TWO REMARKS ON SEXTIC DOUBLE SOLIDS

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1 TWO REMARKS ON SEXTIC DOUBLE SOLIDS IVAN CHELTSOV AND JIHUN PARK Abstract. We study the potential density of rational points on double solids ramified along singular sextic surfaces. Also, we investigate elliptic fibration structures on nonsingular sextic double solids defined over a perfect field of characteristic 5. Keywords : elliptic fibration, number field, perfect field of characteristic 5, potential density, sextic double solid Mathematics Subject Classification. Primary 14E05, 14G05, 14G15, 14J30, 14J Introduction In arithmetic geometry, it is one of important problems to measure the size of the set of rational points of a variety defined over a number field. The most profound work in this area is the Faltings theorem that a nonsingular curve of genus at least two defined over a number field F has only finitely many F-rational points (see [13]). One of higher dimensional analogue of the theorem is conjecture that the set of rational points of a nonsingular variety of general type defined over a number field is contained in a proper Zariski closed subset of the variety. It is natural that we should expect the set of rational points of a Fano variety defined over a number field to be huge in some sense. Definition 1.1. The set of rational points of a variety V defined over a number field F is called potentially dense if there is a finite field extension K of the field F such that the set of K-rational points of the variety V is Zariski dense in V. One of goals in this paper is to study the potential density of double covers of P 3 ramified along singular sextic surfaces defined over number fields that constitute one family of Fano varieties. Let τ : Y P 2 be a double cover branched over a reduced sextic curve R P 2 defined over a number field F. In the case when R is a nonsingular sextic curve, it is unknown whether the set of rational points of Y is potentially dense or not. In the case when the curve R P 2 is a union of six lines intersecting at a single point, the set of rational points on Y is not potentially dense due to [13] because Y is birationally equivalent to C P 1, where C is a hyperelliptic curve of genus 3. The following result was proved in [4], [5]. Theorem 1.2. Suppose that the curve R is singular. Then the set of rational points of the surface Y is potentially dense unless the curve R is the union of six lines intersecting at a single point. Let π : X P 3 be a double cover branched over a reduced sextic surface S P 3 defined over a number field F. We prove the following result: Theorem 1.3. Suppose that the surface S is singular. Then the set of rational points of the 3-fold X is potentially dense if the surface S is not a union of six planes passing through a single line, and if the surface S is not a cone over a nonsingular plane sextic curve. 1

2 2 IVAN CHELTSOV AND JIHUN PARK In the case when S is a union of six planes passing through a single line, the set of rational points of X is not potentially dense due to [13], because X is birational to C P 2, where C is a hyperelliptic curve of genus 3. In the case when S is a cone over a nonsingular plane sextic curve E P 2, the 3-fold X is birational to F P 1, where F is a double cover of P 2 branched over E, and the set of rational points of the 3-fold X is potentially dense if and only if the set of rational points of the surface F is potentially dense. Let us move to the other goal of this paper that we investigate elliptic fibration structures on nonsingular sextic double solids defined over a perfect field of characteristic 5. In the papers [7] and [8], nonsingular sextic double solids, i.e., double cover of P 3 ramified along nonsingular sextic hypersurfaces of P 3, defined over a perfect field of characteristic 0 or p > 5 are proved not to be birationally transformed into elliptic fibrations. In addition, the paper [8] provides a very interesting example that a nonsingular sextic double solid defined over a perfect field of characteristic 5 can be birationally transformed into an elliptic fibration as follows: Example 1.4. Suppose that the base field is F 5 = Z/5Z. Let X be the double cover of P 3 = Proj(F 5 [x, y, z, w]) ramified along the sextic S given by the equation x 5 y + x 4 y 2 + x 2 y 3 z y 5 z 2x 4 z 2 + xz 5 + yz 5 + x 3 y 2 w + 2x 2 y 3 w xyz 3 w xyz 2 w 2 x 2 yw 3 + xy 2 w 3 + x 2 zw 3 + xyw 4 + xw 5 + 2yw 5 = 0. Then X is nonsingular (see [12] and [14]) and Pic(X) = Z by Lemma in [11] or Lemma 3.5 in [10] (see [6] and [15]). Moreover, X contains a curve C given by the equations x = y = 0 whose image