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Showing 31 of 280 problems in Algebraic geometry

NT-ARITH-002

Birch–Swinnerton-Dyer Conjecture

IconicConjecture1965

Let

E / Q

be an elliptic curve. Write

E (Q) ≅ E (Q)_{tors} \oplus Z^{r}

, and let

L (E, s)

be the Hasse–Weil

L

-function of

E

, continued to an entire function by modularity. Then

ord_{s = 1} L (E, s) = r .

NT-GALOIS-031

Fontaine–Mazur Conjecture

LandmarkConjecture1995

Let

p

be a prime, put

G_{Q} = Gal (\overline{Q} / Q)

, and let

ρ : G_{Q} ⟶ GL_{n} (\overline{Q}_{p})

be a continuous irreducible representation, unramified outside finitely many primes and potentially semistable at

p

, meaning semistable after restriction to the Galois group of a finite extension of

Q_{p}

. Then there exist a smooth projective variety

X / Q

, integers

i, j

, and a finite extension

E / Q_{p}

such that

ρ

, after scalar extension to

E

, is a subquotient of

H_{\overset{e}{ˊ} t}^{i} (X_{\overline{Q}}, E) (j)

NT-DIO-036

$abc$ Conjecture

LandmarkConjecture1985

For every

ε > 0

, there is a constant

K_{ε} > 0

such that, whenever coprime positive integers

a, b, c

satisfy

a + b = c

c < K_{ε} rad (ab c)^{1 + ε}, rad (n) = p ∣ n \prod p .

NT-DIO-011

Fermat–Catalan Conjecture

MajorConjecture1995

There are only finitely many sextuples

(a, b, c; m, n, k) \in Z_{> 0}^{3} \times Z_{\geq 2}^{3}

such that

a^{m} + b^{n} = c^{k}, g cd (a, b, c) = 1, \frac{1}{m} + \frac{1}{n} + \frac{1}{k} < 1.

NT-DIO-012

Beal Conjecture

MajorConjecture1993

If positive integers

A, B, C, x, y, z

satisfy

A^{x} + B^{y} = C^{z}, x, y, z > 2,

then

g cd (A, B, C) > 1

NT-DIO-013

Erdős–Straus Conjecture

MajorConjecture1948

For every integer

n \geq 2

, there exist positive integers

x, y, z

such that

\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z} .

ALG-POLY-001

Plane Jacobian Conjecture

LandmarkCanonical finite case1939

F = (F_{1}, F_{2}) : C^{2} \to C^{2}

is a polynomial map and

det (\frac{\partial F _{i}}{\partial x _{j}})_{1 \leq i, j \leq 2} \in C^{\times},

then there is a polynomial map

G : C^{2} \to C^{2}

satisfying

G \circ F = F \circ G = id_{C^{2}}

ALG-GALCOH-018

Serre’s Conjecture II

LandmarkConjecture1962

Let

k

be a perfect field of Galois cohomological dimension at most

2

, and let

G

be a semisimple simply connected linear algebraic group over

k

. Then

H^{1} (k, G) = {1}

; equivalently, every

G

-torsor over

Spec k

has a

k

-rational point.

AG-CYCLE-001

Hodge Conjecture

IconicConjecture1950

Let

X

be a smooth projective variety over

C

. For every integer

p \geq 0

H^{2 p} (X, Q) \cap H^{p, p} (X) = span_{Q} {cl (Z) : Z \subseteq X is a closed algebraic subvariety of codimension p},

where

cl (Z)

is the singular-cohomology cycle class and

H^{p, p} (X)

is the

(p, p)

-summand of the Hodge decomposition of

H^{2 p} (X, C)

AG-CYCLE-003

Tate Conjecture

LandmarkConjecture1963

Let

k

be a finitely generated field,

X / k

a smooth projective variety,

r \geq 0

, and

ℓ \neq = char k

a prime. The image of

cl_{ℓ} : CH^{r} (X) \otimes_{Z} Q_{ℓ} ⟶ H_{\overset{e}{ˊ} t}^{2 r} (X_{\overset{ˉ}{k}}, Q_{ℓ} (r))

equals the invariant subspace

H_{\overset{e}{ˊ} t}^{2 r} (X_{\overset{ˉ}{k}}, Q_{ℓ} (r))^{Gal (\overset{ˉ}{k} / k)},

where

CH^{r} (X)

is the group of codimension-

r

algebraic cycles modulo rational equivalence.

