Week 4 ====== Applications of intersection numbers D1D2, esp. to resolution Y -> X of isolated surface singularities. First introduction to canonical class KX We write S for a nonsingular surface and C in S for a projective curve. The surface S is quasiprojective but not necessarily projective. The first point is to understand C^2 = degree of the normal bundle N_{S|C}. If C is irreducible and moves in an algebraic family (e.g. a linear system), C^2 = CC' where C' is a different curve, with C' ~ C under linear or algebraic or some other equivalence. Then of course CC' >= 0. An important case is when S in PP^n is a projective surface embedded by a linear system H of hyperplanes. Then H moves in a _very ample_ linear system, and H^2 = HH' > 0. In fact then HD > 0 for D any effective curve on S (because the components of D are curves in PP^n and so each has degree > 0). We discuss later the criteria for ampleness, including the fact that D^2 > 0 and DC > 0 for every curve C in S implies that D is ample. We need to get used to the idea of negative curves: C in S with C^2 < 0. This simple case is fairly typical: let S be the surface covered by 2 copies of AA^2, say S0 = AA^2 and S1 = AA^2 glued together on x,y <> 0 by y = x^-1 and t = x^a*z for some a > 0. Then S contains the central curve C = PP^1 obtained by glueing the x-axis of S0 and the y-axis of S1. Moreover, the identification t = x^a*z (and z = y^a*t) is linear in t and z. If we think of S0 projected to its x-axis, the fibres are AA^1 with coordinate z, and these are identified with the fibres of S1 -> AA^1 with the power of x as a transition function. Thus S is an AA^1 fibre bundle over its central curve C iso PP^1. The normal bundle to C in S is N_{S|C} = O_{PP^1}(-a), and C^2 = -a. It is instructive to calculate the global regular polynomial functions on S, that is Ga(S, OS). First, x is not regular: it has zeros on the t-axis on S0, but x = y^-1, and has poles on the z-axis of S1. On the other hand, ui = x^i*z = y^{a-i}t for i = 0,.. a is regular on both pieces S0 and S1. One sees that these generate the ring Ga(S, OS), and that the relations between them are the 2x2 minors of [ u0 u1 .. u_{a-1} ] [ u1 u2 .. u_a ], the same as the relations for the rational normal curve of degree a in PP^a, that is, the image of PP^1 -> PP^a given by S^a(u,v) = u^a : u^{a-1}*v : .. v^a. This means that the above abstract surface S is the blowup of the affine cone over the rational normal curve of degree a (the cyclic quotient singularity 1/a(1,1)). In the case a = 1, this is the standard blowup of the origin in AA^2. See my cyclic chapter: in the general case 1/r(1,b), the resolution of singularities have a chain of k negative curves Ci with Ci^2 = -ai, where r/b = [a1,.. ak]. === There is a general theorem that if S is a projective surface and f: S -> X a generically finite morphism (that is, f(S) is still a surface), and { Ci } are curves contracted to P by f, then ( CiCj )_{ij} is a negative definite matrix. In other words, any nonzero linear combination D = sum ni*Ci has D^2 < 0. The detailed proof is in Chapter A of my Park City lectures. The basic idea: choose h in m in O_{X,P} a regular element, so that H = div h is an effective Cartier divisor (think of it as a hyperplane section of X through P). The ideal IH = OX*h is principal, so that OX(H) iso OX in a Zariski neighbourhood of P, and OS(f^*H) iso OS in a neighbourhood of f^-1P. Therefore the intersection numbers (f^*H).Ci = 0 for all Ci. On the other hand, H is an effective divisor passing through P, so that f^*H contains f^-1P set-theoreticaly, so has f^*H = sum ni*Ci + Ga, where the residual divisor Ga does not have any of the Ci as components. The key point is that the support of f^*H is connected, so that Ga must intersect at least one of the Ci. This may seem obvious, but it depends on a technically advanced result, the Zariski connectedness theorem. See Hartshorne III.11.4. Assuming that point, f^*H.Ci = (sum nj*Cj + Ga).Ci = 0, so that the effective divisor D = sum nj*Cj has D.Ci <= 0 for all i and < 0 for some i. The negative definite assertion then follows by a straightforward argument in quadratic algebra. ===