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NOETHER-LEFSCHETZ THEORY AND QUESTIONS OF SRINIVAS SCOTT NOLLET Abstract. Noether-Lefschetz theory surrounds the question of determining when the restriction map from the Picard group of a variety to the
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NOETHER-LEFSCHETZ THEORY AND QUESTIONS OF SRINIVAS SCOTT NOLLET Abstract. Noether-Lefschetz theory surrounds the question of determining when the restriction map from the Picard group of a variety to the Picard group of a general member of a linear system is an isomorphism, its origins dating back to Lefschetz proof from 1921 of a claim made my Noether in After discussing the Grothendieck-Lefschetz theorems in higher dimensions, we focus on modern results in the three-dimensional case, finishing with a new proof of Moishezon s theorem. Then we will discuss problems in local commutative algebra posed by Srinivas [38] and recent results obtained from Noether-Lefschetz theory. 1. Noether-Lefschetz theory Modern Noether-Lefschetz theory surrounds the following problem: Problem 1.1. For which complex varieties X and line bundles L Pic X is the restriction map r Y : Pic X Pic Y an isomorphism for general Y in the linear system L = PH 0 (L)? For pairs (X, L) as bove, the following question has also received plenty of attention: Problem 1.2. Describe the irreducible families V L of surfaces for which the restriction map r Y fails to be an isomorphism: these are the Noether-Lefschetz components. I will discuss results over C, but in their study of monodromy groups in characteristic p 0 [11, 19], Grothendieck, Deligne and Katz extended Lefschetz pencils, vanishing cycles and the Picard-Lefschetz formula to obtain results on Problem 1.1 meaningful in finite characteristic The Noether-Lefschetz theorem. The statement is as follows: Theorem 1.3. For d 3, the restriction map Pic P 3 Pic Y is an isomorphism for very general Y H 0 (O P 3(d)) Noether s idea. Count dimensions. Example 1.4. Let V O(d) be the family of surfaces Y containing a line L. Let I = {(L, S) : L S} G(1, 3) V be the incidence variety of lines on surfaces along with projections π 2 : I V and π 1 : I G(1, 3), where G(1, 3) is the Grassmann variety of lines. The exact sequence 0 H 0 (I L (d)) H 0 (O P 3(d)) H 0 (O L (d)) 0 Date: October 27, Mathematics Subject Classification. Primary: 14C20, 14C22, 14B07, 14J17, 14J29, 13C20, 13F15. 1 2 SCOTT NOLLET shows that dim π 1 1 (L) = dim O(d) d 1, hence dim I dim O(d) d + 3 since dim G(1, 3) = 4. Therefore dim V dim O(d) d + 3 so that V O(d) is a proper subvariety for d 3. Apparently Noether did many such calculations, leading to his conclusion. Exercise 1.5. Carry out Noether s dimension count for surfaces containing conics Lefschetz proof. Lefschetz easily proved an analogous statement in higher dimensions. The exponential sequence 0 Z O Y O Y 0 yields (1) H 1 (P n, O P n) H 1 (P n, O P n) H2 (P n, Z) H 2 (P n, O P n) α β H 1 (Y, O Y ) H 1 (Y, O Y ) H2 (Y, Z) H 2 (Y, O Y ). For n 3 the cohomology groups in the four corners are zero and α is identified with the restriction map Pic P n Pic Y. The Lefschetz hyperplane theorem says that the maps H k (P n, Z) H k (Y, Z) are isomorphisms for k n 1 and injective for k = n 1. Thus β is an isomorphism for n 3 and therefore α as well: Theorem 1.6. If Y P n is a smooth hypersurface and n 3, then the restriction map Pic P n Pic Y is an isomorphism. When n = 3 the result no longer follows from Diagram (1) because there is no zero in the lower right and the Lefschetz hyperplane theorem no longer applies to β. The statement fails for d = 2 and d = 3 and for d 3 there are infinitely many Noether-Lefschetz components V O P 3(d), so the restriction map is not an isomorphism for Zariski general Y O P 3(d). Lefschetz used a monodromy argument, showing that a typical deformation (along a Lefschetz pencil) takes a Hodge class γ H 1,1 (Y, C) H 2 (Y, Z) representing a non-complete intersection curve into H 0,2 (Y, C) and therefore becomes non-algebraic. Voisin gives