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  "body": "\n## Statement\n\nLet $S$ be a smooth projective [[algebraic surface|KFTECAATKG]] over an algebraically closed field.\nFor [[divisors|Z9DN23DAE2]] $D$ and $E$ on $S$, let $D \\cdot E$ be their [[intersection number|4HV4WTAX9P]].\nLet $H$ be an [[ample divisor|RRK8P6TXG2]] on $S$, and let $D$ be a divisor with $D \\cdot H = 0$.\nThen $D^2 \\le 0$, and $D^2 = 0$ holds if and only if $D \\cdot E = 0$ for every divisor $E$ on $S$.\n\n## Remarks\n\nEquivalently, the intersection form on the real vector space $\\operatorname{NS}(S) \\otimes \\R$, where $\\operatorname{NS}(S)$ is the [[Néron–Severi group|XMZ09SBM0X]], has [[signature|07EZA80NEH]] $(1, \\rho - 1)$, where $\\rho$ is its dimension.\nThe proof uses Riemann–Roch on the surface, which shows that if $D^2 \u003e 0$, then $nD$ or $-nD$ is linearly equivalent to an effective divisor for some $n \\ge 1$.\nA nonzero effective divisor meets the ample $H$ positively, which contradicts $D \\cdot H = 0$.\nA useful consequence is the inequality $(D^2)(H^2) \\le (D \\cdot H)^2$ for $H$ ample and $D$ arbitrary, obtained by applying the theorem to $(H^2) D - (D \\cdot H) H$.\nExample: on $S = \\mathbb{P}^1 \\times \\mathbb{P}^1$ with fibre classes $F_1, F_2$, one has $F_1^2 = F_2^2 = 0$ and $F_1 \\cdot F_2 = 1$; the ample $H = F_1 + F_2$ is orthogonal to $D = F_1 - F_2$, and indeed $D^2 = -2 \u003c 0$.\nOver $\\C$, Hodge theory gives the more general fact that the cup product on $H^2(S, \\R)$ has signature $(2 p_g + 1, h^{1,1} - 1)$, where $p_g = h^{2,0}$.\n\n## Explanations\n\n### L1 human\n\nOn a surface, curves can be intersected and their intersection numbers form a quadratic form.\nThe theorem says that this form has only one positive direction, given by an ample curve, and is negative on everything orthogonal to it.\nSo its shape is like the light cone in special relativity, with one time direction and many space directions.\n\n### L2 human\n\nThe Hodge index theorem is the main structural fact about intersection theory on surfaces.\nIt is used in a proof of the Riemann hypothesis for curves over finite fields, applied to the graph of Frobenius on a product of curves.\nIt is also used in the study of the Néron–Severi lattice of K3 surfaces and in many inequalities between Chern numbers.\nIts higher-dimensional analogues are the Hodge–Riemann bilinear relations and the hard Lefschetz theorem.\n",
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