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  "body": "\n## Example\n\nA *cubic surface* is the zero set in [[projective space|ZT9A6JK8QF]] $\\mathbb{P}^3$ of a homogeneous polynomial of degree $3$ in four variables.\nA concrete smooth one is the Fermat cubic over $\\C$,\n$$X = \\set{[x : y : z : w] \\in \\mathbb{P}^3 : x^3 + y^3 + z^3 + w^3 = 0}.$$\nIt is smooth: the partial derivatives $3x^2, 3y^2, 3z^2, 3w^2$ vanish together only at $x = y = z = w = 0$, which is not a point of $\\mathbb{P}^3$.\nBy the [[adjunction formula|RYB6F7JFGV]], a smooth surface of degree $3$ in $\\mathbb{P}^3$ has canonical divisor $K_X = (3 - 4) H = -H$, where $H$ is a plane section.\nSo $-K_X = H$ is ample, and $X$ is a [[del Pezzo surface|AV2ZAAZXN9]] of degree $K_X^2 = H^2 = 3$.\n\n## Remarks\n\nThe Fermat cubic contains the line $\\set{x = -y, \\ z = -w}$, since then $x^3 + y^3 = 0$ and $z^3 + w^3 = 0$.\nMore generally, for cube roots of unity $\\zeta, \\zeta'$ the lines $\\set{x = -\\zeta y, \\ z = -\\zeta' w}$ lie on $X$, which gives $9$ lines.\nThe other two ways of pairing the four variables give $9$ more each.\nSo there are $27$ lines in all, as the [[twenty-seven lines theorem|VQK5HQFYEM]] predicts.\nEvery smooth cubic surface over an algebraically closed field is the blow-up of $\\mathbb{P}^2$ at six points in general position, so it is a rational surface.\nThe Clebsch surface, given by $\\sum x_i = 0$ and $\\sum x_i^3 = 0$ in $\\mathbb{P}^4$, is a smooth cubic surface on which all $27$ lines are defined over $\\R$.\n\n## Explanations\n\n### L1 human\n\nA cubic surface is the set of solutions of one cubic equation in four homogeneous coordinates.\nThe Fermat cubic $x^3 + y^3 + z^3 + w^3 = 0$ is the most symmetric example.\nIt is smooth, it can be parametrized by the plane, and it contains exactly $27$ straight lines, which can be written down by hand.\n",
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