a clear exposition in her books on Hodge theory [41, 42]. Remark 1.7. Mumford s challenge from the 1960s to find an explicit equation of a smooth quartic S P 3 with Pic S = O S (1) was finally answered by van Luijk in One such equation [40, Remark 3.7] is w(x 3 + y 3 + x 2 z + xw 2 ) = 3x 2 y 2 4x 2 yz + x 2 z 2 + xy 2 z + xyz 2 y 2 z 2. Note that this surface contains the line w = z = 0 in characteristic p = 3, so this surface specializes to a member of the Noether-Lefschetz locus in finite characteristic Higher dimension: Grothendieck-Lefschetz theorems. When dim X 3, the results are excellent. Theorem 1.8. Let X be a smooth projective variety of dimension n 4. Then for any effective ample divisor Y X, the restriction map Pic X Pic Y is an isomorphism. For example, every closed subscheme Y P 4 defined by a homogeneous polynomial has Picard group Pic Y generated by O Y (1). Hartshorne [21, IV, Corollary 3.3] simplified Grothendieck s original proof [18] by assuming X and Y nonsingular, but Lazarsfeld observes that the smoothness of Y was unncessary [25, Remark ]. Grothendieck s NOETHER-LEFSCHETZ THEORY AND QUESTIONS OF SRINIVAS 3 original idea [18, Exposé X] is to consider an open neighborhood U of Y in the formal completion ˆX of X along Y and show that the sequence of induced maps Pic X Pic U Pic ˆX Pic Y are all isomorphisms. The most difficult part is the isomorphism Pic ˆX = Pic U, for which Grothendieck defines effective Lefschetz conditions Leff(X, Y ) that are satisfied by the pair (X, Y ). The last isomorphism is obtained by considering the infinitesimal neighborhoods Y n X defined by ideals IY n. Kodaira vanishing implies that Hi (Y, IY n/in+1 Y i = 1, 2 and therefore the exact sequences 0 IY n/in+1 ) = 0 for Y OY n OY n+1 0 give isomorphisms Pic Y n = Pic Yn+1 for n 0 and hence Pic ˆX = lim Pic Y n = Pic Y Normal varieties and class groups. For X normal, one can consider the strict transform Ỹ X for a desingularization X X. If E X is the exceptional divisor, the for general member Y X of a base point free linear system we have the homomorphism Cl X = Pic(X E) Pic(Y E) = Cl Y. Ravindra and Srinivas [34] prove an almost Lefschetz condition ALeff( X, Ỹ ) which leads to the following analog for class groups. Theorem 1.9. Let X be a normal variety of dimension n 4. Assume L Pic X is ample and V H 0 (X, L) is a base point free linear system. Then the general member Y V is normal and the restriction map r : Cl X Cl Y is an isomorphism Linear systems with base locus. We used Theorem 1.9 in our result for linear systems with base locus [4]. Theorem Let X P N C be a normal variety of dimension n 4 and let Z X be a closed subscheme of codimension 2 with codimension 2 irreducible components Z 1, Z 2,..., Z s. Assume (1) I Z (d 1) is generated by global sections. (2) Z i is not contained in the singular locus of X. (3) Z i has generic embedding dimension at most dim X 1. Then the general member Y H 0 (X, I Z (d)) is normal and the map α : Cl X Z s Cl Y given by (L, a 1,..., a s ) L Y + a i Supp Y i is an isomorphism Dimension three: modern results. Problem 1.1 is harder when dim X = 3. One must take Y L to be very general, avoiding a countable union of Noether-Lefschetz components, which are dense in the Euclidean topology [7, 9]; moreover, the conclusion fails without additional positivity assumptions on L. We briefly survey the work done since 1980; see our survey [2] for more details Carlson, Green, Griffiths and Harris, 1983: Infinitesimal variant for sufficiently ample L on smooth n-fold X using infinitesimal variations of Hodge structures [8]. They prove that in the family of smooth Y L, the infinitesimally fixed part of the middle cohomology groups H p,q (Y ) is precisely the fixed cohomology coming from the ambient space X. 4 SCOTT NOLLET Green, 1984: used Koszul cohomology [14, 15] to show that Noether-Lefschetz components satisfy d 3 codim(v, O P 3(d) ) p g (d) = ( ) d 1 3 [13]. In particular, all Noether- Lefschetz components for quartics have codimension one. His original