in P 3 is a line L contained in the sextic S P 3. For a general enough point p X, there is a unique hyperplane H p P 3 containing π(p) and L. The residual quintic curve Q H p given by S H p = L Q intersects L at a single point q p with mult qp (Q L ) = 5. The two points π(p) and q p determine a line L p in P 3. Define a rational map Ξ L : X Grass(2, 4) by Ξ L (p) = L p. The image of the map Ξ L is isomorphic to P 2, hence we may assume that the map Ξ L is a rational map of X onto P 2. Obviously the map Ξ L is not defined over L, the normalization of its general fiber is an elliptic curve, and a resolution of indeterminacy of the map Ξ L birationally transforms the 3-fold X into an elliptic fibration. This delightful example results from the fact that the characteristic of the base field is 5 and the nonsingular sextic surface S has a line L that a generic hyperplane sections passing through the line L consists of a line and a plane curve of degree 5 intersecting at a single point with multiplicity 5, which is impossible over other fields of characteristics 5. In the present paper, we will show that the birational transform into an elliptic fibration constructed above is the unique type that a nonsingular sextic double solid defined over a perfect field of characteristic 5 may have. Theorem 1.5. A double cover X of P 3 defined over a perfect field of characteristic 5 and ramified along a nonsingular sextic surface S can be birationally transformed into an elliptic fibration if and only if the surface S has a line L such that for a general hyperplane H passing through the line L, the residual quintic curve Q given by S H = Q L intersects L at a point q with mult q (Q L ) = 5. Furthermore, the elliptic fibration is defined in the way of Example 1.4. Acknowledgement. We would like to thank Prof. S. Abhyankar for helpful conversations on resolution of singularities over a perfect field of characteristic 5. The second author has been partially supported by POSTECH BSRI research fund-2004.

3 TWO REMARKS ON SEXTIC DOUBLE SOLIDS 3 2. Potential density. Let π : X P 3 be a double cover branched over a singular reduced sextic surface S P 2 defined over a number field F. Suppose that the surface S is neither the union of six planes passing through a single line nor a cone over a nonsingular plane sextic curve. Lemma 2.1. If the surface S is a cone over a plane sextic curve, then the set of rational points of X is potentially dense. Proof. Suppose that the surface S is a cone over a plane sextic curve R P 2. Then the 3-fold X is birational equivalent to D P 1, where D is a double cover of P 2 branched over R P 2. Moreover, by assumption the curve R is neither a nonsingular curve nor the union of six lines passing through a single point. Therefore, the set of rational points of the 3-fold X is potential dense by Theorem 1.2. Therefore, we may assume that the surface is not a cone to prove Theorem 1.3. Let p be a singular point of the sextic surface S P 3. Replacing F by a finite extension of F, we may assume that the point p is defined over F. We then let L be a general line in P 3 that passes through the point p. In addition, we put C := π 1 (L). Lemma 2.2. The curve C is irreducible. Proof. If the curve C is reducible, then the surface S must be a cone with vertex p because L is a general line passing through p. We consider the rational map ρ : X P 2 defined by the composition of the morphism π and the projection of P 3 to P 2 centered at the point p. We resolve the indeterminancy of the rational map ρ to obtain the following commutative diagram g W X ρ f P 2, where the 3-fold W is nonsingular. Lemma 2.3. If the normalization of the curve C is a rational curve, then the set of rational points of X is potentially dense. Proof. The hypothesis tells us that f : W P 2 is a conic bundle. Let G P 3 be a sufficiently general hyperplane defined over F such that p G and G tangents the surface S somewhere. Put Ĝ = (π g) 1 (G). Then Ĝ is not contained in the fibers of the conic bundle f. Suppose Ĝ is reducible. Then each component of Ĝ is rational, which implies the rationality of the 3-fold X. Indeed, each component of the surface Ĝ is a section of the conic bundle f : W P 2. The set of rational points of a rational variety is potentially dense. Suppose Ĝ is ir reducible.the surface Ĝ is a two-section of the conic bundle f. The set of rational points of the surface Ĝ is potentially dense by Theorem 1.2. Thus we have a conic bundle f : W P 2 with a two-section Ĝ such that the set of rational points of the surface Ĝ is potentially dense. In this case the set of rational points of W is potentially dense.

