A smooth cubic surface \(S \subseteq \PP^3\) is the blowup of \(\PP^2\) at \(6\) points in general position, embedded by the anticanonical system \(\abs{-K_S} = \abs{3H - \sum_{i=1}^{6} E_i}\). So \(\Pic S = \ZZ^7\) with basis \(H, E_1, \dots, E_6\), and \(K_S^2 = 9 - 6 = 3 = \deg S\).
(The 27 lines)
A line on \(S\) is a class \(L\) with \(L^2 = -1\) and \(L \cdot K_S = -1\), and there are exactly \(27\):
the \(6\) exceptional curves \(E_i\);
the \(15\) strict transforms \(H - E_i - E_j\) of the lines through two of the points;
the \(6\) strict transforms \(2H - \sum_{j \neq i} E_j\) of the conics through five of the points.
The count \(6 + 15 + 6 = 27\) is the answer to give, and it is a computation in \(\Pic\), not a geometric miracle: adjunction says a line has \(2g-2 = -2 = L^2 + L\cdot K\), so \(L^2 = -1\) and \(L \cdot K = -1\), and enumerating integer solutions in the basis gives exactly those three families.
A line on \(S\) is a \((-1)\)-curve, so each of the \(27\) can be contracted; different choices of six pairwise disjoint lines realise \(S\) as a blowup of \(\PP^2\) in different ways, which is why the configuration has a large symmetry group, the Weyl group of \(E_6\) of order \(51840\).
That \(S\) is anticanonically embedded is what makes it a del Pezzo surface of degree \(3\); the same analysis with \(9 - d\) blown-up points gives the del Pezzo surfaces of each degree, and the line counts \(27\), \(16\), \(10\) follow the same way.
Dependencies
T-SRFADJThe adjunction formula on a surfaceusesFE-SRFBLOWWhat a blowup does to $\Pic$, to $K$, and to the intersection formuses