BOUNDS ON THE TORSION SUBGROUP SCHEMES OF NÉRON–SEVERI GROUP SCHEMES HYUK JUN KWEON Abstract. Let X,! Pr be a smooth projective variety defined by homogeneous polyno- mials of degree ≤ d over an algebraically closed field. Let Pic X be the Picard scheme of X. Let Pic0 X be the identity component of Pic X. The Néron–Severi group scheme of 0 X is defined by NS X = (Pic X)=(Pic X)red. We give an explicit upper bound on the order of the finite group scheme (NS X)tor in terms of d and r. As a corollary, we give an 1 ab upper bound on the order of the finite group π´et(X; x0)tor. We also show that the torsion subgroup (NS X)tor of the Néron–Severi group of X is generated by less than or equal to (deg X − 1)(deg X − 2) elements in various situations. 1. Introduction In this paper, we work over an algebraically closed base field k. Although char k is arbi- trary, we are mostly interested in the case where char k > 0. The Néron–Severi group NS X of a smooth projective variety X is the group of divisors modulo algebraic equivalence. Thus, we have an exact sequence 0 ! Pic0 X ! Pic X ! NS X ! 0: Néron [27, p. 145, Théorème 2] and Severi [31] proved that NS X is a finitely generated abelian group. Hence, its torsion subgroup (NS X)tor is a finite abelian group. Poonen, Testa and van Luijk gave an algorithm for computing (NS X)tor [29, Theorem 8.32]. The author gave an explicit upper bound on the order of (NS X)tor [21, Theorem 4.12]. As in [32, 7.2], define the Néron–Severi group scheme NS X of X by the exact sequence 0 0 ! (Pic X)red ! Pic X ! NS X ! 0: If char k = 0, then Pic0 X is an abelian variety, so NS X is a disjoint union of copies of Spec k. However, if char k = p > 0, then Pic0 X might not be reduced, and Igusa gave the first example [18]. Thus, the Néron–Severi group scheme may have additional infinitesimal arXiv:2008.01908v2 [math.AG] 27 Sep 2021 p-power torsion. The torsion subgroup scheme (NS X)tor of NS X is a finite commutative group scheme. It is a birational invariant for smooth proper varieties due to [28, p. 92, Proposition 8] and [1, Proposition 3.4]. The first main goal of this paper is to give an explicit upper bound b on the order of (NS X)tor. Let expa b := a . Date: September 28, 2021. 2010 Mathematics Subject Classification. Primary 14C05; Secondary 14C20, 14C22. Key words and phrases. Néron–Severi group, Castelnuovo–Mumford regularity, Gotzmann number. This research was supported in part by Samsung Scholarship and a grant from the Simons Foundation (#402472 to Bjorn Poonen, and #550033). 1 Theorem 5.10. Let X,! Pr be a smooth connected projective variety defined by homoge- neous polynomials of degree ≤ d. Then #(NS X)tor ≤ exp2 exp2 exp2 expd exp2(2r + 6 log2 r): One motivation for studying (NS X)tor is its relationship with fundamental groups. Recall that if k = C, then 1 ab π (X; x0)tor ' H1(X; Z)tor 2 ' Hom H (X; Z)tor; Q=Z ' Hom((NS X)tor; Q=Z): 1 ab However, if char k = p > 0, then π (X; x0)tor is not determined by (NS X)tor. Therefore, an 1 ab upper bound on #(NS X)tor does not give an upper bound on #π´et(X; x0)tor. Nevertheless, 1 ab π (X; x0)tor is isomorphic to the group of k-points of the Cartier dual of (NS X)tor [32, Propo- 1 ab sition 69]. Hence, we give an upper bound on #π´et(X; x0)tor as a corollary of Theorem 5.10. 