DIVISORS ON NONSINGULAR CURVES
BRIAN OSSERMAN
We now begin a closer study of the behavior of projective nonsingular curves, and morphisms
between them, as well as to projective space. To this end, we introduce and study the concept of
divisors.
1. Morphis
PROBLEM SET #4
SELECTED SOLUTIONS AND REMARKS
For reference, I graded problems 3.15, 5.1, 5.4 and 5.10.
3.15 Let X An and Y Am be ane varieties.
k
k
(a) Show that X Y An+m with its induced topology is irreducible.
k
(b) Show that A(X Y ) A(X ) k A(Y ).
=
PROBLEM SET #3
SELECTED SOLUTIONS AND REMARKS
For reference, I graded problems 3.3, 4.7, and the second and third of the nonbook problems.
3.3
a) Let : X Y be a morphism. Then for each P X , induces a homomorphism of
local rings : O(P ),Y OP,X .
P
b) Show
PROBLEM SET #5
SELECTED SOLUTIONS AND REMARKS
2.2 If X is a prevariety with atlas cfw_i : Xi Ui , and U Ui for some i, then a function
f : U k is regular if and only if f i is regular in the quasiane sense on 1 (U ) Xi .
i
Solution: It is immediate from t
PROBLEM SET #6
SELECTED SOLUTIONS AND REMARKS
3.1 A topological space X is Hausdor if and only if the image of the diagonal map X X X
is closed.
Solution: First suppose that X is Hausdor, and (P, Q) X X is not in the diagonal; equivalently,
P = Q. Then th
PROBLEM SET #9
SELECTED SOLUTIONS AND REMARKS
1.8 Show that a nonconstant morphism : P1 P1 is ramied at all points of P1 if and only
if it is ramied at innitely many points, if and only if it factors through the Frobenius
morphism.
Proof: Thinking of P1 a
PROBLEM SET #7
SELECTED SOLUTIONS AND REMARKS
3.1 A topological space X is compact if and only if for every topological space Y , the projection
map X Y Y is a closed map.
Solution: If X is compact, given Z X Y closed, take P Y not in the image of Z ; we
PROBLEM SET #2
SELECTED SOLUTIONS AND REMARKS
Most people did well on this problem set, so I wont make a lot of comments.
I did want to say that its okay to use previous exercises, even if we didnt do
them. For instance, for 1.10, you could use 1.6. As th
PROBLEM SET #1
SELECTED SOLUTIONS AND REMARKS
1.1
a) Let Y be the plane curve y = x2 . Show that A(Y ) is isomorphic to a polynomial ring
in one variable over k.
b) Let Z be the plane curve xy = 1. Show that A(Z ) is not isomorphic to a polynomial
ring in
ABSTRACT VARIETIES VIA ATLASES
BRIAN OSSERMAN
In this expository note we describe how abstract algebraic varieties over an algebraically closed
eld may be dened rigorously via an atlas denition analogous to the usual denition of dierential
manifolds. The
RECOVERING GEOMETRY FROM CATEGORIES
BRIAN OSSERMAN
The fundamental organizing result on ane algebraic varieties over an algebraically closed eld
k is surely that the category they form is equivalent to the (opposite) category of nitely generated
integral
PROJECTIVE VARIETIES
BRIAN OSSERMAN
We now move on to studying projective varieties. Unlike Hartshornes approach, we will treat
them as an example of the more general abstract varieties we have dened.
1. Projective space
We will rst construct projective s
NONSINGULAR CURVES
BRIAN OSSERMAN
The primary goal of this note is to prove that every abstract nonsingular curve can be realized
as an open subset of a (unique) nonsingular projective curve. Note that this encapsulates two facts
in one: that every nonsin
THE RIEMANN-ROCH AND RIEMANN-HURWITZ THEOREMS
BRIAN OSSERMAN
We state without proof the Riemann-Roch theorem, and give some basic applications, including
a proof of the Riemann-Hurwitz theorem. We use throughout the convention that all curves are
projecti
DIFFERENTIAL FORMS
BRIAN OSSERMAN
Dierentials are an important topic in algebraic geometry, allowing the use of some classical
geometric arguments in the context of varieties over any eld. We will use them to dene the genus
of a curve, and to analyze the
PROBLEM SET #8
SELECTED SOLUTIONS AND REMARKS
2.2 (a) Show that if : X Y is a morphism of varieties, and U X is an open subset such
that the composition U Y is an isomorphism, then U = X .
(b) Show that if : X Y is a morphism of varieties, and U X is an o