AG-ANAB-005

Grothendieck Section Conjecture

LandmarkConjecture1983

Let

K

be a field finitely generated over

Q

, put

G_{K} = Gal (\overset{ˉ}{K} / K)

, and let

X / K

be a smooth, projective, geometrically connected curve of genus

g \geq 2

. After choosing a geometric base point, its étale fundamental groups fit into the exact sequence

1 \to π_{1}^{\overset{e}{ˊ} t} (X_{\overset{ˉ}{K}}) \to π_{1}^{\overset{e}{ˊ} t} (X) \to G_{K} \to 1.

The map sending

x \in X (K)

to the conjugacy class (under

π_{1}^{\overset{e}{ˊ} t} (X_{\overset{ˉ}{K}})

) of the section

G_{K} \to π_{1}^{\overset{e}{ˊ} t} (X)

determined by

x

is a bijection.

AG-DIO-006

Bombieri–Lang Conjecture

LandmarkConjecturec. 1971

K

is a number field and

X / K

is a smooth projective variety of general type, then

X (K)

is not Zariski dense in

X

AG-DIO-007

Vojta’s Main Conjecture

LandmarkConjecture1987

Let

X

be a smooth projective variety over a number field

K

, let

D

be a simple-normal-crossings divisor,

A

a big divisor,

S

a finite set of places of

K

r \geq 1

, and

ε > 0

. There is a proper Zariski-closed

Z \subset X

containing the support of

D

such that for every

P \in X (\overset{ˉ}{K}) ∖ Z

with

[K (P) : K] \leq r

m_{D, S} (P) + h_{K_{X}} (P) \leq d_{K} (P) + ε h_{A} (P) + O (1) .

Here

m_{D, S}

is the proximity function, heights use fixed Weil-height choices, the

O (1)

constant may depend on the fixed data

(X, D, A, K, S, r, ε)

but is independent of

P

, and

d_{K} (P) = \frac{lo g N _{K / Q} D _{K (P) / K}}{[ K ( P ) : K ]}

is the normalized logarithmic relative discriminant, with

D_{K (P) / K}

the relative discriminant ideal.

AG-MMP-010

Abundance Conjecture

LandmarkConjecturec. 1985

Let

(X, Δ)

be a projective Kawamata-log-terminal pair over

C

, with

K_{X} + Δ

Q

-Cartier

Q

-divisor. If

K_{X} + Δ

is nef, then it is semiample: some positive integral multiple is Cartier and base-point-free.

AG-MMP-012

Termination of Flips Conjecture

LandmarkConjecturec. 1988

Let

(X_{0}, Δ_{0})

be a projective

Q

-factorial Kawamata-log-terminal pair over

C

. Every sequence

(X_{0}, Δ_{0}) ⇢ (X_{1}, Δ_{1}) ⇢ (X_{2}, Δ_{2}) ⇢ \dots

in which each arrow is a

(K_{X_{i}} + Δ_{i})

-flip terminates after finitely many steps.

