Bounds on the Torsion Subgroups of N\'Eron-Severi Group Schemes

$Bounds on the Torsion Subgroups of N\'Eron-Severi Group Schemes$

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 polynomials 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 πét(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 arbitrary, 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 homogeneous 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 #πét(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 #πét(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 #πét(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 ∈ X(k). Then ét 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 ∞ 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)[p∞]. 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 particular,

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 ét N and let NS X = Pic X/ Pic X. Given x0 ∈ 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 ∈ 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

 a0,0 a0,1 . . . a0,n−1   a1,0 a1,1 . . . a1,n−1  (ai,j)i

f(M) := (ϕ(ai,j))i

3. Numerical Conditions Let ı: X,→ Pr be a closed embedding of a smooth connected projective variety X. Let H ⊂ X be a hyperplane section of X. The goal of this section is to describe an integer m such that

(1) #(NS X)tor ≤ dimk Γ(HilbHPmH X) . For a positive integer m, consider the diagram τ HilbHPmH ←- CDivHPmH X → Pic X (NS X)tor, in which the middle morphism is yet to be defined. The natural embedding CDivHPmH X,→ τ HilbHPmH X is open and closed [20, Theorem 1.13]. The natural quotient Pic X (NS X)tor is faithfully flat [5, VIA, Théorème 3.2]. Suppose that there is a faithfully flat τ morphism CDivHPmH X → Pic X for some large m. Then Γ((NS X)tor) ,→ Γ(CDivHPmH ) is injective by [10, Corollaire 2.2.8], and Γ(HilbHPmH ) Γ(CDivHPmH ) is surjective, so we get (1). Such m will be obtained as a byproduct of the construction of Picard schemes. The theorem and the proof below are a modification of the proof of [19, Theorem 9.4.8]. Theorem 3.1. Let m be an integer such that (2) H1(X, L (m)) = 0 for every numerically zero invertible sheaf L . Then the morphism given by τ (CDivHPmH X)(T ) → (Pic X)(T )

D 7→ OXT (D − mHT ) is faithfully ﬂat. Proof. Let Picσ X be the union of the connected components of Pic X parametrizing invertible sheaves having the same Hilbert polynomial as OX (m). Then for any k-scheme T , (Picσ X)(T ) → (Picτ X)(T ) L 7→ L (−m) 4 gives an isomorphism. Because of the cycle map

σ (CDivHPmH X)(T ) → (Pic X)(T )

D 7→ OXT (D), σ σ σ CDivHPmH X can be regarded as a (Pic X)-scheme. The identity Pic X → Pic X gives σ σ a (Pic X)-point of Pic X, and it is represented by an invertible sheaf L on XPicσ X . Then σ σ σ CDivHPmH X is naturally isomorphic to LinSysL /X×Pic X/ Pic X as (Pic X)-schemes by σ deﬁnition. For every closed point t ∈ Pic X, Lt(−m) is numerically zero, so 1 H (X × {t}, Lt) = 0.

Hence, [19, 9.3.10] implies that LinSysL /X×Picσ X/ Picσ X ' P(Q) for some locally free sheaf σ σ τ Q on Pic X. Since P(Q) → Pic X is faithfully flat, CDivHPmH X → Pic X is also faithfully flat. We now need to find an integer m satisfying (2). Recall the definition of Casteln- uovo–Mumford regularity.

Deﬁnition 3.2. A coherent sheaf F on Pr is m-regular if and only if i r H (P , F (m − i)) = 0 for every integer i > 0. The smallest such m is called the Castelnuovo–Mumford regularity of F . Mumford [26, p. 99] showed that if F is m-regular, then it is also t-regular for every t ≥ m.

Deﬁnition 3.3. Let P be the Hilbert polynomial of some homogeneous ideal I ⊂ k[x0, . . . , xr]. The Gotzmann number ϕ(P ) of P is deﬁned as

ϕ(P ) = inf{m | IZ is m-regular for every r closed subvariety Z ⊂ P with Hilbert polynomial P }. If Z ⊂ Pr is a projective scheme, then let ϕ(Z) be the Gotzmann number of the Hilbert polynomial of IZ .

Given a homogeneous ideal I ⊂ k[x0, . . . , xr], Hoa gave an explicit ﬁnite upper bound on ϕ(HPI ) [16, Theorem 6.4(i)]. We will describe m in terms of Gotzmann numbers of Hilbert polynomials.

