Example sheet 3. Positivity in Algebraic Geometry, L18 Instructions: This is the third and last example sheet. More exercises may be added during this week. The final example class will be held on Thursday April 26, 10-11.30 in MR12. If you desire, you can turn in the solutions of up to 5 exercises from the example sheet. If you wish to do so, please turn your material in to my pigeonhole during the lecture. Exercises which are indicated with Exercise* are those for which I am suggesting writing down a proof. Exercises that are denoted by Exercise n.* or (n)* are to be considered particularly challenging exercises. Caveat: These exercises are supposed to test your understanding of the material explained in class sometimes in non-standard ways and also help you develop a formally correct and complete way to put your ideas on paper. They are also intended as preparation for the final exam that will be held in June. Exercise 1. Let X be a projective scheme and let D be a Cartier divisor on X. By imitating the proof of Asymptotic Riemann-Roch and using induction on the dimension of X, prove that h i (md) = O(m dimx ), i 0, m 0. Exercise 2. (i) Let X be a projective scheme and D an effective Cartier divisor. Assume that O D (D) is ample. Show that D is semiample. (ii) Let X be a projective scheme and let D be a Cartier divisor on X. Show that the following are equivalent: (a) D is ample; (b) for every proper reduced irreducible subvariety V X of positive dimension χ(v, O V (md)) when m ; (c) for every proper reduced irreducible subvariety V X of positive dimension there is a positive integer m and a non-zero section s H 0 (V, O V (md)) such that s vanishes at some point of V. [Hint: show that (a) is equivalent to (b) and (a) is equivalent to (c). For the former equivalence, try to imitate the proof of Nakai s criterion. For the latter equivalence, use part (i).] (iii) Use part (ii) of this question to prove that on a smooth projective curve C a divisor D is ample if and only if deg D > 0. Exercise 3. Let X be a smooth projective surface and let C X be a proper irreducible reduced curve with C K X < 0. (1) Prove that if C 2 0 then C is a smoothly embedded rational curve. (2) Show that if the set of K X -negative curves {C X K X C < 0, C is irreducible and reduced} is finite, then C 2 < 0. (3) Assume that K X is nef and K 2 X > 0. What is the Iitaka dimension of K X? 1
2 Under these assumptions show that if D X is a proper irreducible reduced curve with C K X = 0 then C is a smooth rational curve. Exercise 4. In this question X will always denote a smooth projective surface defined over an algebraically closed field k. (i) Let D be an effective Cartier divisor on X and C a proper irreducible reduced curve on X. Assume that D C < 0. Show that D = D C + C. (ii) Let D be an effective Cartier divisor on X and C a proper irreducible reduced curve on X. Assume that D C < 0 and (D C) C 0. Show that md = m(d C) + mc. Use this result to show that if ψ : X X is the blowup of X at a smooth point, then H 0 ( X, O X(mK X)) = H 0 (X, O X (mk X )), for every m 0. (iii) Let X be smooth and projective. Let C X be a proper smooth rational curve. Assume that C 2 = 0. Show that C is a semiample divisor and that the Iitaka dimension of C is 1. (iv) In this last part, you should assume that K X is big. Prove the following assertions. (a) If K X is not nef, consider the set Neg(K X ) := { C X dim C = 1, C is proper, reduced, irreducible and K X C < 0}. Show that Neg(K X ) is a finite set. (b) If K X is not nef, show that every element of Neg(K X ) is a smooth rational curve of self-intersection 1. [Hint: Show that if C Neg(K X ) and C 2 0 then Neg(K X ) is not a finite set. Use part (iii) for the case C 2 = 0.] (c) State Castelnuovo s criterion. If K X is not nef, let C be an element of Neg(K X ). Show that there exists a smooth surface X 1 and a birational map f 1 : X X 1 which contracts C to a point and is an isomorphism on X \ C. Exercise 5. Let F n be the n-th Hirzebruch surface. (1) Describe the cone given by divisors of Iitaka dimension 2 and the closure of the cone generated by effective divisors on F n. We will call this the big and the pseudoeffective cone, respectively. (2) What is the dual of the pseudoeffective cone? (3) Describe the Iitaka fibration for any class in the pseudoeffective cone. (4) Compute the volume of any class in the pseudoeffective cone. Exercise 6. Let π : X S be a projective morphism of schemes and let D on X be a Cartier divisor. Assume that there exists s S such that D Xs is nef, where X s is the fibre of π over s. Then prove that D is nef when restricted to the fibre over a very general point in S. [Recall that a point s S is very general if it lays in the complement of countably many Zariski closed subsets of X.]
