Exercises

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
\[X\]
and
\[Y\]
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
\[z^2=x^2+y^2\]
at the origin. Show that the resulting surface is nonsingular. Show that the exceptional divisor is isomorphic to
\[\mathbb{P}^1\]
.
Exercise 5. Let
\[S\]
be a non-singular surface defined over a field
\[k\]
. Let
\[{\rm Div}(S)\]
denote the group of divisors of
\[S\]
, i.e., the abelian group generated by the irreducible curves of
\[S\]
. 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
\[D_2-D_1\]
is a principal divisor. The relation 
\[\sim\]
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
\[P\]
. Show that the exceptional curve 
\[E=\pi^{-1}(P)\]
has self intersection 1. (Hint: consider
\[{\cal O}(1)\]
as lines in
\[\mathbb{P}^2\]
and compute
\[\pi^*({\cal O}(-1))\]
·
\[\pi^*({\cal O}(-1))\]
in two ways: when the lines intersect at
\[P\]
and when they do not intersect in
\[P\]
. Use the fact that the pullback of a curve
\[C\]
in
\[\mathbb{P}^2\]
  intersection
\[P\]
with multiplicity
\[m\]
is
\[\tilde{C}+mE\]
, where
\[\tilde{C}\]
is the closure of
\[\pi^{-1}(C\setminus P)\]
. Check Beauville’s Complex Algebraic Surface, Chapter II.)