1 Lecture 1: The conjectures of Lang and Vojta

• http://math.bu.edu/people/abrmovic/IHP

WORK IN CHARACTERISTIC 0

ARITHMETIC ↔^{?} GEOMETRY

1.1 Theorem (Caporaso-Harris-Mazur)

Assume that Lang’s “weak” conjecture holds true. Let K be a number field. Then there is a number B(K, g), such that the number of K-rational points on any curve of genus g > 1 over K is bounded by B(K, g).

Exerc. on many points What’s known on Lang

Varieties:

1.2 Definition

A variety defined over k is a geometrically integral scheme of finite type over k. If k ⊂ k_{1} we write X_{k_{1}} for X ×_{Spec k} Speck_{1}.

1.3 Definition

Let X be a variety. A subscheme Z ⊂ X is called a strict normal crossings divisor if for each point x ∈ Z, there is a regular system of parameters y_{1}, …, y_{dim X} for x in X (in particular the point x ∈ X is supposed to be a regular point on X), such that Z is given on a Zariski neighborhood of x by the equation y_{1} ⋯ y_{l} = 0.

Kodaira dimensions Let X be a variety, and L an invertible sheaf. Denote

R(X, L) = ⊕_{n≥0}H^{0}(X, L^{n}),

the algebra of sections of L.

1.4 Definition

The Iitaka dimension κ(X, L) is defined as follows:

• κ(X, L) = tr. deg.R(X, L) − 1, if this is non-negative;

• κ(X, L) = −∞, otherwise.

why −∞ properties characterize

If κ(X, L) = dim X we say that L is big on X.

Given a smooth variety, denote by ω_{X} its dualizing sheaf. It is common to denote the associated divisor class by K_{X}, and then write ω_{X} = O_{X}(K_{X}). ^{abuse}

1.5 Definition

Let X be a smooth proper variety. Then the Kodaira dimension κ(X) of X is defined to be κ(X, ω_{X}). ^{properties}

1.6 Definition

Let X̄ be a smooth proper variety, and let D ⊂ X̄ be a reduced normal crossings divisor. Denote X = X̄ D. The Logarithmic Kodaira dimension κ̄(X) is defined to be κ(X, ω_{X}(D)).

sense properties

1.7 Definition

Let X be a variety. Let X′ be a nonsingular proper variety birational to X. Then the Kodaira dimension κ(X) of X is defined to be κ(X)′. ^{sense}

1.8 Definition

Let X be a variety. Let X′ → X be a desingularization of X. Then the logarithmic Kodaira dimension κ̄(X) of X is defined to be κ̄(X′). ^{sense}

1.9 Definition

A variety X is said to be of general type if κ(X) = dim X. ^{ω big}

1.10 Definition

A variety X is said to be of logarithmic general type if κ̄(X) = dim X ^{ω(D) big}

Curves: κ(X) = −∞.

Geometry: κ(X) = −∞ ⟺ XK̄ = ℙ^{1} Arithmetic: after a number field extension we have infinitely many points. ^{twists}

0. Geometry: κ(X) = 0 ⟺ XK̄ is elliptic Arithmetic: after a number field extension we have infinitely many points (but fewer). ^{twists}

1. Geometry: κ(X) = 1 ⟺ g(X) > 1 Arithmetic: at most finitely many points (Falt-

ings) Lang-Vojta?

Curves with divisors: ^{Siegel} ℙ^{1} {p_{1}, …, p_{n}}:

E {p_{1}, …, p_{n}}:

K = a number field S = a finite set of finite places O_{K,S} = the ring of S-integers in K

1.11 Lang’s “weak” conjecture

Let X be a variety of general type over a number field K. Then the set of K-rational points X(K) is not Zariski-dense in X. ^{gen. Mordell}

1.12 Lang-Vojta conjecture

Let X → Spec O_{K,S} be a scheme of finite type such that X := X_{K} is a variety of logarithmic general type. Then the set of points X (O_{K,S}) is not Zariski dense in X.

1. in X

2. compatible 3. gen. Siegel

1.13 Lang’s “geometric” conjecture

Let X be a variety of general type over an algebraically closed field k. Then the union of all subvarieties of X which are not of general type is not Zariski dense in X. ^{will come back}

1.14 Lang’s “strong” conjecture

Let X be a variety of general type over a number field K. Then for any number field extension L ⊃ K, the set X(L) Z(X) of rational points on X outside Z(X) is finite. ^{gen. Mordell}

1.15 Theorem (Caporaso-Harris-Mazur)

Assume that Lang’s “weak” conjecture holds true. Let K be a number field. Then there is a number B(K, g), such that the number of K-rational points on any curve of genus g > 1 over K is bounded by B(K, g).

Exerc. on many points What’s known on Lang

Curves: κ(X) = −∞.

Geometry: κ(X) = −∞ ⟺ XK̄ = ℙ^{1} Arithmetic: after a number field extension we have infinitely many points. ^{twists}

0. Geometry: κ(X) = 0 ⟺ XK̄ is elliptic Arithmetic: after a number field extension we have infinitely many points (but fewer). ^{twists}

1. Geometry: κ(X) = 1 ⟺ g(X) > 1 Arithmetic: at most finitely many points (Falt-

ings) Lang-Vojta?

Curves with divisors: ^{Siegel} ℙ^{1} {p_{1}, …, p_{n}}:

E {p_{1}, …, p_{n}}:

Surfaces:

MINIMAL embarrassing

−∞. Geometry: XK̄ ruled. Arithmetic:

0. Geometry: XK̄ Abelian / finite group,

or K3 / finite group

cases enriques

Arithmetic:

1. Geometry: XK̄ “elliptic” surface ^{cases} Arithmetic:

B–S-D CT–S–S-D

2. “general type”.

Lang geom. Miyaoka

1.16 Theorem (Iitaka)

Suppose κ(X) ≥ 0. There is a dominant map ^{rational} X −−→ I(X), where

1. dim I(X) = κ(X)

2. X_{η} a variety, where η ∈ I(X) the generic point.

3. κ(X_{η}) = 0.

1.17 Conjecture

Suppose κ(X) = −∞. Then X is covered by rational curves.

1.18 Conjecture

The Kodaira dimension is constant in a smooth projective family. ^{sing.}

1.19 “Additivity”

X → Y η ∈ Y generic point. Assume X_{η} a variety ^{gen. finite case}

1.20 Theorem (Iitaka)

κ(X) ≤ κ(X_{η}) + dim Y.

1.21 Conjecture (Iitaka)

κ(X) ≥ κ(X_{η}) + κ(Y).

what’s known

Kollár, Kawamata MM

1.22 Log case?