Lemma 3.4. Let L be a numerically zero invertible sheaf on X. Then L ((d − 1) codim X) is generated by global sections.

Proof. See [21, Lemma 3.5 (a)]. Lemma 3.5. Let L be a numerically zero invertible sheaf on X. Then for every i ≥ 1, n ≥ (d − 1) codim X and m ≥ max{ϕ(nH), ϕ(X)}, we have

Hi(X, L (m − n)) = 0. 5 Proof. There is an eﬀective divisor Z on X such that IZ⊂X = L (−n) by Lemma 3.4. i r i r Since IZ and IX are m-regular, H (P , IZ (m)) = 0 and H (P , IX (m)) = 0 for all i ≥ 1. Consider the exact sequence

0 → IX → IZ → ı∗IZ⊂X → 0. Then its long exact sequence shows that i i r H (X, L (m − n)) = H (P , ı∗IZ⊂X (m)) = 0 for every i ≥ 1. We are ready to prove the main theorem of this section. Theorem 3.6. Assume that n ≥ (d − 1) codim X and m ≥ max{ϕ(nH), ϕ(X)}. Then

#(NS X)tor ≤ dimk Γ(HilbHPmH X) . τ Proof. The quotient Pic X → (NS X)tor is faithfully ﬂat. Lemma 3.5 and Theorem 3.1 give τ a faithfully ﬂat morphism CDivHPmH X → Pic X. As a result, their composition gives an injection

Γ((NS X)tor) ,→ Γ(CDivHPmH X) by [10, Corollaire 2.2.8]. Recall that CDivHPmH X is an open and closed subscheme of

HilbHPmH X. Consequently,

#(NS X)tor = dimk Γ(NS X)tor

≤ dimk Γ(CDivHPmH X)

≤ dimk Γ(HilbHPmH X) .

Therefore, if we bound dimk Γ(Y, OY ) for a general projective scheme Y , and explicitly describe Hilbert schemes in projective spaces, then we can bound #(NS X)tor.

4. The Dimension of Γ(Y, OY ) Let Y,→ Pr be a projective scheme deﬁned by a homogeneous ideal I generated by homogeneous polynomials of degree at most d. The aim of the section is to give an upper bound on dimk Γ(Y, OY ) in terms of r and d. First, we reduce to the case where I is a monomial ideal.

r Lemma 4.1. Let be a monomial order. Let Y0 ,→ P be a projective scheme deﬁned by in(I). Then

dimk Γ(Y, OY ) ≤ dimk Γ(Y0, OY0 ). Proof. By [15, Corollary 3.2.6] and [15, Theorem 3.1.2], there is a ﬂat family Y → Spec k[t] such that

(1) Y0 ' Y0 and (2) Yt ' Y for every t 6= 0.

Hence, dimk Γ(Y, OY ) ≤ dimk Γ(Y0, OY0 ) by the semicontinuity theorem [13, Theorem 12.8]. 6 Theorem 4.2 (Dubé [6, Theorem 8.2]). Let I ⊂ k[x0, . . . , xr] be an ideal generated by homogeneous polynomials of degree at most d. Let be a monomial order. Then in(I) is generated by monomials of degree at most

r−1 d2 2 2 + d . 2 Hence, we will focus on the case where I is a monomial ideal. Let m be a monomial and I be a monomial ideal of k[x0, . . . , xr]. Then by abusing notation, we denote (m) (m) := . I I ∩ (m)

Deﬁnition 4.3. Let m ∈ k[x0, . . . , xr] be a monomial, and let I ⊂ k[x0, . . . , xr] be a mono- d d mial ideal. Let M(m, I) be the set of monomials in (m) \ (I ∪ (x0, . . . , xr)). In particular, (m) #M(m, I) = dim . k d d I + (x0, . . . , xr)

Deﬁnition 4.4. Given a monomial m ∈ k[x0, . . . , xr], let µ(m) + 1 be the number of i such that xi | m.

Deﬁnition 4.5. Let I be a monomial ideal with minimal monomial generators m0, m1, . . . , mn−1. Then

µ(I) := µ(m0) + µ(m1) + ··· + µ(mn−1). Suppose that I is generated by monomials of degree at most d. We want to show that r dimk Γ(Y, OY ) ≤ d . By induction on µ(I), we will show a slightly stronger proposition: for every monomial m ∈ k[x0, . . . , xr] whose exponents are at most d, we have #M(m, I) dim Γ r, (^m)/I ≤ . k P d r r+1 r Since Γ(Y, OY ) = Γ(P , (1)]/I) and #M(m, I) ≤ d , it follows that dimk Γ(Y, OY ) ≤ d .

b0 b1 bn−1 Lemma 4.6. For some n ≤ r + 1, let I = (x0 , x1 , . . . , xn−1 ) such that bi > 0 for every i < n. Then  1 if n < r,  dimk Γ(Y, OY ) = b0 . . . bn−1 if n = r, and 0 if n = r + 1.

