Geometry of Algebraic Curves

131 Pages·2011·0.98 MB·English

Geometry of Algebraic Curves
Lectures delivered by Joe Harris
Notes by Akhil Mathew
Fall 2011, Harvard
Contents
Lecture 1 9/2
§1 Introduction 5 §2 Topics 5 §3 Basics 6 §4 Homework 11
Lecture 2 9/7
§1 Riemann surfaces associated to a polynomial 11 §2 IOUs from last time: the
degree of K , the Riemann-Hurwitz relation 13 §3 Maps to projective space 15 §4
X
Trefoils 16
Lecture 3 9/9
§1 Thecriterionforveryampleness 17 §2 Hyperellipticcurves 18 §3 Properties
of projective varieties 19 §4 The adjunction formula 20 §5 Starting the course
proper 21
Lecture 4 9/12
§1 Motivation 23 §2 A really horrible answer 24 §3 Plane curves birational to
a given curve 25 §4 Statement of the result 26
Lecture 5 9/16
§1 Homework 27 §2 Abel’s theorem 27 §3 Consequences of Abel’s theorem 29
§4 Curves of genus one 31 §5 Genus two, beginnings 32
Lecture 6 9/21
§1 Differentials on smooth plane curves 34 §2 The more general problem 36 §3
Differentials on general curves 37 §4 Finding L(D) on a general curve 39
Lecture 7 9/23
§1 More on L(D) 40 §2 Riemann-Roch 41 §3 Sheaf cohomology 43
Lecture 8 9/28
§1 Divisors for g =3; hyperelliptic curves 46 §2 g =4 48 §3 g =5 50
1
Lecture 9 9/30
§1 Low genus examples 51 §2 The Hurwitz bound 52
2.1 Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.2 Step 1(cid:48) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.3 Step 1(cid:48)(cid:48) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.4 Step 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Lecture 10 10/7
§1 Preliminary remarks 57 §2 The next theorem 57 §3 Remarks 58 §4 The
main result 59
Lecture 11 10/12
§1 Recap 62 §2 The bound, again 64 §3 The general position lemma 64 §4
Monodromy 65 §5 Proof of the general position lemma 66
Lecture 12 10/14
§1 General position lemma 68 §2 Projective normality 69 §3 Sharpness of the
Castelnuovo bound 70 §4 Scrolls 71
Lecture 13 10/17
§1 Motivation 74 §2 Scrolls again 75 §3 Intersection numbers 75
Lecture 14 10/19
§1 Abasiclemma 79 §2 Castelnuovo’slemma 81 §3 EqualityintheCastelnuovo
bound 82
Lecture 15 10/21
§1 Introduction 84 §2 Castelnuovo’s argument in the second case 85 §3
Converses 87
Lecture 16 10/26
§1 Goals 89 §2 Inflection points 90 §3 A modern reformulation 91
Lecture 17 11/2
§1 Review 95 §2 The Gauss map 96 §3 Plane curves 97
Lecture 18 11/4
§1 Inflectionarypoints;onelastremark101 §2 Weierstrasspoints102 §3 Examples105
Lecture 19 11/9
§1 Real algebraic curves106 §2 Singularities108 §3 Harnack’s theorem108 §4
Nesting109 §5 Proof of Harnack’s theorem110
Lecture 20 11/16
§1 Basic notions111 §2 The differential of u112 §3 Marten’s theorem115
Lecture 21 11/18
§1 Setting things up again116 §2 Theorems118
Lecture 22 11/30
§1 Recap121 §2 Families122 §3 The basic construction123 §4 Constructing
such families124 §5 Specializing linear series125
Lecture 23 12/2
§1 Two special cases127 §2 The inductive step128 §3 The basic construction128
§4 A relation between the ramifications of U,W 130 §5 Finishing131
Introduction
JoeHarristaughtacourse(Math287y)onthegeometryofalgebraiccurvesatHarvard
in Fall 2011. These are my “live-TEXed” notes from the course.
