωt (S) = t min |S i |.
1≤i≤n

Proof by induction.
First non trivial case : 3 points in C2 not on a line.

Generic S ⊂ C2 with |S| = 3
S = {(0, 0), (0, 1), (1, 0)}.
P 1 (X, Y ) = XY
P 2 (X, Y ) = XY (X + Y − 1)
ω1 (S) = 2,

ω2 (S) = 3.

With
P 2m−1 = X m Y m (X + Y − 1)m−1 , P 2m = X m Y m (X + Y − 1)m ,we deduce
ω2m−1 (S) = 3m − 1,

ω2m (S) = 3m.

Complete intersections of hyperplanesn
Let H 1 , . . . , H N be N hyperplanes
 in general position in C
Nwith N ≥ n and S the set of n intersection points of any nof them. Then, for t ≥ 1,

ωnt (S) = N t.

n = 2, N = 5, |S| = 10.

Linear algebra
The space of polynomials
in n variables of degree ≤ D hasdimension D+n.
n
The number of tuples (τ1 , . . . , τn ) of nonnegative integers
satisfying τ1 + · · · + τn ≤ t − 1 is t+n−1.
n
As a consequence, given a finite subset S of Cn and a positive
integer t, if D is a positive integer such that

 

t+n−1
D+n
|S|
<
,
n
n
then
ωt (S) ≤ D.

Properties of ωt(S)Consequence of Dirichlet’s box principle :ωt (S) ≤ (t + n − 1)|S|1/n .

Subadditivity :ωt1 +t2 (S) ≤ ωt1 (S) + ωt2 (S).

Therefore lim supt→∞ ωt (S)/t exists and is ≤ ωt (S)/t for allt ≥ 1.

An asymptotic invariant
Theorem. The sequence

1
ωt (S)tt≥1

has a limit Ω(S) as t → ∞, and1ω1 (S) − 2 ≤ Ω(S) ≤ ω1 (S).
nFurther, for all t ≥ 1 we haveΩ(S) ≤

ωt (S)·tRemark : Ω(S) ≤ |S|1/n .
M.W. Propriétés arithmétiques de fonctions de plusieurs variables(II). Sém. P. Lelong (Analyse), 16è année, 1975/76 ; Lecture Notes
in Math., 578 (1977), 274–292.

L2- estimates of Hörmander

Lars Hörmander

Existence theorems for the ∂operator.
E. Bombieri. Let ϕ be aplurisubharmonic function in
Cn not identically −∞. Then
there exists a nonzero entire
function F such that(1931 – 2012)

Z|F (z)|2 e−ϕ(z) (1 + |z|2 )−3n dλ(z) < ∞.
nC

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hormander.html

Improvement of L2 estimate by Henri Skoda
Let ϕ be a plurisubharmonic function in Cn not identically−∞. For any  > 0 there exists a nonzero entire function F
such thatZ|F (z)|2 e−ϕ(z) (1 + |z|2 )−n− dλ(z) < ∞.
Cn

Corollary :1
ω1 (S) ≤ Ω(S) ≤ ω1 (S).
nH. Skoda. Estimations L2 pourl’opérateur ∂ et applications
arithmétiques. Springer Lecture
Notes in Math., 578 (1977),
314–323.
Henri Skoda
https://en.wikipedia.org/wiki/Henri_Skoda

Comparing ωt1 (S) and ωt2 (S)Idea: Let P be a polynomial of degree ωt1 (S) vanishing on S
with multiplicity ≥ t1 . If the function P t2 /t1 were an entire
function, it would be a polynomial of degree tt21 ωt1 (S)vanishing on S with multiplicity ≥ t2 , which would yieldωt2 (S) ≤ tt21 ωt1 (S).
However P t2 /t1 is usually not an entire function but
ϕ = tt21 log P is a plurisubharmonic function. By theL2 –estimates of Hörmander – Bombieri – Skoda, eϕ is wellapproximated by a nonzero entire function. This function is apolynomial vanishing on S with multiplicity ≥ t2 .
Theorem. For all t ≥ 1,ωtωt
≤ Ω(S) ≤ ·t+n−1t
M.W. Nombres transcendants et groupes algébriques. Astérisque,
69–70 . Société Mathématique de France, Paris, 1979.

Hilbert’s 14th problem
Let k be a field and K a
subfield of k(X 1 , . . . , X n )containing k. Is the k–algebraK ∩ k[X 1 , . . . , X n ]David Hilbert(1862 – 1943)

finitely generated ?

Oscar Zariski (1954) : true for n = 1 and n = 2.
Counterexample by Masayoshi Nagata in 1959.
http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hilbert.html
http://www.clarku.edu/~djoyce/hilbert/

Hilbert’s 14th problem : restricted case

Masayoshi Nagata(1927 – 2008)

Original 14th problem :Let G be a subgroup of thefull linear group of thepolynomial ring inindeterminate X 1 , . . . , X nover a field k, and let o be theset of elements ofk[X 1 , . . . , X n ] which are
invariant under G. Is o finitely
generated ?

M. Nagata. On the 14-th Problem of Hilbert. Amer. J. Math 81(1959), 766–772.
http://www.jstor.org/stable/2372927

Fundamental Lemma of Nagata
Given 16 independent generic points of the projective planeover a field and a positive integer t, there is no curve of degree
4t which goes through each pi with multiplicity at least t.
In other words for |S| = 16 generic in C2 , we have ωt (S) > 4t.