Proof. Suppose that n < r, and take a nonzero function f ∈ Γ(Y, OY ). Take any i and j such deg g that n ≤ i < j ≤ r. In the open set xi 6= 0, we have f = g/xi such that g ∈ k[x0, . . . , xr] deg h and xi - g. In the open set xj 6= 0, we have f = h/xj such that h ∈ k[x0, . . . , xr] and xj - h. Therefore, in the open set xixj 6= 0, we have deg h deg g gxj = hxi .

Therefore, deg g = deg h = 0. As a result, f is constant meaning that dimk Γ(Y, OY ) = 1. If n = r, then dimk Γ(Y, OY ) = b0 . . . bn−1 by Bézout’s theorem. If n = r + 1, then Y is empty, so dimk Γ(Y, OY ) = 0. 7 Lemma 4.7. Let m be a monomial and I be a monomial ideal such that µ(I) > 0. Then there are monomials m0 and m1, and monomial ideals I0 and I1 such that

M(m, I) = M(m0,I0) q M(m1,I1),

r r r dimk Γ P , (^m)/I ≤ dimk Γ P , (m^0)/I0 + dimk Γ P , (m^1)/I1 ,

µ(I0) < µ(I) and µ(I1) < µ(I). Proof. Since µ(I) > 0, there is a minimal monomial generator n0 of I such that µ(n0) > 0. Let J be the ideal generated by the other minimal monomial generators of I. Then I = (n0) + J 0 0 s and n 6∈ J. We may assume that n = xi n where s > 0 and xi - n. Grouping the monomials in M(m, I) according to whether the exponent of xi is ≥ s or < s yields s s M(m, I) = M (lcm(xi , m), (n) + J) q M (m, (xi ) + J) . Moreover, the short exact sequence s (lcm(xi , m)) (m) (m) 0 → → → s → 0. (n) + J I (xi ) + J implies that ! ! (lcm(^xs, m)) ^(m) r ^ r i r dimk Γ P , (m)/I ≤ dimk Γ P , + dimk Γ P , s . (n) + J (xi ) + J Finally, we have µ((n) + J) ≤ µ(n) + µ(J) < µ(n0) + µ(J) = µ(I) s and similarly µ((xi ) + J) < µ(I).

Lemma 4.8. Let m ∈ k[x0, . . . , xr] be a monomial whose exponents are at most d. Let I ⊂ k[x0, . . . , xr] be an ideal generated by monomials of degree at most d. Then #M(m, I) dim Γ r, (^m)/I ≤ . k P d Proof. This will be shown by induction on µ(I). If µ(I) < 0, then I = (1), so

r dimk Γ P , (^m)/I = 0. Suppose that µ(I) = 0. Then by changing coordinates, we may assume that for some n ≤ r + 1, b0 b1 bn−1 a0 a1 ar I = (x0 , x1 , . . . , xn−1 ) and m = x0 x1 . . . xr such that 0 < bi ≤ d for every i < n, and 0 ≤ aj ≤ d for every j ≤ r. If ai ≥ bi for some i < n, then m ∈ I, so (m)/I = 0. Thus, we may assume that ai < bi ≤ d for every i < n. r Let ı: Y,→ P be the projective scheme deﬁned by I. Then (^m)/I is a subsheaf of ı∗OY . Hence, if n < r, then Lemma 4.6 implies that

r dimk Γ P , (^m)/I ≤ 1. deg m Suppose that n = r. Since the open set deﬁned by xr 6= 0 contains Y , multiplying by xr gives an isomorphism (gm) (gm) ×xdeg m : → (deg m). r I I 8 Moreover, multiplication by m−1 gives an isomorphism (m) k[x , . . . , x ] ×m−1 : (deg m) → 0 r . I b0−a0 br−1−ar−1 (x0 , . . . , xr−1 ) Hence, by Lemma 4.6,

r dimk Γ P , (^m)/I = (b0 − a0)(b1 − a1) ... (br−1 − ar−1). Suppose that n = r + 1. Then since Y is empty,

r dimk Γ P , (^m)/I = 0. Because r−n+1 #M(m, I) = (b0 − a0)(b1 − a1) ... (bn−1 − an−1)d , we have Γ Pr, (^m)/I ≤ #M(m, I)/d in every case.