4 4 IVAN CHELTSOV AND JIHUN PARK The genus of the normalization of the curve C cannot exceed 1. Therefore, we may assume that f : W P 2 is an elliptic fibration. Lemma 2.4. Suppose that there is a multi-section M of of degree d 2 of the elliptic fibration f such that the morphism f M is branched at a point q M such that q is contained in a nonsingular elliptic curve that is a scheme-theoretic fiber of f. Then the divisor q 1 q 2 Pic(C b ) is not a torsion for any two distinct points q 1 and q 2 of M C b, where C b = f 1 (b) and b is a C-rational point in the complement to a countable union of Zariski closed subsets in P 2. Proof. The claim follows from Proposition 2.4 in [3]. Lemma 2.5. Let M be an irreducible divisor in (π g) (O P 3(1)) such that M is defined over the field F, the morphism f M is branched at a point q contained in a nonsingular elliptic curve that is a scheme-theoretic fiber of f. If the set of F-rational points of the surface M is Zariski dense in M, then the set of rational points of X is potentially dense. Proof. See [8]. Lemma 2.6. If the singularities of the surface S is not isolated, the set of rational points of X is potentially dense.. Proof. Let H be a sufficiently general hyperplane in P 3. We consider F = (π g) 1 (H). We easily see that F (π g) (O P 3(1)) and π g F : F H = P 2 is a double cover branched over a singular sextic curve H S. Replacing the field F by its finite extension, we may assume that H and F are defined over the field F. Hence the set of rational points of F is potentially dense by Theorem 1.2. Therefore, the statement immediately follows from Lemma 2.5. From now on we suppose that the surface S has only isolated singularities. Let T be the set of points (x, y) S S such that: (1) the points x and y are distinct; (2) the points x and y are nonsingular points of the surface S; (3) the point x is contained in the hyperplane D P 3 tangent to S at y; (4) the point x is a nonsingular point of the intersection S D; (5) the intersection S D is reduced. Lemma 2.7. Let ψ : T S be the projection to the second factor. Then the image ψ(t ) contains a Zariski open subset of the sextic S P 3. Proof. Let y be a general point on the sextic S P 3 and D be the hyperplane tangent to the surface S at the point y in P 3. To prove the claim we just need to show that D S is reduced, which is nothing but the finiteness of the Gauss map of the surface S at y S. Suppose that the Gauss map of the surface S is not finite at y S. Then there is a line L that pass through y by Theorem 2.3 in [21]. In particular, the surface S contains a one-dimensional family of lines. The line L does not pass through the singular point of S because the point y is sufficiently general and the singularities of the surface S are isolated. But we are assuming that the surface S is not a cone. Hence, the self-intersection number L 2 of the line L is negative by the adjunction formula, which contradicts to the fact that L moves in a one-dimensional family on the surface S.

5 TWO REMARKS ON SEXTIC DOUBLE SOLIDS 5 Let x be a sufficiently general point on S. Then it follows from the Lemma 2.7 that there is a hyperplane D P 3 such that x D, the intersection D S is reduced, π(p) D, and the hyperplane D tangent the surface S at some point y of S. Let L be a line in P 3 passing through π(p) and x, and C = (π g) 1 (L). Then C is a nonsingular elliptic curve, which is a scheme-theoretic fiber of the morphism f : W P 2 over a general point of P 2. Let F = (π g) 1 (D). Then the morphism f F : F D = P 2 is a double cover branched over a plane sextic curve D S which is singular at the point y. Note that we may assume that x, y, D are defined over the field F, replacing F by a finite extension of F. Moreover, the set of rational points of F is potentially dense by Theorem 1.2. Because the divisor F satisfies all the conditions in Lemma 2.5, the set of rational points of X is potentially dense. Summing up Lemmas 2.1, 2.3, 2.6, and the above, we see that Theorem 1.3 is true. Let us conclude this section with the following result. Theorem 2.8. Let τ : V P r be a double cover branched over a singular reduced sextic hypersurface S P r defined over a number field. Suppose that S is not a cone over a nonsingular sextic hypersurface in P k, where k < r. Then the set of rational points of the variety V is potentially dense for every r 2. Proof. The claim follows from Theorems 1.2 and the proof of Theorem 1.3. Indeed, we can prove the claim by induction starting from r = 3. Every step of the proof of Theorem 1.3 is valid in the case r > 3. Moreover, in the case r > 3 the claim of Lemma 2.7 is implied by the finiteness of the Gauss map of a nonsingular hypersurface in P r 1 (see [21]). It should be pointed out that the unirationality of a variety V defined over a number field implies that the set of rational points of V are potentially dense. On the other hand, the following result is proved in [9]. Theorem 2.9. Let τ : V P r be a double cover branched over a sufficiently general hypersurface of degree d. Then there is r(d) N such that V is unirational if r r(d). However, it is unknown whether the variety V of Theorem 2.9 is unirational or not in the case when r is relatively small. 3. Elliptic fibrations Throughout this section, all varieties are always assume to be defined over a perfect field k of characteristic 5. Because the field k is perfect, we may assume that it is algebraically closed. For the problem of elliptic fibration structures on nonsingular sextic double solids over a perfect field of characteristic 5, we use movable log pairs introduced in [2]. Before we proceed, we briefly overview their properties. Definition 3.1. On a variety X a movable boundary M X = n i=1 a im i is a formal finite Q-linear combination of linear systems M i on X such that the base locus of each M i has codimension at least two and each coefficient a i is non-negative. A movable log pair (X, M X ) is a variety X with a movable boundary M X.