Reference: M. Nagata. On the fourteenth problem of Hilbert.
Proc. Internat. Congress Math. 1958, Cambridge University Press,
pp. 459–462.
http://www.mathunion.org/ICM/ICM1958/Main/icm1958.0459.0462.ocr.pdf

Nagata’ contribution

Masayoshi Nagata(1927 – 2008)

Proposition. Let p1 , . . . , pr
be independent generic pointsof the projective plane overthe field. Let C be a curve ofdegree d passing through thepi ’s with multiplicities ≥ mi .
Then√
m1 + · · · + mr < d r
for r = s2 , s ≥ 4.

It is not known if r > 9 is sufficient to ensure the inequality of
the Proposition.
M. Nagata. Lectures on the fourteenth problem of Hilbert. Tata
Institute of Fundamental Research Lectures on Mathematics 31,(1965), Bombay.
http://www.math.tifr.res.in/~publ/ln/tifr31.pdf

Reformulation of Nagata’s ConjectureP
By considering σ C σ where σ runs over the cyclic
permutations of {1, . . . , r}, it is sufficient to consider the casem1 = · · · = mr .
Conjecture. Let S be a finite generic subset of the projective
plane over the field with |S| ≥ 10. Thenpωt (S) > t |S|.

Nagata :• True for |S| a square.
• False for |S| ≤ 9.
Harbourne, Brian. On Nagata’s conjecture, Journal of Algebra
236 2 (2001), 692–702.
https://doi.org/10.1006/jabr.2000.8515

|S| ≤ 9 in C2
Nagata : generic S in C2 with |S| ≤ 9 have|S|ωt (S) p≤ |S|.
t

= 1

2

3

4

5

6

7

8

9

ω1 (S) = 1

1

2

2

2

3

3

3

3

= 1

1

2

1

1

5

8

17 1

ωt (S) = 1

1

3

2

2

12

21

48 3

t

ωt (S)
3
12 21 48
= 1 1
2 2
3
t
2
5
8 17
p
√ √
√ √ √ √
|S| = 1
2
3 2
5
6
7
8 3

|S| = 1 or 2 in C2|S| = 1 : S = {(0, 0)}, P t (X, Y ) = X t ,ωt (S) = t, Ω(S) = 1.

|S| = 2 : S = {(0, 0), (1, 0)}, P t (X, Y ) = Y t ,ωt (S) = t, Ω(S) = 1.

Generic S with |S| = 3 in C2
Given a set S of 3 points in C2 , not on a straight line, we have
3t + 1
for t odd, 2
ωt (S) =
 3t
for t even,2henceΩ(S) = limn→∞3ωt (S)= ·t2

Since ω1 (S) = 2 and n = 2, this is an example with
ω1 (S)< Ω(S) < ω1 (S).
nGeneric S ⊂ C2 with |S| = 4
For a generic S in C2 with |S| = 4, we have ωt (S) = 2t, henceΩ(S) = 2.
Easy for a Cartesian product (square grid) S 1 × S 2 with
|S 1 | = |S 2 | = 2, also true for a generic S with |S| = 4.

More generally, for the same reason, when S is a Cartesian
product S 1 × S 2 withωt (S) = mt
p |S 1 | = |S 2 | = m, we have pand Ω(S) = m = |S|. The inequality Ω(S) ≥ |S| for ageneric S with |S| a square follows (Chudnovsky).

Generic S ⊂ C2 with |S| = 5
Since 5 points in C2 lie on a conic, for a generic S with
|S| = 5 we have ωt (S) = 2t and Ω(S) = 2.
Remark. A polynomial in 2 variables of degree d has(d + 1)(d + 2)
2
coefficients. Hence for 2|S| < (d + 1)(d + 2) we have
ω1 (S) ≤ d.
For |S| = 1, 2 we have ω1 (S) = 1,
for |S| = 3, 4, 5 we have ω1 (S) ≤ 2,
for |S| = 6, 7, 8, 9 we have ω1 (S) ≤ 3.

Generic S ⊂ C2 with |S| = 6 (Nagata)
Given 6 generic points
s1 , . . . , s6 in C2 , consider 6
conics C 1 , . . . , C 6 where S ipasses through the 5 points sj
for j 6= i. This produces apolynomial of degree 12 withmultiplicity ≥ 5 at each si .
Hence ω5 (S) ≤ 12.
In fact ω5t (S) = 12t,Ω(S) = 12/5.

Generic S ⊂ C2 with |S| = 7 (Nagata)
Given 7 points in C2 , there is a cubic passing through these 7
points with a double point at one of them.
Number of coefficients of a cubic polynomial : 10.
Number of conditions : 6 for the simple zeros, 3 for the double
zero.
This gives a polynomial of degree 7 × 3 = 21 with the 7
assigned zeroes of multiplicities 8.
In fact ω8t (S) = 21t, Ω(S) = 21/8.

Generic S ⊂ C2 with |S| = 8 (Nagata)
Given 8 points in C2 , there is a sextic with a double point at 7of them and a triple point at 1 of them.
Number of coefficients of a sextic polynomial :(6 + 1)(6 + 2)/2 = 28.
Number of conditions : 3 × 7 = 21 for the double zeros, 6 forthe triple zero.
This gives a polynomial of degree 8 × 6 = 48 with the 8assigned zeroes of multiplicities 2 × 8 + 1 = 17.
In fact ω17t (S) = 48t, Ω(S) = 48/17.

