Math 2.0

Algebraic geometry Arithmetic and Diophantine geometry Conjecture

Manin conjecture

Builds on Fano varietyAlgebraic number fieldHeight functionCanonical bundlePicard groupZariski topologyRational point

Statement

Let KK be an algebraic number field and let VV be a smooth Fano variety over KK. Assume that V(K)V(K) is dense in VV for the Zariski topology. Let HH be a height on V(K)V(K) relative to the anticanonical divisor −KV-K_V, and let ρ\rho be the rank of the Picard group of VV. Then there are a nonempty Zariski open subset U⊆VU \subseteq V and a constant c>0c > 0 such that the number NU(B)=#{P∈U(K):H(P)≤B}N_U(B) = \#\set{P \in U(K) : H(P) \le B} satisfies NU(B)∼c B(log⁡B)ρ−1as B→∞.N_U(B) \sim c \, B (\log B)^{\rho - 1} \quad \text{as } B \to \infty.

Remarks

Franke, Manin and Tschinkel stated it in 1989, and Batyrev and Manin extended it to other heights; Peyre later gave a conjectural formula for the constant cc. The open set UU is needed, since a subvariety such as a line on a cubic surface can contain more points of bounded height than the rest of the variety. For V=ℙnV = \mathbb{P}^n over ℚ\Q one has ρ=1\rho = 1, and the anticanonical height is H0n+1H_0^{n+1} with H0H_0 the usual height. So the prediction N(B)∼cBN(B) \sim cB follows from Schanuel's count of points of bounded height. The conjecture is proved for flag varieties, toric varieties and several other families, and is open in general. Batyrev and Tschinkel found examples in which removing a Zariski closed set is not enough, and current formulations remove a thin set instead.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Algebraic number field, concept, no evidence levelAlgebraic number fieldconceptCanonical bundle, concept, no evidence levelCanonical bundleconceptFano variety, concept, no evidence levelFano varietyconceptHeight function, concept, no evidence levelHeight functionconceptPicard group, concept, no evidence levelPicard groupconceptRational point, concept, no evidence levelRational pointconceptZariski topology, concept, no evidence levelZariski topologyconceptManin conjecture, conjecture, no evidence levelManin conjectureconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Algebraic number field concept
  2. Canonical bundle concept
  3. Fano variety concept
  4. Height function concept
  5. Picard group concept
  6. Rational point concept
  7. Zariski topology concept
  8. Manin conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Manin_conjecture
About this page
TagXSZ49NF2AE
URIm2://XSZ49NF2AE
Kindconjecture
WritingW0 stub
MSC14G05, 11G50, 14J45
StatusThis copy of the site has no build data. Run m2 status XSZ49NF2AE.
AboutRational point, Fano variety
UsesFano variety, Algebraic number field, Height function, Canonical bundle, Picard group, Zariski topology, Rational point
Same as
Changed2 November 2026

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