Week 4
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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].
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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.
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