Week 5
Towards classification

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The canonical class KX.
For PP^n, K = O(-n-1)
Most immediate argument: s0 = dx1 \w dx2 \w .. dxn bases K with
x1,..xn affine coordinates on AA^n. One calculates that this has a
pole of order n+1 at infinity.

A superior method is to take s = dx1/x1 \w dx2/x2 .. dxn/xn as
n-form with log poles on each coordinate hyperplane It then also has
a log pole on the hyperplane at infinity dx0/x0.

This is superior for a number of reasons: the complement of all the
coordinate hyperplanes is the algebraic group (GG_m)^n, which has
trivial tangent bundle. It is natural to take xi \d/\d xi
(partial \d) as vector fields with log zeros on the complement (log
zero = derivations that preserve the ideal of the complement, or
tangent fields along the complement).

Monomial coordinates on the big stratum (GG_m)^n are define up to a
choice in the group SL(n,ZZ), and the log n-form
   s = dx1/x1 \w dx2/x2 .. dxn/xn
is invariant under this group. Up to a rational unit factor, we can
also take a QQ-basis of monomials, so increase the choice of bases to
GL(n,QQ). (This is useful for working with weighted PPn, or with
monomial coordinates on Abelian orbifolds or more general toric
varieties.)

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Passing to a hypersurface, and the adjunction formula.
The adjunction formula that everyone remembers is: If Y in X are
nonsingular varieties in codimension 1 then
    KY = (KX + Y)|Y.
Here Y is a Cartier divisor on X, so OX(Y) is a line bundle, and
the formula means its restriction
  OY(KY) = OX(KX + Y) tensor_OX OY.
For example, a hypersurface Y(d) in PP^n has KY = OY(d-n-1). In
particular, a plane curve C(d) in PP^2 has KC = OC(d-3), which has
degree
  d(d-3), giving 2g-2 = d(d-3), that is g = (d-1 choose 2).

A curve C on a nonsingular surface has KC = (KS+C)|C. In particular,
C = PP^1, C^2 = a then KS.C = -2-a. The most basic cases are
  -1-curve with C^2 = -1, KS.C = -1, and
  -2-curve with C^2 = -2, KS.C = 0.

Two proofs of the adjunction formula with different flavours.
(1) Popular from manifolds or complex analysic Y in X. The
restriction of the tangent bundle of X to Y gives rise to the exact
sequence of vector bundles on Y
  0 -> TY -> TX|Y -> N_{X|Y} -> 0
where the final bundle is the normal bundle, or as a sheaf
N_{X|Y} = (IY/IY^2)^dual. Taking top wedge gives

 \w^n TX|Y = \w^{n-1}TY tensor N_{X|Y}

that is -KX | Y = -KY + -Y|Y. This has the advantage that it also
works in higher codimension, in terms of the normal bundle of higher
rank. It has the disadvantage that it depends very strongly on
nonsingularity.

(2) Poincare residue Res_{X|Y}
  Y in X codimension 1 with X nonsingular, and ideal IY
  locally given as IY = (f) = OX*f.
Then there is a canonical exact sequence
  0 -> OX(KX) -> OX(KX+Y) -Res-> OY(KY) -> 0
where the map Res takes
  s/f = (dx1 \w dx2 \w .. \dxn)/f |-> 
    Res s/f = (-1)^i*(dx1 \w .. (omit dxi) \w .. dxn) / (\df/\dxi)
(independent of i in 1..n). Then s is a canonical form or complex
volume element on X, and s/f is a form with pole or order 1 or log
pole on Y. The formula omits one factor \w dxi in the numerator, and
replaces the f in the denominator by its partial derivative
\df/\dxi. This is well-defined independently of the coordinates,
because
  df = sum_i (\d f/\d xi) dxi = 0 on Y.
Taking \w with (omit dx1, dx2) \w dx3 \w .. dxn shows that the
expression with \d f/\d x1 equals the expression with \d f/\d x2.

The advantage is that this gives a well-defined Cartier divisor KY
on Y, even if Y has singularities (say in codim >= 2). To explain:
Y is nonsingular where some \d f/\d xi is invertible, so that the
rational canonical differential Res_{X|Y} s is regular on the
nonsingular locus Y^0 in Y.

Hypersurface singularity Y in X (say isolated surface)
   f = x^2 + y^2 + z^{n+1} = 0
then 2*x*dx + 2*y*dy + (n+1)*z^n*dz = 0 on Y, and
  Res{X|Y} (dx dy dz)/f = (dx \w dy)/z^n
                 = const.*(dx \w dz)/y
bases KY = (KX + Y)_{|Y}.

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Blowup and canonical class

If P in X is a nonsingular point of a n-fold X with local
coordinatess x1,..xn then the n-form
  s = dx1 \w dx2 .. \w dxn     (*)
is a local basis of KX at P.

Let si: X1 = Bl_PX -> X be the blowup. The exceptional divisor is
PP^{n-1} with coordinates y1,.. yn with proportionality
  (x1 : .. : xn) = (y1 : .. yn)
where yn <> 0 (say) proportionality means that set yn = 1 and
yi = xi/xn. So this affine piece has local coordinates
  y1,.. y{n-1}, xn  with xi = xn*yi for i in 1..n-1.
The morphism X1 contracts the exceptional divisor E = PP^{n-1}
(on this affine piece AA^{n-1} given by xn=0). If we substitute
the coordinate transformation xi = xn*yi for i in 1..n-1 into s
of (*), we see that
  si^*(s) = xn^{n-1}* dy1 \w .. dy_{n-1} \ dxn
so that (evaluated as an n-form on the blow up X1), it has zero of
order n-1 along the exceptional divisor E (given by xn=0). Thus
  KX1 = si^*KX + (n-1)E
the divisor (n-1)E is the divisor of zeros of the Jacobian
determinant J(x1,.. xn, y1,..y_{n-1},xn)

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Application to Du Val singularities

P in X in Y = CC^3  such as x^2+y^2+z^{n+1} or x^2+y^3+z^4
Bl_P is covered by 3 affine pieces, but representative case
is x = x1*z, y = y1*z, z = z. Write Y1 -> Y for the blowup
and X1 in Y1 for the birational transform, E for the exceptional
PP^2. Then Y1 -> Y as above has KY1 = si^* KY + 2*E
si^* X = X1 + 2*E, so   KX1 = (KY1 + X1)|X1 (adjunction formula)
= si^*((KY +X)|X) = si^*KX.

