Exercise 1. Let F ∈ k[X

UNIVERSIT ´ E NICE SOPHIA ANTIPOLIS Ann´ ee 2015/2016

Master 2 MPA Introduction to algebraic geometry

Exercise sheet 8

Let k be an algebraically closed field of characteristic 0.

Exercise 1. Let F ∈ k[X

, X

, X

] be a homogeneous polynomial of degree d that does not have a multiple factor. For every P ∈ P

denote by µ

(F ) the multiplicity of V (F ) in P . a) Show that one has

X

P∈P²

µ

(F)(µ

(F) − 1) ≤ d(d − 1).

Hint : apply Bezout to V (F ) and V (

0

).

b) Construct an example where the inequality above is an equality.

Exercise 2. Denote by V

the vector space of homogeneous polynomials of degree 3 in k[X

, X

, X

]. Let P

, . . . , P

∈ P

be eight distinct points.

a) Show that the vector space

V

(P

, . . . , P

) := {F ∈ V

| F(P

) = 0 ∀ i = 1, . . . , 8}

has dimension at least two.

b) Suppose that dim

V

(P

, . . . , P

) = 2. Suppose also that at most three of the points P

are on a line l ⊂ P

, and at most six of the points P

are on a conic C ⊂ P

. Show that there exists a polynomial f ∈ V

(P

, . . . , P

) which is irreducible.

c) Under the assumptions of b), let f, g be a base of V

(P

, . . . , P

) such that f and g are irreducible. Set C

= V (f ) and C

= V (g). Show that one of the following holds :

— There exists a point i ∈ {1, . . . , 8} such that µ

(C

, C

) ≥ 2.

— There exists a point P

∈ P

\ {P

, . . . , P

} such that P

∈ C

∩ C

. In the second case show that

V

(P

, . . . , P

) = V

(P

, . . . , P

).

We say that the linear system V

(P

, . . . , P

) has an unassigned base point P

.

Exercise 3. Consider the smooth projective surface

X := {(x

: x

: x

), (y

: y

) ∈ P

× P

| x

y

= x

y

}

a) Let ϕ : X → P

be the morphism defined by the projection on the first factor. Show that if p ∈ P

is distinct from (1 : 0 : 0), then ϕ

(p) is a point. Show that ϕ

((1 : 0 : 0)) is a curve E ⊂ X isomorphic to P

. Show that X \ E ' P

\ (1 : 0 : 0).

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b) Let l ⊂ P

be a line defined by P

i=0

a

x

= 0, and let ϕ

l ⊂ X be the projective set defined by

{(x

: x

: x

), (y

: y

) ∈ P

× P

| x

y

= x

y

,

2

X

i=0

a

x

= 0}

Show that ϕ(ϕ

l) = l. Show that ϕ

l has two irreducible components if and only if (1 : 0 : 0) ∈ l.

c) Let l

, l

⊂ P

be two distinct lines. Show that the projection formula (ϕ

l

).(ϕ

l

) = l

.l

holds. Hint : use that l

and l

are linearly equivalent to lines that do not pass through (1 : 0 : 0).

d) Show that E

= −1. Hint : choose distinct lines that pass both through (1 : 0 : 0), then use the projection formula.

Exercise 4. Let X be an algebraic variety endowed with its structure sheaf O

. If U, V ⊂ X are open non-empty subsets and f ∈ O

(U), g ∈ O

(V ) we define that f ∼ g if f |

= g|

. a) Show ∼ defines an equivalence relation on the set S

∅6=U⊂X

O

(U ) where U ⊂ X are open sets.

b) Denote by K(X) the set of equivalence classes for the relation ∼. For every f, g ∈ K(X) one can define (in a natural way) the sum f + g and the product f · g. Show that K(X ) is a field, called the field of rational functions of X.

c) Let ϕ : X → Y be a morphism of algebraic varieties that is birational, i.e. there exist non- empty Zariski open subsets X

⊂ X and Y

⊂ Y such that ϕ|

: X

→ Y

is an isomorphism.

Show that K(X) ' K(Y ).

d) Suppose that X ⊂ k

is an affine variety and let Γ(X) its function ring. Show that K(X ) = Frac(Γ(X)).

e) Suppose that X ⊂ P

is a projective variety, and let S(X ) be its associated graduated ring.

Show that

K(X ) = S(X )

where S(X )

⊂ Frac(S(X )) are the elements of the form

with deg f = deg g.

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