Now, suppose that µ(I) > 0 and assume the induction hypothesis. Then we have m0, m1, I0 and I1 as in Lemma 4.7. Thus, r r r Γ P , (^m)/I ≤ Γ P , (m^0)/I0 + Γ P , (m^0)/I0 #M(m ,I ) #M(m ,I ) ≤ 0 0 + 1 1 (by the induction hypothesis) d d #M(m, I) = . d Corollary 4.9. Let Y,→ Pr be a projective scheme deﬁned by monomials of degree at most d. Then r dimk Γ(Y, OY ) ≤ d . Proof. Let I be the ideal generated by the deﬁning monomials of Y . Then

r dimk Γ(Y, OY ) = dimk Γ P , (1)]/I #M(1,I) ≤ (by Lemma 4.8 with m = 1) d r = d . Now, we are ready to prove the main theorem of this section. Theorem 4.10. Let Y,→ Pr be a projective scheme deﬁned by homogeneous polynomials of degree at most d. Then r−1 d2 r2 dim Γ(Y, O ) ≤ 2r + d k Y 2 Proof. This follows from Lemma 4.1, Theorem 4.2 and Corollary 4.9.

The artinian scheme Spec Γ(Y, OY ) satisfies the universal property: every morphism f : Y → G to an artinian scheme G uniquely factors through the natural morphism p: Y → Spec Γ(Y, OY ) [13, Exercise II.2.4]. Thus, Y and Spec Γ(Y, OY ) has the same number of connected compo- r nents. Andreotti–Bézout inequality [4, Lemma 1.28] then implies that dimk Γ(Y, OY )red ≤ d . r Thus, we may expect that dimk Γ(Y, OY ) ≤ d , and this is true if Y is defined by monomials by Corollary 4.9. 9 Question 4.11. Let Y,→ Pr be a projective scheme defined by homogeneous polynomials of degree at most d. Is r dimk Γ(Y, OY ) ≤ d ? One the other hand, Mayr and Meyer [25] constructed a family of ideals whose Castelnuovo– Mumford regularities are doubly exponential in r. Thus, the doubly exponential upper bound on dimk Γ(Y, OY ) might be unavoidable.

5. Hilbert Schemes Let X,→ Pr be a smooth connected projective variety deﬁned by polynomials of degree at most d. The aim of this section is to give an explicit upper bound on #(NS X)tor. P Theorem 3.6 implies that it suﬃces to give an upper bound on dimk Γ(Hilb X) for a P polynomial P . Gotzmann explicitly described Hilb Pr as a closed subscheme of a Grass- mannian [9][17, Proposition C.29]. Let Rn := k[x0, . . . , xr]n for n ∈ N. Theorem 5.1 (Gotzmann). Let t ≥ ϕ(P ) be an integer. Then there exists a closed immer- sion given by P r Hilb P (A) → Gr(P (t),Rt)(A) r [Z] 7→ Γ(PA, IZ (t)) for every k-algebra A. Therefore, we have closed immersions P P r VP (t) Hilb X,→ Hilb P ,→ Gr(P (t),Rt) ,→ P Rt

P P r VP (t) where Hilb X,→ Hilb P is the natural embedding and Gr(P (t),Rt) ,→ P( Rt) is the Plücker embedding. We will bound the degree of deﬁning equations of HilbP X VP (t) P in P( Rt). Then Theorem 4.10 will gives an upper bound on dimk Γ(Hilb X). For simplicity, let n + r Q(n) := dim R − P (n) = − P (n). n r The theorem below is a refomulation of the work in [17, Appendix C].

Theorem 5.2. Let t ≥ max{ϕ(P ), d} be an integer. Let A be an k-algebra, and let Sn = P A[x0, . . . , xr]n. Then M ∈ Gr(P (t),Rt)(A) is in (Hilb X)(A) if and only if

(a) FittQ(t+1)−1(St/(S1 · M)) = 0 and r (b) Γ(PA, IXA (t)) ⊂ M.