Conventions are as follows: Each lecture gets its own “chapter,” and appears in the
table of contents with the date. Some lectures are marked “section,” which means that
they were taken at a recitation session. The recitation sessions were taught by Anand
Deopurkar.
Of course, these notes are not a faithful representation of the course, either in the
mathematics itself or in the quotes, jokes, and philosophical musings; in particular, the
errors are my fault. By the same token, any virtues in the notes are to be credited to
the lecturer and not the scribe.
Please email corrections to [email protected].
Lecture 1 Geometry of Algebraic Curves notes
Lecture 1
9/2
§1 Introduction
The text for this course is volume 1 of Arborello-Cornalba-Griffiths-Harris, which is
even more expensive nowadays.
We will be covering a subset of the book, and probably adding some additional
topics, but this will be the basic source for most of the stuff we do. There will be
weekly homeworks, and that will determine the grade for those of you taking the
course for a grade. There will be no final.
There will be a weekly section with Anand Deopurkar.
The course doesn’t meet on Mondays because of the “Basic Notions” seminar. On
those weeks that there won’t be seminars, we might meet then. Every third or fourth
week, the lecture will be cut short on Wednesdays. However, the course will basically
meet on a three-hour-per-week basis.
§2 Topics
Here is what we are going to talk about. We are going to talk about compact Riemann
surfaces, and a compact Riemann surface is the same thing as a smooth projective
algebraic curve (over C). That in turn is really the same thing as a smooth projective
curve over any algebraically closed field of characteristic zero. By abuse of notation,
we will use C to denote any such field as well.
The fact that these are the same thing—that is, that a compact Riemann surface
is an algebraic curve—is nontrivial. It requires work to show that such an object even
admits a nontrivial meromorphic function. Note:
1.1 Proposition. There are compact complex manifolds of dim ≥ 2 that do not admit
nonconstant meromorphic functions.
The miracle in dimension one is that there are nonconstant meromorphic functions,
and enough to embed the manifold in projective space.
There are a number of beautiful topics that we are not going to cover. We will not
talk about singular algebraic curves, in general. We will encounter them (e.g. when
we consider maps of curves to projective space, the image might be singular), but they
will not be the focus of study. When we have a singular curve C in projective space,
we will treat C its normalization, i.e. as an image of a smooth projective curves. We
also will not talk about open (i.e. noncompact) Riemann surfaces. Another even-
more large-scale and interesting topic is the theory of families of curves, and how the
isomorphismclassofacurvechangesaswevarythecoefficientsofthedefiningequations
(e.g. moduli of curves). Finally, we are not going to talk about curves over fields that
are not algebraically closed. There has been a huge amount of work on algebraic curves
over R, but we won’t discuss them.
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Lecture 1 Geometry of Algebraic Curves notes
§3 Basics
Today, we shall set the notation and conventions. Algebraic curves is one of the oldest
subjects in modern mathematics, as it was one of the first things people did once they
learned about polynomials. It has developed over time a multiplicity of language and
symbols, and we will run through it.
Let X be a smooth projective algebraic curve over C. There are many ways of
defining the genus of X, e.g. via the Hilbert polynomial, the Euler characteristic (via
coherent cohomology), and so on. We are just going to take the naive point of view.
1.2 Definition. The genus of X is the topological genus (as a surface).
We can also use:
1. g(X) = 1−χ(O ).
X
2. 1− 1χ (X).
2 top
3. 1 degK +1 (for K the canonical divisor, see below).
2 X X
Given a geometric object, one wishes to define the functions on it. On a compact
Riemannsurface,thereareno nonconstantregularfunctions,bythemaximalprinciple:
everyholomorphicfunctiononX isconstant. Weneedtoallowpoles,andtokeeptrack
of them we will need to introduce the language of divisors.
1.3 Definition. A divisor D on X is a formal finite linear combination of points
(cid:80)
n p ,p ∈ X on the curve. We say that D is effective if all n ≥ 0 (in which case
i i i i
(cid:80)
we write D ≥ 0), and we say that degD = n is the degree of D.
i
As a reality check, one should see that the family of effective divisors of a given
degree d should be the dth symmetric power of the curve with itself, C ; this is C ×
d
···×C (d times) modulo the symmetric group S .
d
Thepointofthisis,ifwehaveameromorphicfunctionf : X → P1,wecanassociate
to it a divisor measuring its zeros and poles.