G.V. Chudnovsky

Gregory Chudnovsky

Conjecture :ω1 + n − 1
ωt
≤ ·
n
t

G.V. Chudnovsky. Singularpoints on complex
hypersurfaces andmultidimensional Schwarz
Lemma. M.-J. Bertin (Ed.),Séminaire de Théorie desNombres Delange-PisotPoitou, Paris, 1979–80, Prog.
Math., vol. 12, Birkhäuser.

True for n = 2 (J-P. Demailly).
https://fr.wikipedia.org/wiki/David_et_Gregory_Chudnovsky

Chudnovsky : n = 2, |S| = 2, 3, 4, 5

Chudnovsky : n = 2, |S| = 5, 6

Chudnovsky : n = 2, |S| = 6, 7

Chudnovsky : n = 2, |S| = 8

Chudnovsky : n = 2, |S| = 9

n = 2, |S| = 6, t = 5, ωt = 12, Ω = 12/5 genericn = 2, |S| = 8, t = 2, ωt = 5, Ω = 5/2

n = 2, |S| = 8, t = 3, ωt = 8, Ω = 8/3

n = 2, |S| = 10, t = 6, ωt = 17, Ω = 17/6
Three sides : multiplicity 3.
Three concurrent lines : multiplicity 2.

Hélène Esnault and Eckart Viehweg

Hélène Esnault

Eckart Viehweg(1948 - 2010)

H. Esnault and E. Viehweg Sur une minoration du degré
d’hypersurfaces s’annulant en certains points. Math. Ann. 263(1983), 75 – 86
Methods of projective geometry : for n ≥ 2,Ω(S) ≥

ωt + 1·t+n−1Jean-Pierre Demailly
Using an appropriate
generalization of the
Poisson–Jensen formula,
proves a new variant of the
Schwarz lemma in Cn .
Jean-Pierre Demailly
1957 – 2022

Consequence :ω1 (S) ω1 (S) + 1 · · · ω1 (S) + n − 1Ω(S) ≥
n!ω1 (S)n−1
Corollary : For n = 1 or 2,Ω(S) ≥

ω1 (S) + n − 1·nDemailly’s Conjecture
Recall the Conjecture of Chudnovsky and the Theorem ofEsnault and Viehweg :Ω(S) ≥

ω1 + n − 1,nΩ(S) ≥

ωt + 1·t+n−1Conjecture of Demailly :Ω(S) ≥

ωt (S) + n − 1·
t+n−1

J–P. Demailly. Formules de Jensen en plusieurs variables etapplications arithmétiques. Bull. Soc. Math. France 110 (1982),
75–102.
https://de.wikipedia.org/wiki/Jean-Pierre_Demailly

Conjecture of André Hirschowitz

André Hirschowitz

Denote by ωt (n, m) the
maximum of ωt (S) over all
finite sets S in Cn with melements.
Conjecture : ωt (n, m) is aslarge as possible.

For every n ≥ 1 there is an integer c(n) such that, for everym ≥ c(n) and, for all t, ωt (n, m) is the smallest integer d such
that




d+n
t+n−1
>m
·
n
n
True for t = 2 and n = 2 and 3, and for t = 3 and n = 2.
A. Hirschowitz. La méthode d’Horace pour l’interpolation à
plusieurs variables. Manuscripta Math. 50 (1985), 337–388.

Abdelhak Azhari
A. Azhari. Démonstration
analytique d’un lemme demultiplicités. C. R. Acad. Sci.
Paris Sér. I Math. 303 (1986),
no. 7, 269–272.
Abdelhak Azhari

A. Azhari. Sur la conjecture de Chudnovsky – Demailly et les
singularités des hypersurfaces algébriques. Ann. Inst. Fourier 40(1990), no. 1, 103–116.
http://www.numdam.org/item?id=AIF_1990__40_1_103_0

Schneider – Lang Theorem (1949, 1966)

Theodor Schneider

Serge Lang

(1911 – 1988)(1927 – 2005)

Let f 1 , . . . , f m be meromorphic functions in C. Assume f 1 and f 2
are algebraically independent and of finite order. Let K be anumber field. Assume f 0j belongs to K[f 1 , . . . , f m ] forj = 1, . . . , m. Then the set
S = {w ∈ C | w not pole of f j , f j (w) ∈ K (j = 1, . . . , m)}is finite.
http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schneider.html
http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lang.html

Hermite – Lindemann Theorem (1882)

Charles Hermite

Carl Louis Ferdinand von Lindemann(1822 – 1901)(1852 – 1939)

Corollary. If w is a nonzero complex number, one at least of
the two numbers w, ew is transcendental.
Consequence : transcendence of e, π, log α, eβ , for algebraic
α and β assuming α 6= 0, log α 6= 0, β 6= 0.
http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hermite.html
http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lindemann.html

Gel’fond – Schneider Theorem (1934)

Aleksandr Osipovich Gelfond

Theodor Schneider(1906 – 1968)(1911 – 1988)

Corollary (Hilbert’s seventh problem). If β is an irrational
algebraic number and w a nonzero complex number, one atleast of the two numbers ew , eβw is transcendental.
√

Consequence : transcendence of eπ , 2 2 , αβ , log α1 / log α2 ,for algebraic α, α1 , α2 and β assuming α 6= 0, log α 6= 0,
β 6∈ Q, log α1 / log α2 6∈ Q.
http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gelfond.html
http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schneider.html