6 6 IVAN CHELTSOV AND JIHUN PARK Remark 3.2. Throughout this section, we always assume that the variety X is nonsingular. To implement theory of movable log pair to the fullest, we need resolution of indeterminancy of rational maps. What we need for this paper is resolution of indeterminancy of birational maps of nonsingular 3-folds defined over a perfect field of characteristic 5, which is proved by S. Abhyankar (See [1]). Because the sextic double solids to be considered in Theorem 1.5 are nonsingular, we are free to use the tools described in what follows. Every movable log pair can be considered as a usual log pair by replacing each linear system by its general element. In particular, for a given movable log pair (X, M X ) we may handle the movable boundary M X as an effective divisor. We can also consider the self-intersection M 2 X of M X as a well-defined effective codimension-two cycle when X is Q-factorial. We call K X + M X the log canonical divisor of the movable log pair (X, M X ). The notions such as discrepancies, (log) terminality, and (log) canonicity can be defined for movable log pairs as for usual log pairs (see [17]). Definition 3.3. A movable log pair (X, M X ) has canonical (terminal, resp.) singularities if for every birational morphism f : W X each discrepancy a(x, M X, E) in K W + f 1 (M X ) = f (K X + M X ) + a(x, M X, E)E E: f-exceptional divisor is non-negative (positive, resp.). A proper irreducible subvariety Y X is called a center of the canonical singularities of a movable log pair (X, M X ) if there is a birational morphism f : W X and an f-exceptional divisor E W such that the discrepancy a(x, M X, E) 0 and f(e) = Y. The set of all the centers of the canonical singularities of the movable log pair (X, M X ) will be denoted by CS(X, M X ). Note that a log pair (X, M X ) is terminal if and only if CS(X, M X ) =. Let (X, M X ) be a movable log pair and Z X be a proper irreducible subvariety such that X is nonsingular along the subvariety Z. Then elementary properties of blow ups along nonsingular subvarieties of nonsingular varieties imply that Z CS(X, M X ) mult Z (M X ) 1 and in the case when codim(z X) = 2 we have Z CS(X, M X ) mult Z (M X ) 1. The following result is a key to our investigation on birational elliptic fibration structures. Theorem 3.4. Let X be a terminal Q-factorial Fano variety with Pic(X) = Z, ρ : X Y a birational map, and π : Y Z a fibration. Suppose that a general enough fiber of π is a nonsingular variety of Kodaira dimension zero. Then the singularities of the movable log pair (X, M X ) is not terminal, where M X = rρ 1 ( π (H) ) for a very ample divisor H on Z and r Q >0 such that K X + M X Q 0. Proof. Because of Remark 3.2, the proof goes in the same way as in [8]. For the detail, see [8]. It is important for us to study the singularities of certain movable log pairs on Fano varieties. It requires us to investigate the multiplicities of certain movable boundaries or their selfintersections. The following result is Corollary 7.8 of [20], which holds even over fields of positive characteristic and implicitly goes back to the classical paper [16].