1.4 Definition. Given f : X → P1, we say that the divisor of X is the sum
(cid:80)
ord (f)p. The positive terms come from the zeros, while the negative terms
p∈X p
come from the poles.
The problem is to deal with interesting classes of functions. Holomorphic functions
do it, while there are too many meromorphic functions to work with them all at once.
Instead, we do the following:
(cid:80)
1.5 Definition. Given a divisor D = n p, we can look at functions which may have
p
poles at each p, but with orders bounded by n . Namely, we look at the space L(D)
p
of rational functions f : X → P1 such that ord (f) ≥ −n for all p ∈ X. This is
p p
equivalently the space of f such that div(f)+D ≥ 0.
We note (without proof) that L(D) is finite-dimensional. We write:
1.6 Definition. 1. (cid:96)(D) = dimL(D).
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Lecture 1 Geometry of Algebraic Curves notes
2. r(D) = (cid:96)(D)−1.
In the literature, both notations (cid:96),r are used.
The basic problem is this: given D, find explicitly these vector spaces L(D), and
in particular the dimension (cid:96)(D) and the number r(D). This is a completely solved
problem, and not just by general theorems like Riemann-Roch. If one is given an
algebraic curves as a smooth projective curve (given by explicit equations), and an
explicit divisor, there is an algorithm to determine the space L(D). We’ll do that in a
week or two.
The one thing to observe is that there’s a certain redundancy here, in this problem.
This is for the following reason: if L(D) is known, and E is another divisor that differs
from D by div(f) for some global meromorphic f : X → P1 (nonconstant), then we
can determine L(E). Namely, there is an elementary isomorphism
L(D) (cid:39) L(E),
given by multiplying by the rational function f. So, in some sense, asking to describe
L(D) is the same as asking to describe L(E). To solve this problem, we need only
study divisors modulo this equivalence relation.
1.7 Definition. We say that D,E are linearly equivalent when there is a global
meromorphic function f such that D−E = div(f). We write D ∼ E.
In some sense, the fundamental object is the space of divisors modulo linear equiva-
lence. Note that the degree of a global rational function is zero, so the degree is defined
modulo linear equivalence. (This states that the number of poles is the same as the
number of zeros, for a global meromorphic functions.)
We are going to realize the space of divisors modulo linear equivalence as a space.
For now:
1.8 Definition. WecallPicd(X)thespaceofdivisorsonX modulolinearequivalence.
So far, we’re just talking about divisors in general on X. There is a particular one
we should keep in mind, the canonical divisor.
1.9 Definition. If ω is a meromorphic 1-form (i.e., something that locally looks like
f(z)dz for a meromorphic function f), we can define the order at a point p ∈ X (via
the order of the coefficient function). In particular, we can define the divisor div(ω)
of a meromorphic 1-form ω.
Note that the ratio of two meromorphic 1-forms ω ,ω is a global meromorphic (or
1 2
rational) function ω /ω . In particular, div(ω ) = div(ω ).
2 1 1 2
The canonical class K of X is the class of the divisor of ω for any meromorphic
X
1-form ω.
We’re now going to turn around and say everything again, in a more modern lan-
guage.
TherearealotofcircumstancesinwhichwewanttoforgetaboutthedivisorD,and
think only of linear equivalence. We would like a terminology that would let us only
specify the divisor mod linear equivalence. This will be the language of line bundles.
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Lecture 1 Geometry of Algebraic Curves notes
(cid:80)
1.10 Definition. Suppose D = n p is a divisor on X. Let O be the sheaf of
p X
regular functions on X; similarly, we define O (D) to be the sheaf of functions with
X
zerosandpolesprescribedbythisdivisorD. Inotherwords,thesectionsofO (D)over
X
an open subset U are the meromorphic functions f : U → P1 such that ord (f) ≥ n
p p
for p ∈ U.