1 ab As far as the author knows, this is the first explicit upper bound on #π´et(X; x0)tor. Theorem 6.6. Let X,! Pr be a smooth connected projective variety defined by homogeneous polynomials of degree ≤ d with base point x0 2 X(k). Then ´et ab #π1 (X; x0)tor ≤ exp2 exp2 exp2 expd exp2(2r + 6 log2 r): N Let π1 (X; x0) be the Nori’s fundamental group scheme [28] of (X; x0). Then we also give a N ab similar upper bound on #π1 (X; x0)tor. The second main goal of this paper is to give an upper bound on the number of generators 1 of (NS X)tor. If ` 6= char k, then (NS X)[` ] is generated by at most (deg X − 1)(deg X − 2) elements by [21, Corollary 6.4]. The main tool of this bound is the Lefschetz hyperplane theorem on étale fundamental groups [11, XII. Corollaire 3.5]. Similarly, we prove that the Lefschetz hyperplane theorem on Picτ X gives an upper bound on the number of generators of (NS X)[p1]. However, the author does not know whether this is true in general. Never- theless, Langer [22, Theorem 11.3] proved that if X has a smooth lifting over W2(k), then the Lefschetz hyperplane theorem on Picτ X is true. This gives a bound on the number of generators of (NS X)tor. Theorem 8.4. If X,! Pr is the reduction of a smooth connected projective scheme X ,! r over W (k), then (NS X) is generated by (deg X − 1)(deg X − 2) elements. PW2(k) 2 tor In Section3, we prove that #(NS X)tor ≤ dimk Γ(HilbQ X; OHilbQ X ) for some explicit Hilbert polynomial Q. Section4 bounds dimk Γ(Y; OY ) for an arbitrary projective scheme Y,! Pr defined by homogeneous polynomials of degree at most d. Section5 gives an upper ´et ab N ab bound on #(NS X)tor. Section6 gives an upper bound on #π1 (X; x0)tor and #π1 (X; x0)tor. τ Section7 discusses the Lefschetz-type theorem on Pic X. Section8 proves that (NS X)tor is generated by less than or equal to (deg X − 1)(deg X − 2) elements if X has a smooth lifting over W2(k). 2. Notation Given a scheme X over k, let OX be the structure sheaf of X. We sometimes denote Γ(X) = Γ(X; OX ). Let Pic X be the Picard group of X. If X is integral and locally noetherian, then Cl X denotes the Weil divisor class group of X. If Y is a closed subscheme 2 of X, let IY ⊂X be the sheaf of ideal of Y on X. If X is a projective space and there is no confusion, then we may let IY = IY ⊂X . Given a k-scheme T and a k-algebra A, let X(T ) = Hom(T;X) and X(A) = Hom(Spec A; X). r Suppose that X is a projective scheme in P . Let HilbQ X be the Hilbert scheme of X corresponding to a Hilbert polynomial Q. Let HilbP X be the Hilbert scheme of X parametrizing closed subschemes Z of X such that IZ has Hilbert polynomial P . In partic- ular, P (n) Hilb X = Hilb n+r X: ( r )−P (n) Now, suppose that X is smooth. Let CDivQ X be the subscheme of HilbQ X parametrizing r divisors with Hilbert polynomial Q. If D ⊂ P is an effective divisor on X, let HPD be the Hilbert polynomial of D as a subscheme. Let Pic X be the Picard scheme of X. Let Pic0 X be the identity component of Pic0 X. Let Picτ X be the disjoint union of the connected components of Pic X corresponding to 0 0 τ τ 0 (NS X)tor. Let Pic X = (Pic X)(k) and Pic X = (Pic X)(k). Let NS X = Pic X=(Pic X)red, 0 ´et N and let NS X = Pic X= Pic X. Given x0 2 X(k), let π1 (X; x0) and π1 (X; x0) be the étale fundamental group and Nori’s fundamental group scheme [28] of (X; x0), respectively. Given a vector space V , let Gr(n; V ) be the Grassmannian parametrizing n-dimensional linear subspaces of V . Let S be a separated k-scheme of finite type, and let X be a S-scheme, with the structure map f : X ! S. Then given a point t 2 S, let Xt be the fiber over t. Let F be a quasi- coherent sheaf on X, then let Ft be the sheaf on Xt which is the fiber of F over t. Let g : T ! S be a morphism. Let p1 : X ×S T ! X and p2 : X ×S T ! T be the projections. Then as in as in [19, Definition 9.3.12], for an invertible sheaf L on X, let LinSysL =X=S be the scheme given by LinSysL =X=S(T ) = D D is a relative effective divisor on XT =T such that ∗ ∗ OXT (D) ' p1L ⊗ p2N for some invertible sheaf N on T : Given a set S and T , let S qT be the disjoint union of S and T . Let #S be the cardinality of S. Let N be the set of nonnegative integers, and N<n be the set of nonnegative integers strictly less than n. We regard an n-tuple a on S as a function a: N<n ! S. Thus, the ith component of a is denoted by a(i). Given a group G, let Gab be the abelianization of G. Suppose that G is abelian.