AG-SING-013

Resolution of Singularities in Positive Characteristic

LandmarkOpen problem1964

Let

k

be a perfect field of positive characteristic and

X

an integral separated scheme of finite type over

k

. There exist a smooth integral

k

-scheme

Y

and a proper birational morphism

f : Y \to X

that is an isomorphism over the regular locus

X_{reg}

AG-DIFF-021

Grothendieck–Katz $p$ -Curvature Conjecture

LandmarkConjecture1969–1972

Let

K

be a number field, let

X / K

be a smooth connected variety, and let

(E, \nabla)

be an algebraic vector bundle with integrable connection on

X

. After extending these data over

O_{K} [S^{- 1}]

for some finite set

S

of finite places, reduce at a place

p \in / S

of residue characteristic

p

. For a local derivation

D

on the reduction

X_{p}

, define the

p

-curvature by

ψ_{p} (D) = \nabla_{D}^{p} - \nabla_{D^{[p]}},

where

D^{[p]}

is the

p

-fold iterate of

D

as a derivation. If

ψ_{p} (D) = 0

for every local derivation

D

and all but finitely many

p

, then, after any embedding

K ↪ C

, the analytic monodromy of

(E, \nabla)

X (C)

has finite image; equivalently, the connection becomes trivial after a finite étale cover of

X_{C}

AG-HYP-022

Green–Griffiths–Lang Conjecture

LandmarkConjecture1980

Let

X

be a smooth projective complex variety of general type. There exists a proper Zariski-closed subset

Y ⊊ X

such that every nonconstant holomorphic map

f : C \to X

satisfies

f (C) \subseteq Y

AG-CYCLE-002

Generalized Hodge Conjecture

MajorConjecture1969

Let

k, c \geq 0

be integers and let

X

be a smooth projective complex variety. If

V \subseteq H^{k} (X, Q)

is a rational Hodge substructure such that

V_{C}^{p, q} = 0

whenever

p < c

q < c

, where

V_{C} = V \otimes_{Q} C = ⨁_{p + q = k} V_{C}^{p, q}

, then there exists a closed algebraic subset

Z \subseteq X

of codimension at least

c

such that

V \subseteq ker (H^{k} (X, Q) ⟶ H^{k} (X ∖ Z, Q)) .

AG-CYCLE-004

Lefschetz Standard Conjecture $B$

MajorConjecture1968

Let

X

be a smooth projective variety of dimension

d

over an algebraically closed field, let

H^{*}

be a Weil cohomology theory with characteristic-

0

coefficient field, and let

L (α) = h ⌣ α

for the class

h

of an ample divisor. For every

i < d

, the inverse of the hard-Lefschetz isomorphism

L^{d - i} : H^{i} (X) \sim H^{2 d - i} (X)

is induced by an algebraic correspondence in

CH^{i} (X \times X) \otimes Q

, where

CH^{i}

denotes codimension-

i

algebraic cycles modulo rational equivalence.

AG-UNLIKELY-008

Zilber–Pink Conjecture for Abelian Varieties

MajorConjecture1999–2005

Let

A

be a complex abelian variety. For

s \geq 0

, let

A^{[s]}

be the union of all torsion cosets

τ + B

, where

τ \in A

is a torsion point and

B \subseteq A

is an abelian subvariety with

codim_{A} B \geq s

. If

X \subseteq A

is an irreducible closed subvariety not contained in any proper torsion coset, then

X \cap A^{[d i m X + 1]}

is not Zariski dense in

X

AG-SURF-009

Nagata Conjecture

MajorConjecture1959

Let

r \geq 10

, let

p_{1}, \dots, p_{r}

be very general points of

P_{C}^{2}

, meaning outside a countable union of proper Zariski-closed subsets of the configuration space, and let

m_{1}, \dots, m_{r} \geq 0

. If a plane curve of degree

d

has multiplicity at least

m_{i}

at every

p_{i}

, then

d r > i = 1 \sum r m_{i} .

AG-MMP-011

Nonvanishing Conjecture

MajorConjecturec. 1988

Let

(X, Δ)

be a projective Kawamata-log-terminal pair over

C

with

K_{X} + Δ

Q

-Cartier

Q

-divisor. If

K_{X} + Δ

is pseudo-effective, then there is an effective

Q

-divisor

D \geq 0

with

D \sim_{Q} K_{X} + Δ

AG-EMBED-014

Hartshorne Complete-Intersection Conjecture

MajorConjecture1974

X \subset P_{C}^{n}

is a smooth closed subvariety with

dim X > 2 n /3

, then

X

is a complete intersection in

P^{n}

: its homogeneous ideal is generated by

codim_{P^{n}} X

homogeneous polynomials.

