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  "body": "\n## Statement\n\nLet $K$ be an [[algebraic number field|MWBDFS2KJ5]] and let $V$ be a smooth [[Fano variety|FMHA649HJ9]] over $K$.\nAssume that $V(K)$ is dense in $V$ for the [[Zariski topology|8SQK7RTB63]].\nLet $H$ be a [[height|F1FF505PWH]] on $V(K)$ relative to the anticanonical divisor $-K_V$, and let $\\rho$ be the rank of the [[Picard group|3Z9DCWYXQZ]] of $V$.\nThen there are a nonempty Zariski open subset $U \\subseteq V$ and a constant $c \u003e 0$ such that the number\n$$N_U(B) = \\#\\set{P \\in U(K) : H(P) \\le B}$$\nsatisfies\n$$N_U(B) \\sim c \\, B (\\log B)^{\\rho - 1} \\quad \\text{as } B \\to \\infty.$$\n\n## Remarks\n\nFranke, 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 $c$.\nThe open set $U$ 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.\nFor $V = \\mathbb{P}^n$ over $\\Q$ one has $\\rho = 1$, and the anticanonical height is $H_0^{n+1}$ with $H_0$ the usual height.\nSo the prediction $N(B) \\sim cB$ follows from Schanuel's count of points of bounded height.\nThe conjecture is proved for flag varieties, toric varieties and several other families, and is open in general.\nBatyrev and Tschinkel found examples in which removing a Zariski closed set is not enough, and current formulations remove a thin set instead.\n\n## Explanations\n\n### L1 human\n\nA Fano variety, such as projective space or a smooth cubic surface, usually has infinitely many rational points.\nThe Manin conjecture predicts how many of them have height at most $B$: roughly a constant times $B$ times a power of $\\log B$.\nThe power of the logarithm is fixed by the geometry, namely by the number of independent divisor classes.\nIt turns a counting problem in number theory into a statement about the geometry of the variety.\n",
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      "first_stated": "1989",
      "posed_by": "Yuri Manin, with Jens Franke and Yuri Tschinkel"
    },
    "kind": "conjecture",
    "msc": [
      "14G05",
      "11G50",
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  "msc": [
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