EXERCICES ALGEBRAIC GEOMETRY MASTER 2
Affine varieties (1) LetR be a Noetherian ring.
(a) Let I ⊂ R be an ideal. Show that the quotient ring R/I is also Noetherian.
(b) Suppose thatR is an integral domain and letR⊂Q(R) be its field of fractions. Let 0∈/S ⊂R be a subset, and define
T=R[S−1] ={a
b ∈Q(R)|a∈R, b= 1 or a product of elements inS}.
Show thatT is also a Noetherian ring. Hint: show that an ideal in T is determined by its intersection with the subringR⊂T.
(2) Show that the radical of an idealI⊂R defined by
√
I={r∈R | ∃n∈N∗ rn∈I}
is an ideal inR.
(3) Find the radicals of the following ideals
• (x3)⊂k[x]
• (xy)⊂k[x, y]
• (x3y5)⊂k[x, y]
• (x2(x2−y2))⊂k[x, y]
(4) Give an example of a radical ideal, which is not prime.
(5) Consider the following ideals ink[x, y, z]
J = (xy, xz, yz) andJ0 = (xy,(x−y)z).
(a) Determine the two setsZ(J) andZ(J0).
(b) Are they irreducible ? (c) Show thatJ =I(Z(J)).
(d) Determine the radical√
J0. Give the explicit expression of the powers of generators of√
J0 in terms of generators ofJ0. (6) Consider the idealJ = (xz−y2, x3−yz)⊂k[x, y, z].
(a) Show thatJ is not prime. Hint: find an expression ofx(z2−x2y)in terms of the generators ofJ.
(b) Show that there is a decomposition into closed subsets Z(J) = (Z(J)∩Z(x))∪ Z(J)∩Z(z2−x2y)
. (c) DetermineZ(J)∩Z(x).
(d) We denoteC =Z(J)∩Z(z2−x2y). Show that there is a surjective map
ϕ:A1→C⊂A3, ϕ(t) = (t3, t4, t5).
(e) Deduce thatC is irreducible.
Date: September 2018.
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(7) (***) A unique factorization domain (UFD) is defined to be an integral domain Rsuch that every non-zeror∈Rcan be written uniquely as
r=up1. . . pn, withuunit andpiirreducible.
(a) Let R be a UFD and let r ∈ R be irreducible. Then the ideal (r) generated byris a prime ideal.
(b) If K is a field, then the ring K[X] is a UFD. Hint: use Euclidian division inK[X]
(c) (**) Let Rbe a UFD. A polynomial f ∈R[X] is said to be primitive if the greatest common divisor of its coefficients is 1. Then (Gauss Lemma) a product of primitive polynomials is primitive.
(d) (***) IfR is a UFD, then the ringR[X] is a UFD.Hint: use unique factorization inQ(R)[X]and Gauss lemma on primitive polynomials.
(e) Deduce that the ringK[X1, . . . , Xn] is a UFD.
(8) (**) Let K denote the field of fractions of the ring k[X] and consider k[X, Y] = k[X][Y] as a subring of K[Y]. Recall that k[Y] is a principal ideal domain (every ideal in k[Y] is principal).
(a) Let I⊂K[Y] be an ideal and putJ =I∩k[x, y]. Show thatJ is an ideal ink[x, y] and that the ideal generated byJ in K[Y] equals I.
(b) Letp∈K[Y] and define
L={d∈k[X]|d·p∈k[X, Y]}.
Show thatLis an ideal ink[X].
(c) Deduce that for any p∈ K[Y] there exists a p ∈ k[X, Y] such that (p)∩k[X, Y] = (p).
(d) Deduce that iff, g∈k[X, Y] have a common factor inK[Y], thenf, g have a common factor ink[X, Y].
(9) (**) Same notation as in the previous exercise. Let f, g ∈ k[X, Y] be irreducible elements, not multiples of one another.
(a) Show thatf andghave no common factor inK[Y], whereK=k(X).
(b) Show that there exist polynomialsh ∈k[X] and a, b ∈ k[X, Y] such that
h=af+bg.
(c) Deduce that the zero setZ(f, g) is a finite set.
Topological spaces and Zariski topology
(10) (**) Consider An as a topological space with the Zariski topology. LetX denote the closure of a setX inAn. Show the equality
Z(I(X)) =X.
(11) (**) ConsiderPn as a topological space with the Zariski topology. Let X denote the closure of a setX inPn. Show the equality
Zp(IH(X)) =X.
(12) (*) Let X be a topological space and let (Ui)i∈I be an open cover of X.
Show thatY ⊂X is closed if and only ifY∩Ui is closed inUifor alli∈I.
(13) (*) Let X be an irreducible topological space and let U1, U2 be two non- empty open subsets ofX. ThenU1∩U2 is non-empty.
(14) (*) Let f : X →Y be a continuous map between two topological spaces.