This is a local version of the space L(D) (i.e. L(D) = Γ(X,O (D)), and is much
X
larger.
The point is that this sheaf O (D) is locally free of rank one. In other words, it
X
is a holomorphic line bundle. We will basically identify holomorphic line bundles with
locally free sheaves of O -modules of rank one; this is standard in algebraic geometry.
X
Remark. If D ∼ E, then O (D) (cid:39) O (E) as line bundles. So, in some sense, we can
X X
realize the space of divisors modulo linear equivalence as the space of line bundles on
X. That’s how Picd(X) is typically defined.
There’s one more thing we should define. There is a canonical divisor class, and
the associated line bundle can be easily defined. Namely, we just have to take the
holomorphic cotangent bundle T∗; this corresponds to the divisor associated to any
X
global meromorphic 1-form.
We said at the outset that compact Riemann surfaces correspond to smooth pro-
jective curves. How do we go from one to another? How do we describe maps from X
to projective space Pn? This is where the notion of divisors and line bundles plays an
essential role. We could write all this in the classical language of divisors, but we’ll use
the modern language of line bundles; you can think in the former way if you wish.
Let L be a line bundle on X. Suppose σ ,...,σ are global sections of L, say
0 r
linearly independent. (So if L = O (D), then each σ corresponds to an element of
X i
L(D), i.e. a meromorphic function on X with appropriate zeros and poles.) Suppose
that they have no common zeros. In that case, there is induced a map
X → Pr, x (cid:55)→ [σ (x),...,σ (x)].
0 r
Here the σ (x) sure aren’t numbers, but we can still make sense of this. Namely, the σ
i i
are not numbers, but they are elements of the fiber of L over p. Given an r+1-tuple of
elements of a one-dimensional C-vector space, we get a uniquely determined element of
Pr. Another way to see it is that a line bundle is locally trivial, and we can use a local
trivialization to think of the σ (x) as functions so that the σ (x) are actual numbers. If
i i
you chose a different trivialization, you would get a different vector [σ (x),...,σ (x)],
0 r
but it would be the same up to scalars. In fact, if we changed the σ around, then we
i
would just change the embedding by some automorphism group of Pn.
So,
Up to automorphisms of Pn, the map X → Pr is uniquely determined by
the subspace of H0(L) spanned by the {σ }.
i
We thus get:
1.11 Proposition. There is a correspondence between pairs (L,V) where L is a line
bundle on X of degree d and V ⊂ H0(L) is an r+1-dimensional space of sections with
8
Lecture 1 Geometry of Algebraic Curves notes
no common zeros, and the space of nondegenerate maps of degree d, X → Pr (up to
automorphisms of Pr).
So if we’re looking for maps to projective space, it’s the same as looking for line
bundles and sections. Here the word “nondegenerate” means that the image is not
contained in a hyperplane. If the image is contained in a hyperplane, then the sections
σ ,...,σ used to define the map would satisfy a nontrivial linear relation.
1 r
1.12 Definition. A linear series of degree d and dimension r on X is a pair (L,V):
1. L is a holomorphic line bundle on X, of degree d (i.e. the associated divisor has
degree d).
2. V is an r+1-dimensional space of sections, contained in H0(L).
We are no longer requiring that there be no common zeros. This is to make the space
compact.
This will be denoted gr: here, again, the d refers to the degree, r the dimension.
d
Here “g” comes from the old word for a divisor, a “group.”
A gr can be thought of as a family of effective divisors. For each σ ∈ V ⊂ H0(L),
d
we can associate the divisor of zeros div(σ). This means that P(V) corresponds to a
family of effective divisors on X, parametrized by the projective space P(V). To recap,
each nonzero section has a divisor (or divisor of zeros), and the section is defined up
to rescaling by its divisor (because there are no nonconstant holomorphic functions on
X). When you think of a linear series in this form, it is often denoted D.
We can give a more intrinsic description of the map associated to a linear system.