Proofs of the corollaries
Hermite - Lindemann. Let K = Q(w, ew ). The twofunctions f 1 (z) = z, f 2 (z) = ez are algebraically independent,of finite order, and satisfy the differential equations f 01 = 1,
f 02 = f 2 . The set S contains {`w | ` ∈ Z}. Since w 6= 0, thisset is infinite ; it follows that K is not a number field. Gel’fond - Schneider. Let K = Q(β, ew , eβw ). The twofunctions f 1 (z) = ez , f 2 (z) = eβz are algebraicallyindependent, of finite order, and satisfy the differential
equations f 01 = f 1 , f 02 = βf 2 . The set S contains{`w | ` ∈ Z}. Since w 6= 0, this set is infinite ; it follows that
K is not a number field. Schneider’s Theorems on elliptic functions (1937)
Corollary (Schneider). Let ℘ be an elliptic function ofWeierstrass with algebraic invariants g 2 , g 3 . Let w be acomplex number, not pole of ℘. Then one at least of the two
numbers w, ℘(w) is transcendental.
Proof. Let K = Q(g 2 , w, ℘(w), ℘0 (w)). The two functions
f 1 (z) = z, f 2 (z) = ℘(z) are algebraically independent, offinite order. Set f 3 (z) = ℘0 (z). From ℘0 2 = 4℘3 − g 2 ℘ − g 3
one deduces
f 01 = 1,

f 02 = f 3 ,f 03 = 6f 2 2 − (g 2 /2).

The set S contains{`w | ` ∈ Z, `w not pole of ℘}which is infinite. Hence K is not a number field. The transcendence machinery
The prototype of transcendence methods is Hermite’s proof of
the transcendence of e.
The proof of the Schneider – Lang Theorem follows thefollowing scheme :Step 1 Construct an auxiliary function f with many zeroes.
Step 2 Find a point z 0 where f (z 0 ) 6= 0.
Step 3 Give a lower bound for |f (z 0 )| using arithmeticarguments.
Step 4 Give an upper bound for |f (z 0 )| using analytic
arguments.
We are interested here mainly (but not only) with the last part(step 4) which is of analytic nature.

Schwarz Lemma in one variable
Let f be an analytic function
in a disc |z| ≤ R of C, withat least N zeroes in a disc|z| ≤ r with r < R. Then

Hermann Amandus Schwarz|f |r ≤

(1843 – 1921)

3r
R

N
|f |R .

We use the notation|f |r = sup |f (z)|.
|z|=rWhen R > 3r, this improves the maximum modulus bound|f |r ≤ |f |R .
http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schwarz.html

Schwarz Lemma in one variable : proof
Let a1 , . . . , aN be zeroes of f in the disc |z| ≤ r, counted withmultiplicities. The function
g(z) = f (z)NY(z − aj )−1
j=1

is analytic in the disc |z| ≤ R. Using the maximum modulus
principle, from r ≤ R we deduce |g|r ≤ |g|R . Now we have|f |r ≤ (2r)N |g|r

and

|g|R ≤ (R − r)−N |f |R .

Finally, assuming (wlog) R > 3r,
3r
2r
≤ ·
R−r
R


Schneider – Lang Theorem in several variables :cartesian products (1941, 1966)
Let f 1 , . . . , f m be meromorphic functions in Cn withm ≥ n + 1. Assume f 1 , . . . , f n+1 are algebraicallyindependent of finite order. Let K be a number field. Assume(∂/∂z i )f j belongs to K[f 1 , . . . , f m ] for j = 1, . . . , m andi = 1, . . . , n. If e1 , . . . , en is a basis of Cn , then the set
S = {w ∈ Cn | w not pole of f j , f j (w) ∈ K (j = 1, . . . , m)}does not contain a cartesian product{s1 e1 + · · · + sn en | (s1 , . . . , sn ) ∈ S 1 × · · · × S n }where each S i is infinite.

Schneider’s Theorem on Euler’s Beta function
Let a, b be rational numbers,
not integers. Then thenumberB(a, b) =Leonhard Euler(1707 – 1783)

Γ(a)Γ(b)Γ(a + b)is transcendental.

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Euler.html

Further results by Th. Schneider and S. Lang on abelianfunctions and algebraic groups.

Schwarz lemma in several variables : cartesian
products (grids)

Let f be an analytic function in a ball |z| ≤ R of Cn . Assume
f vanishes with multiplicity at least t on a set S 1 × · · · × S n
where each S i is contained in a disc |z| ≤ r with r < R andhas at least s elements.
Then
 st
3r
|f |R .
|f |r ≤
R

Cartesian products
Schwarz Lemma for Cartesian products can be proved byinduction.
§4.3 of M.W.. Diophantine Approximation on Linear Algebraic
Groups. Grund. Math. Wiss. 326 Springer-Verlag (2000).

Another proof, based on integral formulae, yields a weaker
result : for R > 3r,
n  st3r
R − 3r|f |R .
|f |r ≤
2r
R
The conclusion follows from a homogeneity argument :replace f by f N (and t by N t) and let N → ∞.
Chap. 7 of M.W.. Nombres transcendants et groupes algébriques.
Astérisque, 69–70 (1979).

Nagata’s suggestion (1966)

Masayoshi Nagata(1927 – 2008)

In the conclusion of theSchneider – Lang Theorem,
replace the fact that S does
not contain a cartesian
product S 1 × · · · × S n where
each S i is infinite by the factthat S is contained in an
algebraic hypersurface.

