UNIVERSIT ´E NICE SOPHIA ANTIPOLIS Ann´ee 2015/2016
Master 2 MPA Introduction to algebraic geometry
Exercise sheet 6
Letkbe an algebraically closed field.
Exercice 1.LetXbe a topological space of finite dimension. Suppose that we haveX =Sn i=1Xi
whereXi⊂X is a closed set. Show that we have dimX = maxi=1,...,ndimXi.
Exercice 2. Let X ⊂kn and Y ⊂kn be irreducible affine sets, and let W be an irreducible component of X∩Y. We want to show that
dimW ≥dimX+ dimY −n.
a) Show that the result is true ifY =V(f) wheref ∈k[X1, . . . , Xn] is a non-constant polyno- mial.
b) Let
∆ :={(x, y)∈kn×kn |x=y}
be the diagonal inkn×kn. Show thatX∩Y is isomorphic to (X×Y)∩∆.
c) Show that the dimension ofX×Y is equal to dimX+ dimY. Use b) and a) to conclude.
Exercice 3. Show that the morphism
ϕ:k3→k3, (x, y, z) 7→ (x,(xy−1)y,(xy−1)z) is surjective, but the set
V :={(x, y, z)∈k3| dimϕ−1((x, y, z))≥1}
is not closed in the Zariski topology.
Exercice 4. Let X and Y be algebraic varieties, and let ϕ : X → Y be a closed surjective morphism.
a) Suppose that dimY = 1. Show that dimϕ−1(y) = dimX−1 for everyy∈Y. b) Suppose that dimY = 2. Show that
V :={y∈Y | dimϕ−1(y) = dimX−1}
is empty or a finite union of points.
Exercice 5. We denote by (x1, x2) the standard coordinates of a point in k2 and by (y1:y2) the homogeneous coordinates of a point inP1. Consider the set
X :={((x1, x2),(y1:y2))∈k2×P1 |x1y2=x2y1}.
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a) Show thatX is an algebraic set. Hint : consider the open setsUi ⊂k2×P1defined byyi6= 0.
b) We consider the map defined by the projection on the first factor, i.e. the map p:X →k2, ((x1, x2),(y1:y2)) 7→ (x1, x2).
Show thatpis surjective. Show thatp−1((x1, x2)) is a point if (x1, x2)6= (0,0) andp−1((0,0))' P1. We callX the blow-up ofk2in the origin.
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