Given a linear system (L,V) on X without common zeros, we can describe the associ-
ated map X → Pr as the map specifically sending X to the projectivization P(V∗), so
that a point p ∈ X is sent to the hyperplane of sections in V vanishing on p (and this
hyperplane belongs to P(V∗)).
Lastly (and this is really crucial), when the genus of X is ≥ 1, we have enough
holomorphic differentials on X to get a map to projective space.
1.13 Proposition. ThedimensionofthespaceL(K)ofholomorphic1-formsisexactly
the genus g. Moreover, there are no common zeros of the canonical line bundle L(K),
so there is a canonical map X → Pg−1 associated to the linear series of all 1-forms.
We’ve left out a lot of details (and even a lot of definitions), but now we want to
do something. Let’s give a false proof of Riemann-Roch in the next five minutes.
Given a divisor D, we’re (as before) interested in the space L(D) of rational func-
tions with the desired poles. Suppose for simplicity D is a sum of distinct points,
D = p +p +···+p . The vector space L(D) consists of meromorphic functions that
1 2 d
have at most a simple pole at the points p but are otherwise regular. How might you
i
define such a function?
Choose a local coordinate z around p ; given a function f ∈ L(D), we can write it
i i
near p as ai +f for some holomorphic f (defined in a neighborhood of p ), because
f is onily alzliowed0to have simple poles. Th0is polar part ai says a lot aboutif. In fact,
zi
f is determined up to addition of scalars by specifying these polar parts {a }, again
i
9
Lecture 1 Geometry of Algebraic Curves notes
because there are no nonconstant holomorphic functions on X. In other words, there
is a natural map
L(D) → Cd,
whose kernel consists of the constant functions. We get in particular,
(cid:96)(D) ≤ 1+d.
If you want to describe L(D) (and in particular its dimension), you want to know the
image. This raises the question:
Given a ,...,a , when is there a global meromorphic f : X → P1 with
1 d
polar part ai at p , and holomorphic elsewhere?
zi i
In other words, we need to find constraints on the {a } for them to form a family of
i
polarpartsofafunctioninL(D). Here’sthepoint: iff ∈ L(D),andω isaholomorphic
1-form, we can consider the meromorphic differential fω. This has potentially simple
poles at the {p }, but is holomorphic elsewhere. In particular, the sum of the residues
i
is zero.
(Recall that the sum of the residues of a meromorphic differential on a compact
Riemann surface is zero.)
So,ifω = g (z )dz locally,nearp (usingthelocalcoordinatez aroundp ),thenthe
i i i i i i
residueoffω atp isa g (p ). Itfollowsthatifthe{a }ariseasasystemofpolarparts,
i i i i i
then we want (cid:80)a g (p ) = 0. Thus the image of the map L(D) → Cd is contained
i i i
in the orthogonal complement of the space of holomorphic differentials (where each ω
maps to (g (p )) ∈ Cd as before). We get a total of g linear conditions on the {a }, over
i i i
a family of holomorphic differentials, suggesting that
(cid:96)(D) ≤ 1+d−g
But this is false. It’s possible that a differential might not give a serious relation on
the {a }. For instance, a differential at the points {a }. The correct statement is
i i
(cid:96)(D) ≤ 1+d−(g−(cid:96)(K −D)), (1)
because the dimension of the image of L(D) → Cd is contained in the orthogonal
complement of the image of the 1-forms in Cd.
However, a 1-form is the dual of a vector field. So the degree of a 1-form is just
minus the degree of the corresponding vector field, and that is the topological Euler
characteristic. In particular, the degree of a 1-form is the opposite of the topological
Euler characteristic, i.e. 2g−2.
Now we want to apply (1) to K −D, and we get
(cid:96)(K −D) ≤ 1+(2g−2−d)−g+(cid:96)(D),
where we have used the degree of K to get the degree of K−D as 2g−2−d. Now we
add this to (1). We get (cid:96)(D)+(cid:96)(K −D) ≤ (cid:96)(K −D)+(cid:96)(D), and as a result we must
have equalities in both inequalities. We “get”:
10