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Nagata.html

Bombieri’s Theorem (1970)
Let f 1 , . . . , f m be meromorphic functions in Cn withm ≥ n + 1. Assume f 1 , . . . , f n+1 are algebraicallyindependent and of finite order. Let K be a number field.
Assume (∂/∂z i )f j belongs to K[f 1 , . . . , f m ] for j = 1, . . . , mand i = 1, . . . , n.
Then the set
S ={w ∈ Cn |w not pole of f j ,f j (w) ∈ K (j = 1, . . . , m)}
Enrico Bombieri

is contained in an algebraic
hypersurface.

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Bombieri.html

Bombieri – Lang (1970)

Let f be an analytic function in a ball |z| ≤ R of Cn . Assume
f vanishes at N points z i (counting multiplicities) in a ball|z| ≤ r with r < R. Assume min |z i − z k | ≥ δ.
z i 6=z k

Then
3r
R

M



2n−2


|f |r ≤
with
M =N

δ
6r

|f |R

.

Lelong number
E. Bombieri. Algebraic values of meromorphic maps. Invent. Math.
10 (1970), 267–287.
E. Bombieri and S. Lang. Analytic subgroups of group varieties.
Invent. Math. 11 (1970), 1–14.
P. Lelong. Intégration sur unensemble analytique complexe,
Bulletin S.M.F. 85 (1957),
239–262,
Pierre Lelong(1912 – 2011)https://fr.wikipedia.org/wiki/Pierre_Lelong

Towards a Schwarz lemma in several variables
Let S be a finite subset of Cn and t a positive integer. Let M
be a positive number with the following property.
There exists a real number r such that for R > r, if f is ananalytic function in the ball |z| ≤ R of Cn which vanishes withmultiplicity at least t at each point of S, then
|f |r ≤

c(n)r
R

M
|f |R ,

where c(n) depends only on the dimension n.
Question: what is the largest possible value for M ?
Answer: use the property for f a nonzero polynomial
vanishing on S with multiplicity t. We deduce that f has
degree at least M .

Schwarz lemma in several variables
Let S be a finite set of Cn and t a positive integer. Thereexists a real number r such that for R > r, if f is an analytic
function in the ball |z| ≤ R of Cn which vanishes withmultiplicity at least t at each point of S, then n ωt (S)e r|f |r ≤
|f |R .
R

This is a refined asymptotic version due to Jean-Charles
Moreau.
The exponent ωt (S) cannot be improved : take for f a
non–zero polynomial of degree ωt (S).

C.S. Seshadri memorial lectures — July 31, 2020

School of Mathematics!" #" #$#%&'() *$*+()&, ,$!-.($#
5@9: ABC DEDEF G 6* )#-#/012034
5" 6" '071899: !"# $%&'()*+ ,-#&.(#./
6" ,8;;0971<< !$.(#)(*) 0-#-1+(2&/=" >191?8 !"# 345/
67*(.+-# -8 %(9' .(2: ;+22 <% => 1+#&?
5'% 2%9.7*%& ;+22 <% '%2) @+( A--1
B-* A--1 46 (#) C(&&;-*) D2%(&% 9-#.(9.E
0+2+#) C+2(#:(*!D+2(#:(* F 1(.'?.+8*?*%&?+#/

http://conferences.math.tifr.res.in/lectures/seshadri-lectures/seshadri-lectures.html

Conjeevaram Srirangachari Seshadri

Jean–Pierre Demailly(1932-2020)(1957-2022)

http://conferences.math.tifr.res.in/lectures/seshadri-lectures/demailly.pdf

A brief survey on Seshadri constants
Jean-Pierre Demailly
Institut Fourier, Université Grenoble Alpes & Académie des Sciences de Paris

Memorial Lectures forProfessor Conjeevaram Srirangachari Seshadriorganized by the TIFR School of Mathematics
July 31, 2020, 4 PM IST

J.-P. Demailly, Seshadri Memorial Lectures, July 31, 2020

A brief survey on Seshadri constants

1/16

The starting point of Seshadri constant theory
The paper that started the whole theory :C.S. Seshadri, Annals of Mathematics, Vol. 95 (May 1972) 511–556

Seshadri’s ampleness criterion

Definition of the Seshadri constants
Definition (Demailly, 1990)
Let X be a projective nonsingular variety and L a nef (orpseudo-ample) line bundle over X. Given a point x ∈ X, one
defines the Seshadri constant ε(L, x) of L at x to be
L·C
ε(L, x) =inf.
all alg. curves C3x multx (C)This is a very interesting numerical invariant that measures in a
deep manner the “local positivity” of the line bundle L at point x.

Equivalent definition (already observed in Seshadri’s
paper !)Let π : X̃ → X be the blow-up of X at x ∈ X, and E the
exceptional divisor in X̃. Then, for L ∈ Pic(X) assumed to be nef,
one hasε(L, x) = sup{γ ≥ 0 / π ∗ L − γE is nef on X̃}.

Reformulation of Seshadri’s ampleness criterion
Reformulation of Seshadri’s ampleness criterion
A nef line bundle L ∈ Pic(X) is ample if and only if one hasε(L) := inf x∈X ε(L, x) > 0.
A direct consequence of the fact (π ∗ L − γE)n = Ln − γ n ≥ 0 isthat
ε(L, x) ≤ (Ln )1/n ,∀x ∈ X.
L·C
A curve C is said to be submaximal if< (Ln )1/n .
multx (C)A large part of the investigations on Seshadri constants, especiallyin the case of surfaces, rests upon the study of submaximal curves.
Remark. In [D, 1990], over K = C, the Seshadri constant is related
to more analytic invariants. For instance, if L is ample, it can beshown that ε(L, x) is the supremum of γ ≥ 0 for which L possesses
a singular Hermitian metric h with ΘL,h ≥ 0, that is smooth on
X r {x} with a logarithmic pole of Lelong number γ at x.

