A short summary of lectures in Week 1
==== Week 1. Varieties, quotient singularities ===
What is a variety? At the most basic level, V is a set with a notion
of locally defined regular function. An informal introduction to
this is given in [UAG]. An irreducible algebraic variety over an
algebraically closed field k = kbar has a function field k(V), and f
in k(V) is regular at P in V if it can be written as f = g/h, with
g,h polynomials and h(P) <> 0. An affine variety has affine
coordinate ring k[V] = polynomial functions on V, and the regular
functions at P in V is the local ring O_{V,P}. An "abstract variety"
is a topological space with locally defined regular functions, that is
covered by open pieces Ui with the property that each Ui is
isomorphic to an affine variety with its Zariski topology. It is
better to restrict attention to projective varieties or
quasi-projective varieties to avoid questions of separation and
completion.
Quotient V/G by a group action = the underlying set is the orbit
space under G, and the sheaf of functions consists of elements of
the function field k(V)^G invariant under G. Then regularity is
defined in terms of functions regular at the orbit of points.
Those pages of Mumford, Abelian varieties, Chap. 2, Section 7, pp.
65-69, give the results precisely and concisely.
====
Every algebraic geometer will need Abelian varieties sooner or later.
The books of Mumford and of Jim Milne are excellent entry
points. As commentary, I give a brief summary of why Mumford needs
this.
1. What is an Abelian variety A? An irreducible projective variety A
together with an element 0, a binary operation A x A -> A written
additively a1,a2 -> a1+a2, with a minus map A -> A written a -> -a,
making A into an Abelian group.
2. Over CC, A is analytic diffeomorphic to a complex torus CC^g/La
with La (Lambda) = ZZ^2g a discrete group of translations acting
cocompactly. Riemann's theorem says that CC^g/La can be embedded
in projective iff and only if there is a positive definite Hermitian
pairing La x La -> ZZ with positive definite imaginary part. (In
other words, not every complex torus has global meromorphic
functions). The Hermitian pairing of Riemann's theorem allows the
construction of theta functions, that embed A in PP^n.
3. The dual Abelian variety A^dual is another Abelian variety whose
points correspond 1-to-1 with divisor classes of degree 0 on A. It
is one of the basic examples of a moduli space. Over CC, in the
complex analytic topology, it is the group H^1(A, O_A^x) of line
bundles up to isomorphism. The exponential exact sequence (in the
complex topology)
H^1(A, ZZ) -> H^1(A, OA) -> H^1(A, O_A^x) -> H^2(A, ZZ)
with the first map given by exp(2pi i .) shows that it is also a
complex torus, since H^1(A, ZZ) = ZZ^2g and H^1(A, OA) = CC^g.
4. To construct A^dual purely in algebraic geometry, consider the
hyperplane section L of A in PP^n as a divisor, and its translations
T_a^*(L) under the group action. As a moves, the classes
T_a^*(L) tensor L^-1
run through the divisor classes of degree 0. The kernel ker L of
a |-> T_a^*(L) tensor L^-1 is a finite subgroup of A, and A^dual
is constructed as the group quotient A/ (ker L).
====
A curious phenomenon is that you need A to be a projective variety in
order to ensure that the orbit of the group action (or any finite
set) is contained in an affine subvariety. I discuss later in the
course the Kleiman criterion for projectivity, and some easy
counter-example to the statement on finite sets in an affine.
====
Quotient of affine variety by a finite group includes that quotient
singularities that we study at different points in later lectures.
The important special case of the cyclic group mu_r acting on CC^2
by 1/r(1,a) (that is, the diagonal matrix action x -> ep*x,
y -> ep^a*y for ep in mu_r) is treated in my notes on cyclic surface
singularities, and I return to this in subsequent lectures.
Another example is that of the Binary Dihedral group. This is the group
BD_{4r} in SL(2,CC) generated by
< a, b > where a = 1/2*r(1,-1) and b = matrix [0 1 \\ -1 0 ].
You see that a^r = b^2 = -1, and b*a*b^-1 = a^-1, so this is like the
dihedral group except that it contains the diagonal matrix -1 in its
centre.
To construct the quotient CC^2/BD_{4*r}, consider first the invariant
subring of the big diagonal subgroup = 1/2*r(1,-1). This is
generated by
X = x^{2*r}, Y = x*y, Z = y^{2*r} with X*Z = Y^{2*r}.
Now the second generator b acts by
X <-> Z and Y -> -Y.
So the ring of invariant polynomials is generated by
u = X+Z, v = Y*(X-Z), w = Y^2.
The relation between these is obtained by taking v^2 and
getting rid of the cross-term:
v^2 = u^2*w - 4*w^{r+1}.
This calculation was first done by Felix Klein around 1870. The
equation can be written as x^2 = y^2*z + z^(n-1) and is also
called the Du Val singularity Dn. For more on this, see my notes
on Du Val singularities, and later in the course.