Exercice 1. Let J ⊂ k[X 0 , . . . , X n ] be a homogeneous ideal. Show that the radical rac(J) is homogeneous.

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

Master 2 MPA Introduction to algebraic geometry

Exercise sheet 4

Let k be an algebraically closed field.

Exercice 1. Let J ⊂ k[X 0 , . . . , X n ] be a homogeneous ideal. Show that the radical rac(J) is homogeneous.

Exercice 2. Let f ₀ , . . . , f _m ∈ k[X ₀ , . . . , X _n ] be homogeneous polynomials having all the same degree d. Suppose that the vanishing locus V _P (f ₀ , . . . , f _m ) is empty.

a) Show that the map

f : P ⁿ → P ^m , x = (x 0 : . . . : x n ) 7→ (f 0 (x) : . . . : f m (x)) is well-defined.

b) Show that f is continuous (for the Zariski topology).

Exercice 3.

a) Let J ⊂ k[X ₀ , . . . , X _n ] be an ideal generated by homogeneous polynomials g ₁ , . . . , g _r . We denote by J ^[ ⊂ k[X ₁ , . . . , X _n ] the ideal generated by the deshomogenized polynomials g ₁ ^[ , . . . , g _r ^[ . Show that

V (J ^[ ) = V _p (J ) ∩ k ⁿ

where k ⁿ ⊂ P ⁿ is the complement of the hyperplane H = {x 0 = 0}.

b) Let J ⊂ k[X 1 , . . . , X n ] be ideal, and let J ^# ⊂ k[X 0 , . . . , X n ] be the ideal generated by {g ^# | g ∈ J arbitrary}

Set Z := V (J ) ⊂ k ⁿ ⊂ P ⁿ , and denote by ¯ Z ⊂ P ⁿ its Zariski closure. Show that Z ¯ = V _P (J ^# ).

Exercice 4. We consider the quadric Q ⊂ P ³ defined by Q := V _P (X 0 X 1 − X 2 X 3 ).

a) For every (λ : µ) ∈ P ¹ the equations

µX 1 + λX 2 = 0, µX 3 + λX 0 = 0

define a line P ³ . Show that these lines are contained in Q. Construct a second family of lines contained in Q.

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b) Show that for every point x ∈ Q there exists a unique line of each family passing through this point.

c) Show that two distinct lines in the same family are disjoint, and that two lines from different families intersect in a unique point.

d) Let D ₁ , D ₂ , D ₃ ⊂ P ³ be three projective lines that are pairwise disjoint. Show that there exists a unique quadric Q that contains D ₁ ∪ D ₂ ∪ D ₃ .

Hint : start by proving that for 9 arbitrary points in P ³ there exists a homogeneous polynomial of degree two F such that the projective set X = V _P (F ) contains these nine points. Show that if X intersects a line l in three distinct points, then l is contained in X .

Exercice 5. Let R be a ring, and let S ⊂ R be a multiplicative subset. Denote by f : R → S ⁻¹ R the natural map.

a) Show the universal property of the localised ring S ⁻¹ R : if g : R → B is a morphism of rings such that g(s) is invertible for every s ∈ S, then there exists a unique morphism of rings g ⁰ : S ⁻¹ R → B such that g ⁰ ◦ f = g.

b) Let p ⊂ R be a prime ideal and denote by S ⁻¹ p ⊂ S ⁻¹ R the ideal generated by f (p). Let S ¯ ⊂ R/p be the image of S by the quotient map R → R/p. Show that ¯ S ⊂ R/p is a multiplicative set and

S ¯ ⁻¹ (R/p) ' S ⁻¹ R/S ⁻¹ p.

Exercice 6.

a) Let f ∈ k[X ₁ , . . . , X _n ] be a non-zero polynomial. Show that the localised ring k[X ₁ , . . . , X _n ] _f is isomorphic to

k[X 1 , . . . , X n , T ]/(1 − T f ).

b) Let X be an affine set and Γ(X) its function ring. Let f 1 , f 2 ∈ Γ(X) be polynomials such that V X (f 1 ) = V X (f 2 ). Show that

Γ(X ) f

' Γ(X ) f

.

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