w2 such that the intersection of P and F is the union of 3 lines, one can recover the fact that K2=45. Let us give some details of that computation: By a generic point of the cubic F goes 6 lines. Let s be a point of S and let Ls be the corresponding line on the cubic F. Let Cs be the divisor on S parametrizing lines that cut the line Ls. The self-intersection of Cs is equal to the intersection number of Cs and Ct for t a generic point. The intersection of Cs and Ct is the set of lines on F that cuts the disjoint lines Ls and Lt. Consider the linear span of Ls and Lt : it is an hyperplane into P4 that cuts F into a smooth cubic surface. By well known results on a cubic surface, the number of lines that cuts two disjoints lines is 5, thus we get (Cs) 2 =Cs Ct=5. As K is numerically equivalent to 3Cs, we obtain K 2 =45. c) The natural composite map: S -> G(2,5) -> P9 is the canonical map of S. It is an embedding. See also Hodge theory References Algebraic surfaces Complex surfaces