Proof. Note that M ⊂ St is a projective A-module of rank P (t). Nakayama’s lemma and [17, Proposition C.4] imply that St/(S1 · M) is locally generated by Q(t + 1) elements. Thus, [7, Proposition 20.6] implies that FittQ(t+1)(St/(S1 · M)) = A. Hence, by [7, Proposition 20.8], M satisfies (a) if and only if S1 ·M ∈ Gr(P (t+1),Rt+1)(A), which by [17, Proposition C.29] P is equivalent to M ∈ (Hilb Pr)(A). Let Z be the A-scheme defined by the polynomials in M. Then M satisfies (b) if and only if Z ⊂ XA. Therefore, M ∈ Gr(P (t),Rt)(A) satisfies both (a) and (b) if and only if P M ∈ (Hilb X)(A). 10 For simplicity, we replace Rt by a k-vector space V , and P (t) by a nonnegative integer n ≤ dim V . Let e0, e1, . . . , edim V −1 be an ordered basis of V . Given a: N

Deﬁnition 5.3. Given a: N

Deﬁnition 5.4. Given a: N

Let b0, b1,..., b(dim V )n−1 be all the functions N

Proof. Let a: N

zb det Jb = = det Jb ∈ Γ(Ua). za det Ja −1 −1 In particular, za[j7→i]/za = xi,j, meaning that za Ka = J. Thus, im J ⊂ im za L, so it suﬃces −1 to prove that im za Kb ⊂ im J for every b: N

 zb[m7→0]   det Jb[m7→0]  n−1 1 X w = . = . = C v ∈ im J m z  .   .  m,j j a j=0 zb[m7→dimk V −1] det Jb[m7→dim V −1] Lemma 5.6. Let m and q be nonnegative intergers. Let W be a k-vector space and let u: V q → W be a linear map. For every k-algebra A, let q H(A) = {M ∈ Gr(n, V )(A) | Fittdim W −m(WA/uA(M )) = 0}, so H is a subfunctor of Gr(n, V ). Then there is a closed scheme F ⊂ Gr(n, V ) deﬁned by homogeneous polynomials of degree m such that H is represented by G ∩ Gr(n, V ).

Proof. For i < q, let ui be the ith component of u, so u = u0 ⊕ · · · ⊕ uq−1. Let ⊕n(dim V )n ⊕n(dim V )n ⊕n(dim V )n Λ := u0 (L) u1 (L) ··· uq−1 (L) . Vn Let V,→ P ( V ) be the closed subscheme deﬁned by the m × m minors of Λ. We will show that H is represented by F ∩ Gr(n, V ). Take Ua as in the proof of Lemma 5.5. For a k-algebra A, take an A-module M ∈ Ua(A). Then M is represented by a k-algebra morphism f : Γ(Ua) → A. Then Lemma 5.5 implies −1 that M = im f(za L). Thus, q −1 −1 uA(M ) = u0,A im f(za L) + ··· + uq−1,A im f(za L) ⊕n(dim V )n −1 ⊕n(dim V )n −1 = im f u0 (za L) + ··· + im f uq−1 (za L) −1 = im f(za Λ). Hence, we have a free resolution

−1 n f(z Λ) W ... Aqn(dim V ) a W A 0. A u(M q)

s −1 Consequently, Fittdim W −m(WA/u(M )) is generated by m × m minors of f(za Λ). As a result, M ∈ H(A) if and only if M ∈ F (A). Since M is arbitrary, this imples that H is represented by F ∩ Gr(n, V ). Lemma 5.7. Let U be a linear subspace of V . For every k-algebra A, let

H(A) = {M ∈ Gr(n, V )(A) | UA ⊂ M}, so H is a subfunctor of Gr(n, V ). Then H is represented by the intersection of a linear space Vn and Gr(n, V ) in P ( V ). Proof. See [12, p. 66]. P VP (t) Hence, we can bound the degree of the deﬁning equation of Hilb X,→ P( Rt). P VP (t) Theorem 5.8. The closed embedding Hilb X,→ P( Rt) is deﬁned by homogeneous polynomials of degree at most P (t + 1) + 1. Proof. The Plücker relations are quadratic [12, p. 65]. Thus, it follows from Theorem 5.2, Lemma 5.6 with u(f0, f1, . . . , fr) = x0f0 + x1f1 + ··· + xrfr and Lemma 5.7. 12 Therefore, an upper bound on Gotzmann numbers will give an explicit construction of the Hilbert scheme. Such a bound is given by Hoa [16, Theorem 6.4(i)].