Relation to the Nagata conjecture
The concept of Seshadri constant is already highly non trivial on
rational surfaces. For instance, the famous Nagata conjecture, has
attracted lot of work by Hirschowitz, Harbourne, Biran, Bauer,
Szemberg, Dumnicki and others. It can be reformulated :Nagata conjecture (1959), reformulated
Let x1 , . . . , xp be p very general points in P2 , p ≥ 9. Then themultipoint Seshadri constant of O(1) on P2 satisfies1
ε(O(1), x1 , . . . , xp ) = √ .
p
A simple counting argument implies that ε(O(1), x1 , . . . , xp ) ≤ √1p ,and the main difficulty is to find good configurations of points to
get lower bounds. In case p = q 2 is a perfect square, a square grid
works, hence equality. For 4 < p < 9, one is in the Del Pezzo case,and the equality turns out to be strict.

Connection with C.S. Seshadri constant
For a generic set S of s points
in Pn , Seshadri’s constant(S) is related to Ω(S) by

(S)n−1 =

Ω(S)
·
s

Conjeevaram Srirangachari Seshadri(1932-2020)

Duality between the cone of effective curves and the cone ofample divisors.

C.S. Seshadri’s criterion
Let X be a smooth projective variety and L a line bundle on
X. Then L is ample if and only if there exists a positive
number  such that for all points x on X and all irreducible
curves C passing through x one has
L · C ≤  multx C.

R. Hartshorne. Ample
subvarieties of algebraic
varieties. Springer Lecture Notes
in Math., vol. 156, Springer(1970).
Robin Hartshorne

C.S. Seshadri constant at a point
Let X be a smooth projective variety and L a nef line bundle
on X. For a fixed point x ∈ X the real number
(X, L; x) := inf

L·C
multx Cis the Seshadri constant of L at x.
The infimum is over all irreducible curves passing through x.

Jean–Pierre Demailly
1957–2022

J.-P. Demailly. Singular
Hermitian metrics on positive
line bundles. Complex algebraic
varieties (Bayreuth, 1990),
Lecture Notes Math. 1507,
Springer-Verlag, (1992) 87–104.

Seshadri’s constant and Nagata’s Conjecture

Beata Maria Strycharz-Szemberg

Tomasz Szemberg

Beata Strycharz-Szemberg & Tomasz Szemberg. Remarks on
the Nagata conjecture, Serdica Mathematical Journal 30 2-3(2004), 405– 430.
http://www.math.bas.bg/serdica/2004/2004-405-430.pdf

The Nagata – Biran Conjecture

Masayoshi Nagata

Paul Biran(1927 – 2008)

Let X be a smooth algebraic surface and L an ample line
bundle on X of degree d. For sufficiently large r, the Seshadri
constant of a generic set Z = {p1 , . . . , pr } ⊂ X satisfiesd(X, L; Z) = √ ·rZero estimates
Recall step 2 of the transcendence machinery :Step 2 Find a point z 0 where f (z 0 ) 6= 0.

Context : zero estimates, multiplicity estimates, interpolation
estimates on an algebraic group.
Results of C Hermite, C.L. Siegel, Th. Schneider, K. Mahler,
A.O. Gel’fond, R. Tijdeman, W.D. Brownawell, D.W. Masser,
G. Wüstholz, P. Philippon, J-C. Moreau, D. Roy,
M. Nakamaye, N. Rattazzi, S. Fischler. . .

Michael Nakamaye and Nicolas Ratazzi

Michael Nakamaye

Nicolas Ratazzi

M. Nakamaye and N. Ratazzi. Lemmes de multiplicités etconstante de Seshadri. Math. Z. 259, No. 4, 915-933 (2008).
http://www.math.unm.edu/research/faculty_hp.php?d_id=96
http://www.math.u-psud.fr/~ratazzi/

Stéphane Fischler and Michael NakamayeStéphane Fischler

Michael Nakamaye

S. Fischler and M. Nakamaye. Seshadri constants andinterpolation on commutative algebraic groups. Ann. Inst.
Fourier 64, No. 3, 1269-1289 (2014).
http://www.math.u-psud.fr/~fischler/
http://www.math.unm.edu/research/faculty_hp.php?d_id=96

S. David, M. Nakamaye, P. Philippon

Bornes uniformes pour le nombre de points rationnels decertaines courbes, Diophantine geometry, 143–164, CRM
Series, 4, Ed. Norm., Pisa, 2007.
http://www.math.unm.edu/~nakamaye/Pisa.pdf

S. David, M. Nakamaye, P. Philippon
Au Professeur C. S. Seshadri à l’occasion de son 75èmeanniversaire.
Nous commençons par une étude indépendante des jets dessections de fibrés amples sur une surface lisse, puis sur le carré
d’une courbe elliptique, utile pour le Théorème 4.2. Ceci nous
permet en particulier d’introduire dans le présent contexte lesconstantes de Seshadri, dont l’utilisation en géométrie
diophantienne nous semble devoir être positivement stimulée.
http://www.math.unm.edu/~nakamaye/Pisa.pdf

Homogeneous ideals of R = K[X 0, . . . , X n]
Let K be a field of zero characteristic.
For p = (α0 : · · · : αn ) ∈ Pn (K), denote by I(p) the
homogeneous ideal generated by the polynomials αi X j − αj X i(0 ≤ i, j ≤ n) in the polynomial ring R = K[X 0 , . . . , X n ].
For S = {p1 , . . . , ps } ⊂ Pn (K), set
I(S) = I(p1 ) ∩ · · · ∩ I(ps ).
This is the ideal of forms vanishing on S. The least degree ofa polynomial in I(S) is ω1 (S).