argument used a spectral sequence to deduce a vanishing of a Koszul cohomology group, but in 1988 he gave a slicker proof using a filtration [16] Griffiths and Harris, 1985: proved Theorem 1.3 by degenerating a general degree d surface to a union of a plane union a smooth surface of degree d 1 and computing the Picard group of the central fiber of a desingularization of the total family [17] Ein, 1985: extended Noether s theorem from line bundles to vector bundles of higher rank [12]. If T H 0 (E) is a t-dimensional subspace and E is a rank r bundle, one obtains a map T O X E and dependency loci D k where the rank of this map is at most k. If 2(r + 3 t) n, then D t 2 is empty and Y = D t 1 is smooth: assuming sufficient ampleness, Ein computed Pic Y in terms of Pic X Lopez, 1989: For very general surfaces S P 3 containing a smooth curve C, Pic S is freely generated by C and O S (1) [27] Ciliberto, Harris, Miranda and Green, 1988: showed that the Noether-Lefschetz components are dense in the Euclidean topology [9] Ciliberto and Lopez, 1991: constructed components of varying codimensions [10] Joshi, 1995: used ideas in unpublished notes of Mohan Kumar and Srinivas to prove a new infinitesimal variant for smooth threefolds, obtaining a result for general singular surfaces [24] Ravindra and Srinivas, 2009: proved that for X normal, Cl X Cl Y is an isomorphism for very general Y if L ample and K(L) globally generated [35] Brevik and Nollet, 2011: proved a version for class groups and base locus (similar to Theorem 1.10) for X = P 3 [1] Moishezon s theorem. The best result for smooth complex threefolds was obtained by Moishezon in his general study of algebraic homology classes [28]. He adapted the argument of Lefschetz to prove the following remarkable theorem: Theorem Let X P N C be a smooth threefold and let Y X be a very general hyperplane section. Then the restriction Pic X Pic Y is an isomorphism if and only if (a) b 2 (Y ) = b 2 (X) or (b) h 2,0 (X) h 2,0 (Y ). When X = P 3, condition (a) (equivalently H 2 (X, C) = H 2 (Y, C)), picks up the missing case L = O(1) and Y P 3 is a plane. Since (a) fails for sufficiently positive L, the most important case is (b). The Hodge condition h 2,0 (X) h 2,0 (Y ) means h 2 (O Y ) h 2 (O X ) or the Serre dual h 0 (K Y ) h 1 (K X ), but in view of the exact sequence 0 H 0 (K X ) H 0 (K X L) H 0 (K Y ) H 1 (K X ) 0 NOETHER-LEFSCHETZ THEORY AND QUESTIONS OF SRINIVAS 5 arising from adjunction and Kodaira vanishing, this is equivalent to h 0 (K X ) h 0 (K X L). Since L is very ample and dim X 0, this is equivalent to H 0 (K X L) 0 by the following result for varieties X of positive dimension: Fact If A, L Pic X, L very ample, then H 0 (L A) 0 h 0 (A) h 0 (L A). The hypothesis H 0 (K X L) 0 is notably weaker than those of several theorems in the previous section. It is also essentially the hypothesis for the variant in Voisin s book on Hodge theory [41, 42]. Adapting the argument of Griffiths and Harris [17], we give a new proof of Theorem 1.11 (b) when L = O(1) is a product of very ample line bundles: Theorem Let X be a smooth complex threefold. If A, B Pic X are very ample and H 0 (K X A B) 0, then r : Pic X Pic Y for very general Y A B. Proof. Similar to the proof of Griffiths and Harris for X = P 3, we focus on a general linear pencil P 1 A B containing a smooth surface S and a reducible surface T P at t = 0 with D = T P a smooth curve. Our situation is more difficult because Pic X Pic P need not be an isomorphism and Pic 0 X need not be zero. The total family M X P 1 is singular over the central degenerate fiber 0 = t P 1 at the points of intersection S P T, but the corresponding total family of strict transforms M X P 1 where X X is the blow-up at S T is nonsingular near t = 0. The central fiber becomes T P, where P P is the blowup along P T S. Claim 1: Pic M 0 = Pic X ZO M( P ) M0. Combining the hypothesis H 0 (K X L) 0 and Fact 1.12 with the exact sequence 0 H 0 (K X B) H 0 (K X B A) H 0 (K P B) 0 yields H 0 (K P B) 0. Combining this with Fact 1.12 and