Suppose that the image off is dense inY and thatX is irreducible. Then Y is also irreducible.
(15) (*) A topological space is quasi-compact if one can extract from any open covering a finite open covering. Show that a Noetherian topological space is quasi-compact. Hint: Use descending chain of closed subsets.
(16) (*) Describe the closed sets of the topology on A2=A1×A1 which is the product of the Zariski topologies on the two factors. Find a closed subset in the Zariski topology of A2 not of this form.
Projective varieties
(17) (*) Fori= 0, . . . , nshow that the two maps
ui:An→Ui={[a0, . . . , an]∈Pn |ai6= 0} ⊂Pn andϕi:Ui→An, as defined in the lecture, are inverse to each other.
(18) (**) Show that an ideal I⊂k[X0, . . . , Xn] is homogeneous if and only if it can is generated by homogeneous polynomials.
(19) (**) Show that a non-zero polynomial f ∈ k[X0, . . . Xn] is homogeneous if and only if the ideal (f) contains at least one non-zero homogeneous polynomial.
(20) (*) Use Hilbert Basis Theorem to show that a homogeneous ideal I ⊂ k[X0, . . . , Xn] can be generated by a finite set of homogeneous polynomials.
(21) (**) LetI⊂k[X0, . . . , Xn] be a homogeneous ideal and letX =Zp(I)⊂Pn be its projective zero set andC(X)⊂An+1 be its associated cone.
(a) IfX 6=∅, thenC(X) =Za(I)⊂An+1 andI(C(X)) =IH(X) (b) IfX =∅, thenC(X) ={0}andZa(I) ={0}or ∅.
(22) (**) Let X ⊂An be a non-empty affine algebraic set defined by an ideal X =Z(I). Define the ideal ˜I⊂k[X0, . . . , Xn] by defining its homogeneous part of degreedas
I˜d={X0df(X1 X0
, . . . ,Xn X0
)|f ∈I,deg(f)≤d}.
(a) Show that ˜I=L
d≥0I˜d is a homogeneous ideal.
(b) Show thatX0∈/p I.˜
(c) Show that ifI is prime, then ˜I is also prime.
(d) Given a homogeneous idealJ ⊂k[X0, . . . , Xn] such thatX0∈/√ J we defineJ ⊂k[X1, . . . , Xn] as
J ={f(1, X1, . . . , Xn)∈k[X1, . . . , Xn]|f ∈J}.
Show thatJ is an ideal ink[X1, . . . , Xn].
(e) Show thatZ(J)6=∅.
(f) Show that ifJ is prime, thenJ is also prime.
(g) For any proper ideal I⊂k[X1, . . . , Xn], we have ˜I=I.
(h) For any prime idealJ withX0∈/√
J, we haveJ˜ =J.
(i) Show that the projective zero setZp( ˜I) is the closure X ⊂ Pn of X in the Zariski topology ofPn. We viewAn as the open subset in Pn given as the complement of the hyperplane at infinityH∞=Zp(X0).
Hint: Show that one can reduce to the case I radical. If I is radical, show the equalityIH(X) = ˜I and apply Exercise (11).
(23) (*) Use the notation of the previous exercise.
(a) Show that there is a natural isomorphism ofk-algebrasS(X)→A(X).
(b) Let K(X) be the field of rational functions of the projective variety X ⊂Pn and let K(X) be the field of rational functions of the affine variety X ⊂An. Show that there is a natural k-linear isomorphism K(X)∼=K(X).
Regular functions and morphisms
(24) (*) Consider the affine curve defined as the zero set C=Z(f)⊂A2 of the polynomial f(X, Y) =Y2−X3. Consider the polynomial map
ϕ:A1→C, T 7→(T2, T3).
(a) Show thatϕis a bijective map. Describe the inverse mapϕ−1 :C → A1.
(b) Show thatϕis a morphism, but not an isomorphism.
(c) Consider the restriction ϕ0:A1\ {0} →C\ {(0,0)}. Show thatϕ0is an isomorphism.
(25) (**) Show that the two mapsui andϕiof Exercise (13) are isomorphisms.
(26) (*) Consider the polynomial mapϕ:A1→A3 given byT 7→(T, T2, T3).
(a) Show that the image of ϕis an algebraic subsetC⊂A3. (b) Show thatϕ:A1→C is an isomorphism.
(c) Try to generalize.
(27) (**) Letnbe a positive integer. Consider the polynomial mapϕn:A1→A2 given byT 7→(T2, Tn).
(a) Show that ifnis even, then the image of ϕn is isomorphic to A1 and ϕn is 2-to-1 outside (0,0).
(b) Show that ifnis odd, theϕnis a bijective map. Give a rational inverse toϕn.
(28) (**) Let Cdenote the closed subsetZ(X3+X2−Y2)⊂A2. (a) Show that C is irreducible.