Initial degree
Generally, when J is a nonzero homogeneous ideal of R, defineω(J) as the least degree of a polynomial in J.
Since J is homogeneous,
J=

M

Jm

m≥0

we have
ω(J) = min{m ≥ 0 | J m 6= 0}
and ω(J) is also called the initial degree of J.
Since J 1 J 2 is generated by the products P 1 P 2 with P i ∈ J i ,it is plain that ω(J 1 J 2 ) = ω(J 1 ) + ω(J 2 ), henceω(J t ) = tω(J).

Symbolic powers
For t ≥ 1, define the symbolic power I (t) (S) by
I (t) (S) = I(p1 )t ∩ · · · ∩ I(ps )t .
This is the ideal of forms vanishing on S with multiplicities≥ t. Hence
ω(I (t) (S)) = ωt (S).

We have I(S)t ⊂ I(pi )t for all i, hencet

I(S) ⊂s\

I(pi )t = I (t) (S).

i=1

From I(S)t ⊂ I (t) (S) we deduce ωt (S) ≤ tω1 (S).

Symbolic powers
For a homogeneous ideal J in the ring R = K[X 0 , . . . , X n ]and m ≥ 1, define the symbolic power J (m) as follows. Write
primary decompositions of J and J m as
\
\
J=
Qi ,
Jm =
Q0j ,
i

j

where Qi is homogeneous and Pi primary, Q0j ishomogeneous and P0j primary. We set
J (m) =\

Q0jj

where the intersection is over the j with P0j contained insome Pi .

Symbolic powers

Notice that J m ⊂ J (m) .

Example of a fat points ideal. For J = ∩j I(pj )mj ,\J (m) =
I(pj )mmj .
jThe containment problem
Find all m, t withJ (m) ⊂ J t .

Brian Harbourne. Asymptotic invariants of ideals of points. (2009).
Special Session on Geometry, Syzygies and Computations
Organized by Professors S. Kwak and J. Weyman KMS-AMS joint
meeting, December 16–20, 2009.
Slides.
www.math.unl.edu/~bharbourne1/KSSNoPauseRev.pdf

Ω(J) for a homogeneous ideal J
Cristiano Bocci

Brian Harbourne

For a homogeneous ideal J of K[X 0 , . . . , X n ], and t ≥ 1,
define ωt (J) = ω(J (t) ). Thenωt (J)Ω(J) = limt→∞texists and satisfies
ωt (J)Ω(J) ≤
for all t ≥ 1.
t

Alternate proof of Ω(S) ≥ω1(S)(2001, 2002)
n

Theorem (Ein-Lazarfeld-Smith, Hochster-Huneke). Let
J be a homogeneous ideal in K[X 0 , . . . , X n ] and t ≥ 1. Then
J (tn) ⊂ J t .
Consequence : From I(S)t ⊃ I(S)(tn) we deducetω1 (S) ≤ ωtn (S)andω1 (S)
ωtn (S)≤→ Ω(S) as t → ∞.
ntn

J (tn) ⊂ J t by L. Ein, R. Lazarfeld and K.E. Smith

The proof by Lawrence Ein, Robert Lazarfeld and Karen E.
Smith uses multiplier ideals.

L. Ein, R. Lazarfeld and K.E. Smith. Uniform behavior of symbolic
powers of ideals. Invent. Math. 144 (2001), 241–252.

J (tn) ⊂ J t

R. Lazarfeld. Positivity inalgebraic geometry I – II.
Ergeb. Math. 48–49,
Springer, Berlin (2004).

J (tn) ⊂ J t by M. Hochster and C. Huneke
The proof by Melvin Hochster and Craig Huneke uses
Frobenius powers and tight closure.
Melvin Hochster

Craig Huneke

M. Hochster and C. Huneke. Comparison of symbolic and ordinary
powers of ideals. Invent. Math. 147 (2002), 349–369.

Briançon-Skoda Theorem
Melvin Hochster

Craig Huneke

For an m–generated ideal a in the ring of germs of analytic
functions at 0 ∈ Cn , the ν–th power of its integral closure iscontained in a, where ν = min{m, n}.
M. Hochster and C. Huneke. Tight closure, invariant theory, andthe Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990),
31–116.

The resurgence of Bocci and Harbourne
Define%(J) = supHence, ifnmro| J (m) 6⊂ J r .

m> %(J), then J (m) ⊂ J r .
r

By L. Ein, R. Lazarfeld and K.E. Smith, %(J) ≤ n.
C. Bocci and B. Harbourne. Comparing powers and symbolic
powers of ideals. J. Algebraic Geom. 19 (2010), no. 3, 399–417.
C. Bocci and B. Harbourne. The resurgence of ideals of points andthe containment problem. Proc. Amer. Math. Soc. 138 (2010), no.
4, 1175–1190

The resurgence of Bocci and Harbourne
Denote by reg(J) the Castelnuovo– Mumford regularity of J.
Theorem (Bocci, Harbourne). We havereg(J)
ω(J)≤ %(J) ≤
·
Ω(J)
Ω(J)
Further, if ω(J) = reg(J), then
J (m) ⊂ J t ⇐⇒ tω(J) ≤ ω(J (m) ).