7 TWO REMARKS ON SEXTIC DOUBLE SOLIDS 7 Theorem 3.5. Let X be a 3-fold and M X an effective movable boundary on X. Suppose that a nonsingular point o on X belongs to CS(X, M X ). Then the inequality mult o (M 2 X ) 4 holds and the equality holds only when mult o (M X ) = 2. Proof. See [19] or [20]. From now on, we attack Theorem 1.5. Let π : X P 3 be a sextic double solid ramified along a nonsingular sextic hypersurface S P 3. Suppose that the surface S contains a line L such that for a general hyperplane H passing through the line L, the residual quintic curve Q given by S H = Q L intersects L at a single point q with mult q (Q L ) = 5. For a general enough point p X, there is a unique hyperplane H p P 3 containing π(p) and L. The residual quintic curve Q H p given by S H p = L Q intersects L at a single point q p with mult qp (Q L ) = 5. The two points π(p) and q p determine a line L p in P 3. We then define a rational map Ξ L : X P 2 in the way described in Example 1.4. Therefore, the 3-fold X can be birationally transformed into an elliptic fibration. For Theorem 1.5 we have to prove that the elliptic fibration described above is the only type that a nonsingular sextic double solid may have. We consider a fibration τ : Y Z whose general fiber is a nonsingular elliptic curve. Suppose that we have a birational map ρ of X onto Y. We then put M X = 1 n M with M = ρ 1 ( t (H Z ) ), where H Z is a very ample divisor on Z and n is the natural number such that M nk X. By Theorem 3.4 we see that the set CS(X, M X ) is non-empty. However, it does not contain a point of X. Indeed, if it contains a point p on X, from Theorem 3.5 we would obtain a contradictory inequality 2 = H K 2 X = H M 2 X mult p (M 2 X) 4, where H is a general enough effective anticanonical divisor passing through the point p. Therefore, the set CS(X, M X ) must contain a curve C X. Lemma 3.6. The intersection number K X C is 1. In particular, the curve π(c) P 3 is a line and C = P 1. Proof. Let H be a general enough divisor in the linear system K X. Then we have which implies K X C 2. 2 = H K 2 X = H M 2 X mult C (M 2 X)H C K X C, Suppose K X C = 2. Then Supp(M 2 X ) = C and mult C(M 2 X ) = mult2 C (M X) = 1, which means that for two different divisors M 1 and M 2 in the linear system M we have mult C (M 1 M 2 ) = n 2, mult C (M 1 ) = mult C (M 2 ) = n, and set-theoretically M 1 M 2 = C. For a general enough point p C the linear subsystem D of M passing through the point p has no base components because the linear system M is not composed from a pencil. Let D 1 and D 2 be general enough divisors in D. Then in set-theoretic sense p D 1 D 1 = M 1 M 2 = C, which is a contradiction. Consequently, the intersection number K X C must be 1 and hence the curve π(c) P 3 is a line and C = P 1. Because the proof above have not used irreducibility, we see that the set CS(X, M X ) consists of a single curve C.

8 8 IVAN CHELTSOV AND JIHUN PARK Lemma 3.7. The line π(c) is contained in the sextic surface S. Proof. Suppose π(c) S. Let H be the linear subsystem in K X of surfaces containing the curve C. The base locus of H consists of the curve C and the curve C such that π(c) = π( C). Choose a general enough surface D in the pencil H. The restriction M X D is not movable, but M X D = mult C (M X )C + mult C(M X ) C + R D, where R D is a movable boundary. The surface D is nonsingular. Thus, on the surface D, we have C 2 = C 2 = 2. Immediately, the inequality (1 mult C(M X )) C 2 (mult C (M X ) 1)C C + R D C 0 implies C CS(X, M X ). It contradicts CS(X, M X ) = {C}. A hyperplane section of the nonsingular sextic surface S passing though the line π(c) consists of the line π(c) and a quintic plane curve. For a general hyperplane section, the line π(c) and the residual quintic curve intersects at five distinct points if the characteristic of the base field is bigger than 5. However, over a field of characteristic 5, it may happen that they intersect only at a single point. Lemma 3.8. For a general enough hyperplane H passing through the line L := π(c), the residual quintic curve Q given by S H = L Q intersects the line L at a single point q with mult q (Q L ) = 5. Proof. Let H be a general enough hyperplane in P 3 passing through the line L. Then the curve D = H S = L Q is reduced, where Q is a quintic curve. The curve D is singular at which the line L intersects the quintic curve Q. Note that the intersection number L Q on H is 5. Suppose that the curve Q intersects the line L at two distinct points p 1 and p 2. Then the hyperplane H is tangent to the sextic S at the points p 1 and p 2. Hyperplanes passing through the line L form a pencil whose proper transforms on the 3-fold X are K3 surfaces in K X passing through C. Therefore, the lines tangent to the surface S at a general point of L span whole P 3. Let L 1 and L 2 be general enough lines in H passing through the points p 1 and p 2 respectively. Then each line L j is tangent to S at the point p j. Therefore, the proper transform L j X of the curve L j is an irreducible curve with K X L j = 2. It is also singular at the point p j = π 1 (p j ). Consider the proper transform H of the surface H on X and a general surface M in the linear system M. Then M H = mult C (M)C + R, where R is an effective divisor on H such that C Supp(R). Moreover, 2n = M L j mult pj ( L j ) mult C (M) + mult p (M) mult p ( L j ) 2n, p (M\C) L j