the exact sequence 0 H 0 (K P ) H 0 (K P B) H 0 (K D ) H 1 (K P ) 0 yields H 1 (O D ) = H 0 (K D ) H 1 (K P ) = H 1 (O P ). Since H 1 (O V ) is naturally the tangent space to the Picard variety Pic 0 V at the origin, the inclusion leads us to a closed immersion Pic 0 P Pic 0 D of Picard varieties. In particular the embedding of P by B is not the Veronese surface embedded by quadrics nor is it a ruled surface, hence B O P has a pencil consisting of irreducible curves [27, II.2.4], from which it follows that if L Pic P and L D = OD for general D in such a pencil, then L = O P is trivial [27, II.2.3]. Looking at the countably many representatives L Pic P/ Pic 0 P {0} in the Néron-Severi group, it follows that L D = D for a proper algebraic subset of D B and hence Pic P Pic D is injective for very general D B O P. Reversing the roles of P and T we also obtain injective Pic T Pic D for very general D A O T and for very general (P, T ) A B we obtain a commutative diagram (2) Pic X Pic P Pic T Pic D in which all restriction maps are injective (the maps from Pic X are injective by [34]). 6 SCOTT NOLLET We show Diagram (2) is Cartesian. For this we fix P and L Pic P/ Pic 0 P and show that if L D Pic T for general T B, then L Pic X. The idea of the proof here is that the set of T with L D Pic T is closed and if is all of B, then by unicity we can find a continuous family of lines bundles L t Pic T t. Restricting to a pencil P 1 B, representability of the relative Picard scheme gives a line bundle on the total family over the pencil, but if the pencil has base locus C = T 0 T 1, then the total family is isomorphic to X X, the blow up along C. This gives a line bundle A on X modulo the exceptional divisor of the blow up, and with minor modification we show that A P = L. Working over the countable representatives for Pic P/ Pic 0 P we find that the diagram is Cartesian for very general T and hence Pic P Pic D Pic T = Pic X, which computes Pic M 0. With similar arguments we show for very general S that Pic M 0 = Pic X O M( P ) O M0. With Claim 1 in hand, the rest of the proof flows along the lines of Griffiths and Harris proof [17]. Each Noether-Lefschetz component V A B is the image of a relative Hilbert scheme component W Hilb of curves whose divisor classes are not in the image of Pic X and we need to show that the map π : W A B is not dominant. If it were dominant, then there is a pencil P 1 A B as constructed in Step 1 for which π 1 P 1 P 1 is dominant. We can then find an integral curve E 0 W dominating P 1 and after normalizing a smooth curve E P 1. Pulling back the family M P 1 back to a new family Z E, we use Claim 1 to deduce that image is proper. 2. Questions of Srinivas The local ring A = O X,x of a point x on a complex algebraic variety X is a geometric local domain. If A is normal, then so is the completion R =  and the natural map A R is flat, hence the Mori map [39] (3) ι : Cl A Cl R given by p P A=p e(p, p)p is a well-defined injective homomorphism where e(p, p) is the ramification index of the field extension K(A/p) K(B/P ). It is essentially the pull-back map along Spec R Spec A after removing singularities. Srinivas [38] asks about the images of Inclusion (3) for fixed R as A varies over geometric normal local domains satisfying R = Â: Question 2.1. Let R be the completion of a normal geometric local ring. Which subgroups of Cl R arise as images Cl A Cl R where R = Â? The following example of Srinivas [38] shows that Question 2.1 is interesting. Example 2.2. The complete local ring R = C[[x, y, z]]/(x 2 + y 3 + z 7 ) has class group Cl R = C [39], but for every geometric local ring A with  = R, the image Cl A Cl R is necessarily finitely generated [38, Example 3.9]. Srinivas reasons that if A = O X,x for a surface X and Y X is a resolution of singularities, then the induced map Pic Y Cl A is surjective. Since Pic 0 Y is projective, it has trivial image in the affine group G a = C, therefore Cl A C factors through the finitely generated Neron-Severi group. After asking Question 2.1, Srinivas