Theorem 5.9 (Hoa). Let I ⊂ k[x0, . . . , xr] be a nonzero ideal generated by homogeneous polynomials of degree at most d ≥ 2. Let b be the Krull dimension and c = r + 1 − b be the codimension of k[x0, . . . , xr]/I. Then b−1 3 b2 ϕ(HP ) ≤ dc + d . I 2 Now, we are ready to give an upper bound:

Theorem 5.10. Let X,→ Pr be a smooth connected projective variety deﬁned by homogeneous polynomials of degree ≤ d. Then

#(NS X)tor ≤ exp2 exp2 exp2 expd exp2(2r + 6 log2 r).

Proof. If X is a projective space or dim X ≤ 1, then (NS X)tor is trivial. Therefore, we may assume that d ≥ 2, 2 ≤ dim X ≤ r − 1. Let n = dr and m = (dr)r22r−1 . Then n ≥ (d − 1) codim X, and Theorem 5.9 with 2 ≤ b ≤ r implies that r2r−1 3 2 r−1 max{ϕ(nH), ϕ(X)} ≤ (dr)r−1 + dr ≤ (dr)r 2 = m. 2 Therefore, Theorem 3.6 implies that

#(NS X)tor ≤ dimk Γ(HilbHPmH X) . Let t = (dr)r422r−2 . Theorem 5.9 implies that r2r−1 3 2 r−1 max{ϕ(mH), d} ≤ mr−1 + m ≤ mr 2 = t. 2

Therefore, Theorem 5.8 implies that HilbHPmH X is deﬁned by polynomials of degree at VP (t) VP (t) most D = P (t + 1) + 1 in P( Rt). Let N = dimk Rt, and let P be the Hilbert 1295 polynomial of ImH . Because r ≥ 3 and t ≥ 6 dr, t + r + 1 D = P (t + 1) + 1 ≤ + 1 ≤ tr. r and dim Rt dim R t+r tr/4 N = ≤ 2 t = 2( r ) ≤ 2 . P (t) Therefore, Theorem 3.6 and Theorem 4.10 implies that N 2 N2N−1 N2N #(NS X)tor ≤ Γ(HilbHPmH X) ≤ 2 (D /2 + D) ≤ (2D) . Furthermore, r N2N N N N N tr/2 2t log2(2D) ≤ N2 (1 + log2 D) ≤ N · 2 · (1 + r log2 t) ≤ 2 · 2 · 2 ≤ 2 r 2t 5 2r−2 6 2r−2 logd log2 log2 2 = r logd t = r 2 (1 + logd r) ≤ r 2 . As a resulut,

(3) #(NS X)tor ≤ exp2 exp2 exp2 expd exp2(2r + 6 log2 r − 2). 13 In the rest of this section, we drop the condition that X is connected. Let Y0,Y1,...,Yn−1 be the connected components of X.

Theorem 5.11. Let X,→ Pr be a smooth projective variety deﬁned by homogeneous polynomials of degree ≤ d. Then

#(NS X)tor ≤ exp2 exp2 exp2 expd exp2(2r + 7 log2 r). Proof. Since Pic X = Pic Y0 × Pic Y1 · · · × Pic Yn−1, we have (NS X)tor = (NS Y0)tor × (NS Y1)tor · · · × (NS Yn−1)tor. r r The Andreotti–Bézout inequality [4, Lemma 1.28] implies that n ≤ d and deg Yi ≤ d for every i. Moreover, every Yi is deﬁned by homogeneous polynomials of degree at most deg Yi by [14, Proposition 3]. Without loss of generality, we may assume that

#(NS Y0)tor = max{#(NS Yi)tor | 0 ≤ i < n}. Then

dr #(NS X)tor ≤ (#(NS Y0)tor) dr ≤ (exp2 exp2 exp2 expdr exp2(2r + 6 log2 r − 2)) (by (3)) r r622r−2 r ≤ exp2 exp2 exp2 (d ) d 7 2r ≤ exp2 exp2 exp2 expd r 2

= exp2 exp2 exp2 expd exp2(2r + 7 log2 r). 6. Application to Fundamental groups

Let X be a smooth connected projective variety with base point x0 ∈ X(k). If k = C, then ét ab the torsion abelian group (Pic X)tor is the dual of the profinite abelian group π1 (X, x0) . More explicitly, πét(X, x )ab = Hom((Picτ X) , / ) = lim(Picτ X)[n]∨. 1 0 tor Q Z ←− n>0 This can be generalized to algebraically closed fields k of arbitrary characteristic by us- τ N ing Pic X and Nori’s fundamental group scheme π1 (X, x0). