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  "body": "\n## Statement\n\nLet $k$ be an [[algebraically closed field|0PF6TM8TWT]].\nLet $X \\subset \\mathbb{P}^3$ be a smooth cubic surface over $k$.\nThis means that $X$ is the zero set in [[projective space|ZT9A6JK8QF]] of a homogeneous polynomial $F$ of degree $3$.\nMoreover, the partial derivatives of $F$ have no common zero on $X$.\nThen $X$ contains exactly $27$ lines of $\\mathbb{P}^3$.\n\n## Remarks\n\nOne proof identifies $X$ with the [[blow-up|C4KGYAFPN7]] of $\\mathbb{P}^2$ at six points $p_1, \\ldots, p_6$ in general position, so that $X$ is a [[del Pezzo surface|AV2ZAAZXN9]] of degree $3$.\nThe lines on $X$ are exactly the curves $E$ with $E^2 = -1$ and $-K_X \\cdot E = 1$.\nThey are the $6$ exceptional curves over the $p_i$ and the strict transforms of the $15$ lines through two of the points.\nThe other $6$ are the strict transforms of the conics through five of the points, and $6 + 15 + 6 = 27$.\nEach of the $27$ lines meets exactly $10$ of the others, and the symmetry group of the incidence configuration is the Weyl group of type $E_6$, of order $51840$.\nSmoothness is needed: the cone over a smooth plane cubic is a singular cubic surface.\nIt contains infinitely many lines, one through the vertex and each point of the base curve.\nThe result goes back to Cayley and Salmon in 1849.\n\n## Explanations\n\n### L1 human\n\nOne might expect a curved surface given by a cubic equation to contain no straight lines at all.\nBut every smooth cubic surface contains exactly $27$ straight lines, no more and no fewer.\nFor the Fermat cubic $x^3 + y^3 + z^3 + w^3 = 0$ they can be listed: $x = -\\zeta y$, $z = -\\zeta' w$ with cube roots of unity $\\zeta, \\zeta'$, and the analogous lines for the other pairings of the variables.\n\n### L2 human\n\nThe theorem is one of the first results of enumerative geometry and a model for counting curves on varieties.\nThe lines are the $(-1)$-curves of the del Pezzo surface of degree $3$.\nSimilar counts hold for the other degrees, for instance $16$ such curves on a del Pezzo surface of degree $4$ and $240$ on one of degree $1$.\nOver the real numbers a smooth cubic surface has $3$, $7$, $15$ or $27$ real lines.\n",
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