(b) Consider the polynomial mapϕ:A1→Cgiven byT 7→(T2−1, T3− T). Isϕan isomorphism ?
(c) Is the restriction ϕ0:A1\ {1} →C\ {(0,0)} an isomorphism ? (29) (*) (Projective transformation) Let A be an invertible (n+ 1)×(n+ 1)
matrix with coefficients in the field k. Define a map [A] : Pn → Pn by [b0, . . . , bn]7→[A(b0, . . . bn)t], where (b0, . . . bn)t denotes the column vector of (b0, . . . bn).
(a) Show that the map [A] is well-defined.
(b) Show that [A] is an isomorphism with inverse [A]−1= [A−1].
(30) (*) (Projection) ConsiderH0, . . . , Hn−l−1independent linear forms onPn, i.e. the Hi are homogeneous polynomials of degree 1 ink[X0, . . . Xn].
(a) DefineW =Zp(H0, . . . , Hn−l−1)⊂Pn. Show thatW is isomorphic to Pl.
(b) Let X ⊂ Pn be a subvariety such that X ∩W = ∅. Consider the morphismπW :X →Pn−l−1 defined by
p7→πW(p) = [H0(p), . . . , Hn−l−1(p)].
Show thatπW is well-defined and thatπW is a morphism. Show that πW only depends on the linear subspaceW and not on the linear forms Hi.
(c) Letl= 0 and suppose thatW ={p}withp= [0, . . . ,0,1] andHi=Xi
fori= 0, . . . , n−1. Thenπp:Pn\{p} →Pn−1defined byX0, . . . , Xn−1
is a well-defined morphism. Show thatπp(q) is the intersection point of the linepq with the projective spaceZ(Xn)∼=Pn−1.
(31) (**) (product of affine subvarieties) Let X ⊂An and Y ⊂ Am be closed subvarieties. Show that the product
X×Y ={(x, y)∈An×Am |x∈X, y∈Y} is a closed subvariety inAn+m.
Hint: To show thatX×Y is irreducible, suppose thatX×Y =S1∪S2 with Si closed and show thatTi={p∈X | {p} ×Y ⊂Si } is closed inX. (32) (*) Show the following generalization of the previous statement. Let X ⊂ An and Y ⊂ Am be locally closed subvarieties. Then X ×Y is a locally closed subvariety inAn+m.
(33) (*) (universal property of products) Let X ⊂An and Y ⊂Am be closed subvarieties.
(a) Show that the projections p1 :X×Y →X and p2 :X ×Y →Y are morphisms.
(b) LetZ be any variety. Show that a morphism φ: Z → X×Y is the same as two morphismsf :Z →X andg:Z →Y.
(34) (*) The quadric surface in P3.
(a) Show that the Segre embedding of P1×P1 gives an isomorphism of P1×P1 with the quadric
Q=Z(X0X3−X1X2)⊂P3.
(b) What are the images in Q of the two families of lines {p} ×P1 and P1× {p} ?
(c) Show that there are two lines of Q passing through the point p = [1,0,0,0] and that the complement U of these two lines is the image ofA1×A1under the Segre embedding.
(d) Show that under the projection (exercise 30)πp :Q99K P2 the open subsetUmaps isomorphically to a copy ofA2and the two lines through pare mapped to points of P2.
(35) Show that a morphism f : Pn → Pm with m ≥ n is a polynomial map, which means that f is given by m+ 1 homogeneous polynomials of the same degree without common zeros. Hint: Use the local description of a morphism and show that given two open subsets the local description of f on each open extends to their union.
(36) Let X ⊂ An and Y ⊂ Am be affine subvarieties. Let x1, . . . , xn and y1, . . . , ymbe coordinates onAn andAm. Then
(a) The affine coordinate ring of the product X×Y equals A(X×Y) =A(X)⊗kA(Y).
Hint: viewA(X)as ring of polynomial functions on X
(b) Let I(X) ⊂ k[x1, . . . , xn] and I(Y) ⊂ k[y1, . . . , ym] be the ideals of polynomials vanishing onX and Y. Then
I(X×Y) =I(X)⊗k[y1, . . . , ym] +k[x1, . . . , xn]⊗I(Y).
Here + denotes the sum of the two ideals.
Rational maps
(37) (**) (Cremona transformation) Let Φ be the rational mapP299KP2given by
[x0, x1, x2]7→[x0x1, x0x2, x1x2].
(a) Determine the domain of Φ.
(b) Show that Φ is a dominant rational map.
(c) Show that Φ is birational by giving an inverse map.
(38) (**) LetQ=Z(F)⊂Pn be a quadric, i.e. F is a homogeneous polynomial of degree 2. Show that Q is rational. Hint: Use projection π with center p= [0, . . . ,0,1]and writeF =F2+F1Xn+F0Xn2withFi∈k[X0, . . . , Xn−1] of degree i. Assumep∈Q. Give the birational inverse map ofπin terms of the Fi