AG-ADJ-015

Fujita Conjecture

MajorConjecture1985

Let

X

be a smooth projective complex variety of dimension

n

K_{X}

its canonical line bundle, and

L

an ample line bundle. Then

K_{X} \otimes L^{\otimes (n + 1)}

is globally generated and

K_{X} \otimes L^{\otimes (n + 2)}

is very ample.

AG-CY-016

Clemens Conjecture

MajorConjecture1986

Let

X \subset P_{C}^{4}

be a very general smooth quintic hypersurface, meaning one outside a countable union of proper Zariski-closed subsets of the parameter space. For every

d \geq 1

X

contains only finitely many irreducible rational curves of degree

d

; every such curve is a smooth embedded

P^{1}

with normal bundle

O_{P^{1}} (- 1) \oplus O_{P^{1}} (- 1)

AG-RAT-017

Irrationality of a Very General Cubic Fourfold

MajorOpen problemc. 1969

A very general smooth cubic hypersurface

X \subset P_{C}^{5}

, meaning one outside a countable union of proper Zariski-closed subsets of the parameter space, is not rational over

C

AG-RAT-018

Kuznetsov’s Rationality Conjecture for Cubic Fourfolds

MajorConjecture2010

Let

X \subset P_{C}^{5}

be a smooth cubic fourfold. In the bounded derived category

D^{b} (Coh X)

of coherent sheaves on

X

, put

A_{X} = ⟨ O_{X}, O_{X} (1), O_{X} (2) ⟩^{⊥},

where

^{⊥}

denotes the full right-orthogonal subcategory. Then

X

is rational if and only if there is a complex K3 surface

S

with an exact

C

-linear triangulated equivalence

A_{X} ≃ D^{b} (Coh S)

AG-CONE-019

Morrison–Kawamata Cone Conjecture

MajorConjecture1993–1997

Let

(X, Δ)

be a projective

Q

-factorial Kawamata-log-terminal pair over

C

with

K_{X} + Δ \equiv 0

, and put

Nef^{e} (X) = Nef (X) \cap Eff (X)

inside

N^{1} (X)_{R}

, where

Eff (X)

is the cone generated by classes of effective Cartier divisors. There is a rational polyhedral cone

Π \subset Nef^{e} (X)

such that

Nef^{e} (X) = g \in Aut (X, Δ) ⋃ g^{*} Π,

and interiors of

g^{*} Π

and

h^{*} Π

are disjoint unless the cones coincide.

AG-ZERO-020

Bloch Conjecture for Surfaces with $p_g=0$

MajorConjecture1975

Let

X

be a smooth projective complex surface with

p_{g} (X) = h^{0} (X, K_{X}) = 0

. The Albanese map

alb_{X} : CH_{0} (X)^{0} ⟶ Alb (X) (C)

from degree-zero zero-cycles modulo rational equivalence is an isomorphism.

NUM-001

Twelve-stage ninth-order explicit Runge–Kutta method

MajorCanonical finite casec. 1964

Determine whether there exist real coefficients

a_{ij}

(

1 \leq j < i \leq 12

) and

b_{i}

(

1 \leq i \leq 12

), with

c_{i} = \sum_{j < i} a_{ij}

, such that the explicit Runge--Kutta one-step map

k_{i} = f (t + c_{i} h, y + h j < i \sum a_{ij} k_{j}), y^{+} = y + h i = 1 \sum 12 b_{i} k_{i}

has classical order

9

: for every integer

q \geq 1

, every sufficiently smooth

f : R \times R^{q} \to R^{q}

, and every exact solution of

y^{'} = f (t, y)

, one step initialized at

y (t)

satisfies

y^{+} - y (t + h) = O (h^{10})

h \to 0