3 [Hint: You can assume and should use the statement given at the end of Lecture 12 which shows that in the analogous situation, if we take D to be ample, then D is ample when restricted to the fibre over a general point in S.] Exercise 7. Let X be a projective scheme and let D, E be ample Cartier divisors on X. Let us consider X X and let us denote by pr i : X X X the projection onto the i-th factor, i = 1, 2. (1) Show that the divisor pr 1aD + pr 2bE is ample for a, b > 0. We shall denote by (ad, be) the divisor pr 1aD+pr 2bE and by O X X (ad, be) the associated invertible sheaf O X X (pr 1aD +pr 2bE). For a coherent sheaf G on X X, we shall denote by G(aD, be) the sheaf G O X X (pr 1aD + pr 2bE). (2) Show that H 0 (X X, O X X (ad, be)) H 0 (X, O X (ad)) H 0 (X, O X (be)). (3) Let F be a coherent sheaf on X X. Show that there exists m = m(f, D, E) s.t. H i (X X, F(aD, be)) = 0, i > 0, a, b m. (4) Let us denote by the diagonal in X X, := {(x, y) X X x = y}. We shall denote by I the ideal of in X X. By considering the short exact sequence 0 I (ad, be) O X X (ad, be) O (ad, be) 0, show that there exists m = m(d, E) s.t. H 0 (X X, O X X (ad, be)) surjects onto H 0 (, O (ad, be)), a, b m. (5) By identifying O (ad, be) with an appropriate sheaf of X, show that there exists m = m(d, E) s.t. the multiplication map H 0 (X, O X (ad)) H 0 (X, O X (be)) H 0 (X, O X (ad + be)) is surjective a, b m. (6) Show that for an ample divisor H on X, the section ring of H, R(X, H) = n 0 H 0 (X, O X (nh)) on X is finitely generated. Exercise 8. Let X be an irreducible and reduced projective scheme and let D be a semiample Cartier divisors on X. (1) Assume that D is base point free. Prove that there exists m N >0 such that the mappings H 0 (X, O X (ad)) H 0 (X, O X (bd)) H 0 (X, O X ((a + b)d)) determined by the multiplications are surjective a, b m. [Hint: reduce the proof to the case where D is ample, by using the Iitaka fibration of D which is an algebraic fibre space in this case.] (2) More generally, for any coherent sheaf F on X, there exists m N >0 such that the mappings H 0 (X, F(aD)) H 0 (X, O X (bd)) H 0 (X, F((a + b)d)) determined by the multiplications are surjective a, b m.
4 [Hint: as above, reduce the proof to the case where D is ample, by using the Iitaka fibration of D and the push-forward of F to the base of the fibration.] We aim to prove that the section ring of D, R(X, D) = n 0 H 0 (X, O X (nd)) is finitely generated. (3) Suppose first that kd is base point free. Then show that R(X, D) (k) = n N H 0 (X, nkd) is finitely generated. (4) By using part (2) and taking F to be in turn each of O X (D), O X (2D),..., O X ((k 1)D), show that the section ring is finitely generated. Exercise 9. Let X be a an irreducible projective variety of dimension n. Let D be a Cartier divisor on X. Let x X be a smooth closed point. (1) Compute the dimension over k of m k 1 x /m k x. We will indicate this number by f n (k). (2) Compute j i=1 f n(i) asymptotically. (3) Assume that vol(d) > 0 and fix a number 0 < t < vol(d). Use the short exact sequence 0 m j x O X (md) O X (md) O X /m j xo X (md) 0 to show that for any m 0, there exists mt < j < m vol(d) suitably chosen and E md of multiplicity > j. (4) Assume that D satisfies h i (md) = O(m dimx 1 ), i > 0, e.g. D is nef. Fix a positive rational number α s.t. 0 < α n < D n. Show that for any smooth closed point y X there exists m y N >0 s.t. for any m m y there exists E md and mult y E m α. (5) Let H on X be a big and nef divisor such that H n > n n. Then for any smooth closed point x X there exists a Q-divisor s.t. Q H and mult x H > n. [Recall that the multiplicity of a Cartier divisor D at a point x is the minimum integer n s.t. g m n x where g is a local equation for D at x. When 0 D = i a id i is a Q-divisor, then the multiplicity is computed linearly mult x D = i a imult x D i.] Exercise 10. (i) A Cartier divisor D on X is big if its Iitaka dimension is maximal. Show that the following are equivalent: (a) D is big; (b) there exists C > 0 s.t. lim sup h0 (md) C; mdim X (c) there exist m > 0, an ample Cartier divisor H and an effective Cartier divisor E such that md H + E. (ii) A R-Cartier R-divisor D on X is big D = a i D i for some positive a i and some big Cartier divisors D i. Show that the following are equivalent: (a) D is big; (b) for some ample Cartier divisor A on X, there exists a positive real number t and an effective R-divisor E such that D R ta + E; (c) for some ample R-Cartier R-divisor A on X, there exists an effective R-divisor E such that D A + E.
5 (iii) Let X be a projective scheme. Let D a big R-Cartier R-divisor on X. Show that there exists finitely many proper subvarieties of codimension 1, E 1,..., E k X such that if C X is an irreducible reduced proper curve such that D C < 0, then C is contained in one of the E i. Assume now that D is ample when restricted to any proper codimension 1 subvariety of X. Let A be an ample divisor on X. Show that there exists ɛ > 0 such that D ɛa is nef on X. (iv) Let X be a projective scheme. Let D be a class in N 1 (X) R and assume that D dim V [V ] > 0, for every irreducible reduced subvariety V X of positive dimension. (a) Show that D is nef. (b) Show that there exists ample R-Cartier R-divisors A, A such that: the numerical equivalence classes of D + A, D A, A + A are ample and rational, and (D + A ) n > n (D + A ) n 1 (A + A ), where n = dim X. Conclude that D A must be big. (c) Show D ɛa is nef for 0 < ɛ 1. For the purpose of this question, you can assume that you know inductively that D is ample when restricted to any proper subscheme of X. [Hint: Use part (iii).] (d) Show that D is ample. (vv) To every irreducible reduced subvariety of positive dimension V X we can associate a subset φ V N 1 (X) R defined as follows φ V := {H N 1 (X) R H dim V [V ] = 0}. Show that the boundary of the nef cone Nef(X) := Nef(X) \ int(nef(x)), where int(nef(x)) indicates the topological interior, is contained in the union of all sets φ V when V varies among all possible irreducible reduced subvariety of positive dimension of X.