KX1 = si^*KX is a _crepant resolution_.
X a bit singular, but KX|X^0 on the nonsingular locus extends
to a line bundle on X (by the rational differential, the Poincaré
residue) X1 ditto, and taking residue then pullback commutes.

If C in S is an irreducible nonsingular curve in a nonsingular
surface, the adjunction theorems says deg KC = (KS+C)C = 2g-2.
A simple consequence, that is key to the MMP is that KS.C < 0 
and C^2 < 0 has just one solution, a -1-curve: C iso PP^1
C^2 = -1, and KS.C = -1.

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What is the classification of surfaces?
One overall interpretation of the classification of surfaces is to
ask the single question: what are the counterexamples to KS ample?
First, KS could be negative on some curves KS.C < 0
    alternative: KS is nef.
It turns out that the KS.C < 0 can be classified. Mori's "Theorem
on the Cone" and "Classification of Extremal Rays" (that I treat
later) say that if KS is not nef then one of the following hold:
  either there exists a -1-curve C (with KC = -1)
  or S has a PP^1-bundle structure S -> B over base curve
    (with KS.fibre = -2)
  or S iso PP^2 (with KS. line = -3)
A -1-curve can be contracted to a nonsingular point S -> S1, and
we can ask the same Q for S1. Doing this inductively until we
reach a surface with no -1-curves is called running a MMP. All of
the components of this result were well understood by the Italian
geometers around 1900, and reworked by Zariski, Kodaira and the
Shafarevich seminar in the 1950s and 1960s. However, before the
ideas of Mori, it would have been impossible to state and prove
the result as the compact package outlined above.

This more-or-less deals with all the obstructions (local and
global) to KS nef. This is Mori theory or the MMP for surfaces.

If KS is nef, but exists KS.C = 0. Then the same argument
based on the adjunction formula gives C^2 < 0 implies
C iso PP^2 and C^2 = -2, so C is a -2 curve.

A negative definite bunch of -2-curves contracts to one or more
Du Val singularities.

This more-or-less deals with all the _local_ obstructions to KS
nef but not ample. The rest is the classification of surfaces,
involving surfaces with KS = 0 or KS = 0 restricted to the fibres
of a fibre space. We will see that this is not so hard to state
but the proof involves some technical some subtleties and
difficulties.

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Linear systems, cohomology and RR on surfaces

The point I especially need is the coarse statement of
RR: for a nonsingular surface and D a divisor with
D^2 > 0,  chi(nD) = H^0(nD) - H^1(nD) + H^0(nD)
   is a quadratic function, with leading term 1/2*D^2,
   and linear term -1/2*D*KX.    (RR-)

The full statement is
   chi(D) = chi(OS) + 1/2*D*(D-KS)
but the coarse statement already gets us a long way.
I need to say something to introduce the coherent
cohomology groups.

Cartier divisor -> Locally free sheaf of OX-modules
                      of rank 1
    /mod linear equivalence

Important special case: D an effective Cartier divisor
on a nonsingular surface. Then
 0 -> OS(-D) -> OS -> OD -> 0.
We know RR for curves, so have an intuitive idea of what
OD1(D2) means. Messing around with this gives a method
of proof of the coarse RR statement (RR-).

Divisor class groups, N^1 and N_1.

We work with n-folds X that are nonsingular, or at least normal.
In the latter case, we have two different definitions of divisors:
Cartier and Weil divisors. We identify a Cartier divisor on X with
a Weil divisor D sum ni*Pi that is _locally principal_, so that at
every P in X there is a rational function f in k(X) with divisor
of poles D+ and zeros D- the positive and negative parts of D.
They form groups
  CDiv X subset WDiv X.

A divisor corresponds to a _divisorial sheaf_ OX(D), defined by
the same formula as RR for curves
  D -> OX(D) = sheaf of rational function with div f + D >= 0.
One technical point we get for free from this definition is that
OX(D) satisfies Serre's condition S2 or is "saturated in codim 2":
In fact the condition div f + D >= 0 that we impose on f in k(X)
refers only to the order of valuations at prime divisors in X,
so that an inclusion OX(D) subset F that is equal in codim 2 is
actually equal. This condition implies that OX(D) has depth 2
at each point. When dim X = 2, this means that a divisorial sheaf
is Cohen-Macaulay.

Linear equivalence of divisors is defined as usual, and
D1 lineq D2 if and only if OX(D1) iso OX(D1).

The quotient of the group CDiv X by linear equivalence is
CDiv X/lineq = Pic X. We have seen this as the group of line
bundles or H^1(OX^x). For Weil divisors we write
Cl X = WDiv X/lineq.

Different theories in algebraic geometry use many other
equivalence relations on divisors. We can divide by _algebraic
equivalence_, or over CC we can divide Pic X by homological
equivalence, or by homological equivalence modulo torsion.
(This means identifying elements of Pic X that have the same
class H^1)

The Néron-Severi group of X is the quotient
of Div X by algebraic equivalence. Over CC, we can also divide by

The index theorem

The background to the Kleiman criterion

Towards Mori's Theorem on the Cone