backs off somewhat, instead asking about the minimal possible images of the Mori map. NOETHER-LEFSCHETZ THEORY AND QUESTIONS OF SRINIVAS 7 Question 2.3. Let R be the completion of a normal geometric local ring. (a) If R is Gorenstein, is R the completion of a geometric UFD? (b) Does there exist geometric ring A with R =  and Cl A = ω A? Remark 2.4. These questions are only interesting for singularities. If A is a regular local ring, then so is R =  so that Cl R = 0 and consequently any ring A with  = R satisfies Cl A = 0, hence is a UFD. Remark 2.5. If A is the codimension r quotient of a regular local ring B, then the dualizing module ω A = Ext r A(B, A) Cl R is independent of B and the image in Cl R is independent of A. Thus Question 2.1 (b) asks about the minimal image. Moreover ω A Cl R is zero if and only if R is Gorenstein [30], so (a) is a special case of (b) Minimal images. We note some progress made on Question Grothendieck, 1968: solved Samuel s conjecture, proving that a local complete intersection ring that is factorial in codimension 3 is a UFD [18, XI, Cor. 3.14] Hartshorne and Ogus, 1973: if R has an isolated singularity, depth R 3 and embedding dimension at most 2 dim R 3, then R is a UFD [22] Heitmann, 1993: characterized completions of UFDs [23], but his constructions rarely produce geometric rings Srinivas, 1987: Part (a): Yes for rational double points [37] Parameswaran and van Straten, 1993: Part (b): Yes if dim R = 2 [33] Parameswaran and Srinivas, 1994: Part (a): Yes for local complete intersections of dimensions two and three with isolated singularity [32] Brevik and Nollet, 2016: Part (a): Yes for hypersurface singularities of dimension 2 [3] and local complete intersection singularities of dimension 3 [4] Proof for hypersurfaces. We illustrate our method for hypersurface singularities. Let f C[x 1, x 2,... x n ] be the equation of a hypersurface V which normal at the origin p, corresponding to the maximal ideal m = (x 1,..., x n ), and let R = C[[x 1, x 2,... x n ]]/(f) be the completion of A = O V,p. The singular locus D of V is given by the ideal (f) + J f, where J f = (f x1,..., f xn ). Primary decomposition in C[x 1,..., x n ] gives J f = q i p i m where q i is p i -primary and we have sorted into components that meet the origin and those that do not. Denote by K the intersection on the left and J the intersection on the right; localizing at m we find that (J f ) m = K m because J m = (1). Now if K = (k 1,..., k r ) C[x 1,..., x n ], then the closed subscheme Z defined by the ideal I Y = (f, k1, 3... kr) 3 is supported on the components of the singular locus of V that contain the origin, hence codim(z, P n ) 3 by normality of V at the origin. The very p i m q i 8 SCOTT NOLLET general hypersurface Y containing Z satisfies Cl Y = 0 by Theorem 1.10 or above, so Cl O Y,p = 0 as well and O Y,p is a UFD [20, Prop. 6.2]. Moreover, Y has local equation g = f + a 1 k a r kr 3 for units a i, and hence f g K 3. Since K m = (J f ) m, their completions are equal in C[[x 1,..., x n ]]. Therefore f g Jf 3 mj f 2 and a result of Ruiz [36, V, Lemma 2.2] tells us that ÔY,p = C[[x 1,..., x n ]]/(g) = C[[x 1,..., x n ]]/(f) = R General images. While Question 2.3 has received much attention, Question 2.1 remains wide open. To understand it better, we call an element α Cl R a geometric divisor if it is in the image of the inclusion (3) for some geometric local ring A with R = Â. In view of Example 2.2, we pose the following: Question 2.6. Which statements hold for the completion R of a normal geometric ring? (a) Given any finitely generated group G Cl R, there a geometric local ring B with B = R and G = Cl B. (b) Given α 1,..., α r Cl R, there is a geometric local ring B with B = R and α i Cl B for 1 i r. (c) Every α Cl R a geometric divisor. (d) The geometric divisors form a subgroup of Cl R. Note the easy implications (a) (b) (c) (d). Our ignorance about the nature of geometric divisors is revealed in part (c): could the
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