Optimality

Following Bocci and Harbourne, we have
ω1 (S)= n.
|S|<∞ Ω(S)sup

Conjecture of Cristiano Bocci and Brian Harbourne
Let S be a finite subset of P2 . Define %(S) = %(J) for
J = I(S).
Conjecture.
%(S) ≤ 2

ω1 (S)·
ω1 (S) + 1

This conjecture implies Chudnovsky’s conjecture : from
ω1 (S)
ω1 (S)≤ %(S) ≤ 2Ω(S)ω1 (S) + 1
one deduces

ω1 (S) + 1≤ Ω(S).
2

The containment problem (continued)
Let M be the homogeneous ideal (X 0 , . . . , X n ) in R.
Fact. In characteristic zero, the ideal J = I(S) satisfies
J (2) ⊂ MJ.
∂
P ∈ J. Use Euler’s formula
Proof. Let P ∈ J (2) . Hence ∂Xi(deg P )P =

n
X
i=0

Xi

∂
P.
∂X i

Question. For which m, t, j do we have J (m) ⊂ Mj J t ?
Remark. Since Mj J t ⊂ J t , the condition J (m) ⊂ Mj J t
implies J (m) ⊂ J t .
B. Harbourne and C. Huneke. Are symbolic powers highly evolved ?
J. Ramanujan Math. Soc. 28A (2013), 247–266.
arxiv:1103.5809.

Conjecture of Brian Harbourne and Craig Huneke
Chudnovsky’s result
ω1 + n − 1
ωt
≤ ·
n
t
for n = 2 follows fromJ (2t) ⊂ Mt J tfor any homogeneous ideal of points J = I(S) in
K[X 0 , X 1 , X 2 ].
Generalization for n ≥ 2.

Conjecture of Brian Harbourne and Craig Huneke

Let J = ∩j I(pj )mj be a fat points ideal in R.
Conjecture (Harbourne and Huneke). For all t > 0,
J (tn) ⊂ Mt(n−1) J t .

Evolutions
Andrew Wiles

Richard Taylor

Matthias Flach

Evolutions are certain kinds of ring homomorphisms that arose
in proving Fermat’s last Theorem (A. Wiles, R. Taylor,
M. Flach).
An important step in the proof was to show that in certain
cases only trivial evolutions occurred.

Evolutions
D. Eisenbud and B. Mazur showed the question of triviality
could be translated into a statement involving symbolic
powers. They then made the following conjecture incharacteristic 0 :Conjecture (Eisenbud–Mazur) Let P ⊂ C[[x1 , . . . , xd ]] bea prime ideal. Let M = (x1 , . . . , xd ). Then P(2) ⊂ MP.

Evolutions

Heuristically, the main conjecture of Harbourne and Huneke
can be thought of as a generalization of the conjecture of
Eisenbud and Mazur.

B. Harbourne and C. Huneke. Are symbolic powers highly evolved ?
J. Ramanujan Math. Soc. 28, (2011)arxiv:1103.5809.

Brian Harbourne
OberwolfachLinear Series on Algebraic
Varieties : 2010-10-03 –2010-10-09

Brian Harbourne, Sandra Di
Rocco, Tomasz Szemberg,
Thomas Bauer
M. Dumnicki, B. Harbourne, T. Szemberg and H. Tutaj-Gasińska.
Linear subspaces, symbolic powers and Nagata type conjectures.
Adv. Math. 252 (2014), 471–491.
https://owpdb.mfo.de/detail?photo_id=13201

Marcin Dumnicki

Chudnovsky’s conjecture
ω1 (S) + n − 1Ω(S) ≥
holds
nfor generic finite subsets in P3 .

M. Dumnicki. Symbolic powers of ideal of generic points in P3 .
J. Pure Applied Algebra 216 (2012), 1410–1417.

Thomas Bauer and Thomasz Szemberg

Th. Bauer and T. Szemberg. The effect of points fattening in
dimension three. Recent advances in Algebraic Geometry. A
volume in honor of Rob Lazasfeld’s 60th Birthday LMS,
Cambridge University Press 2015.

Further references
M. Baczyńska, M. Dumnicki, A. Habura, G. Malara,
P. Pokora, T. Szemberg, J. Szpond and H. Tutaj-Gasińska.
Points fattening on P1 × P1 and symbolic powers of
bi–homogeneous ideals. J. Pure Applied Algebra 218 (2014),
1555–1562.

C. Bocci, S.M. Cooper and B. Harbourne. Containment resultsfor ideals of various configurations of points in PN . J. Pure
Applied Algebra 218 (2014), 65–75.

Further references
Giuliana Fatabbi, Brian Harbourne, Anna Lorenzini,
Inclics, galaxies, star configurations and Waldschmidt
constants (2013)
Cristiano Bocci, Susan Cooper, Elena Guardo,
The Waldschmidt constant for squarefree monomial ideals(2015)
Marcin Dumnicki, Lucja Farnik, Halszka Tutaj-Gasinska,
Asymptotic Hilbert Polynomial and a bound for Waldschmidt
constants (2015)

March 3, 2023

On the degree of hypersurfaces
with given singularities
Michel Waldschmidt
Professeur Émérite, Sorbonne Université,Institut de Mathématiques de Jussieu, Paris
http://www.imj-prg.fr/~michel.waldschmidt/