Exercise 1. Let n be a non-negative integer. Define the n-dimensional affine and projective spaces.
Exercise 2. Give the definitions of affine and of projective algebraic varieties. Define the concepts of irreducibility and smoothness (non-singular) for algebraic varieties. Give examples of singular and non-singular algebraic varieties. Give examples of irreducible and of reducible algebraic varieties. Give examples of algebraic varieties that cannot be defined by a single equation. Did you choose affine of projective examples? Define the dimension of an algebraic variety. What is the dimension of an irreducible projective algebraic variety that can be defined by one equation in the projective space of dimension n?.
(You can see these concepts in Hartshorne, Chap I or in Silverman’s book, The arithmetic of elliptic curves, Chap I.)
Exercise 3. Given
projective algebraic varieties. Define what it is for a map 
\[\varphi:X\to Y\]
to be rational (resp. birational, resp. a morphism).
Exercise 4. Blow-up the cone
at the origin. Show that the resulting surface is nonsingular. Show that the exceptional divisor is isomorphic to
Exercise 5. Let
be a non-singular surface defined over a field
. Let
\[{\rm Div}(S)\]
denote the group of divisors of
, i.e., the abelian group generated by the irreducible curves of
. Given a rational function
\[f\in k(S)\]
, we can define the divisor
\[{\rm div}(f):=\sum_{\Gamma\subset S}{\rm ord}_{\Gamma}(f)\Gamma\]
. The latter are called principal divisors/ We say that the two divisors
\[D_1, D_2\]
are linearly equivalent if
is a principal divisor. The relation 
is an equivalence relation in
\[{\rm Div}(S)\]
. Let
\[{\rm Pic}(S):={\rm Div}(S)/\sim\]
. Prove that the map
\[{\rm deg}:\Gamma\subset \mathbb{P}^2\mapsto {\rm deg}(\Gamma)\]
gives an isomorphism
\[{\rm Pic}(\mathbb{P}^2)\simeq Z\]
Exercise 6. Let 
\[\pi:X\to \mathbb{P}^2\]
be the blow-up at a point
. Show that the exceptional curve 
has self intersection 1. (Hint: consider
\[{\cal O}(1)\]
as lines in
and compute
\[\pi^*({\cal O}(-1))\]
\[\pi^*({\cal O}(-1))\]
in two ways: when the lines intersect at
and when they do not intersect in
. Use the fact that the pullback of a curve
with multiplicity
, where
is the closure of
\[\pi^{-1}(C\setminus P)\]
. Check Beauville’s Complex Algebraic Surface, Chapter II.)