9 TWO REMARKS ON SEXTIC DOUBLE SOLIDS 9 which implies M L j C set-theoretically. However, on H the curves L 1 and L 1 span two pencils with the base loci consisting of the points p 1 and p 2, respectively. Therefore, we see R = due to the generality in the choice of two curves L 1 and L 2. Hence, set-theoretically M H = C for a general divisor H K X passing through the curve C and a divisor M M with H Supp(M). Let p be a general point on the surface H and M p be the linear system of surfaces in M containing p. Then M p has no base components because the linear system is not composed from a pencil. Therefore, for a general divisor M in M p p M H = C because H Supp( M), which contradicts the generality of the point p H. Lemma 3.9. There is a birational map α : P 2 Z such that the diagram X ρ Ξ L P 2 α is commutative, where Ξ L is the rational map defined in the beginning of this section. Y Proof. Let g : W X be the blow up along the curve C.We then get Z K W = g ( K X ) E, where E is the g-exceptional divisor. Let L be a curve on W such that π g(l ) is a line tangent to S at some general point of π(c). Then M W L 2 2 mult C (M X ) 0, where M W = g 1 (M X ). Because such curves as L span a Zariski dense subset in W, we obtain mult C (M X ) = 1. Each elliptic curve L is a fiber of the elliptic fibration Ξ L g : W P 2. Thus M W lies in the fibers of Ξ L g, which implies the claim. References [1] S. Abhyankar, Resolution of singularities of embedded algebraic surfaces, Princeton University Press (1998) [2] V. Alexeev, General elephants of Q-Fano 3-folds, Comp. Math. 91 (1994), [3] F. Bogomolov, Yu. Tschinkel, Density of rational points on Enriques surfaces, Math. Res. Lett. 5 (1998), [4] F. Bogomolov, Yu. Tschinkel, On the density of rational points on elliptic fibrations, J. Reine Angew. Math. 511 (1999), [5] F. Bogomolov, Yu. Tschinkel, Density of rational points on elliptic K3 surfaces, Asian J. Math. 4 (2000), [6] F. Call, G. Lyubeznik, A simple proof of Grothendieck s theorem on the parafactoriality of local rings, Commutative algebra: syzygies, multiplicities and birational algebra (South Hadley, 1992), Contemp. Math. 159 (1994), [7] I. Cheltsov, A Fano 3-fold with a unique elliptic structure, Mat. Sb. 192 (2001), [8] I. Cheltsov, J. Park, Sextic double solids, preprint (2004) [9] A. Conte, M. Marchisio, J. Murre, On unirationality of double covers of fixed degree and large dimension; a method of Ciliberto, Algebraic geometry, de Gruyter, Berlin (2002), [10] A. Corti, A. Pukhlikov, M. Reid, Fano 3-fold hypersurfaces, L.M.S. Lecture Note Series 281 (2000), [11] I. Dolgachev, Weighted projective varieties, Lecture Notes in Math. 956 (1982), [12] D. Eisenbud, D. Grayson, M. Stillman, B. Sturmfels, Computations in algebraic geometry with Macaulay 2, Algorithms and Computation in Mathematics 8 Springer-Verlag (2001) Berlin τ

10 10 IVAN CHELTSOV AND JIHUN PARK [13] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), [14] D. Grayson, M. Stillman, Macaulay 2: a software system for algebraic geometry and commutative algebra avaivable at [15] A. Grothendieck, et al., Séminaire de géométrie algébrique 1962, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, IHES (1965) [16] V. Iskovskikh, Yu. Manin, Three-dimensional quartics and counterexamples to the Lüroth problem, Mat. Sb. 86 (1971), [17] Y. Kawamata, K. Matsuda, K. Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai (1985), , Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987 [18] L. Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), [19] A. Pukhlikov, Birational automorphisms of Fano hypersurfaces, Invent. Math. 134 (1998), [20], Essentials of the method of maximal singularities, L.M.S. Lecture Note Series 281 (2000), [21] F. Zak, Tangents and secants of algebraic varieties, Mathematical Monographs 127 (1993), AMS, Providence, Rhode Island. Ivan Cheltsov School of Mathematics, University of Edinburgh, Kings Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK; cheltsov@yahoo.com Jihun Park Department of Mathematics, POSTECH, Pohang, Kyungbuk , Korea; wlog@postech.ac.kr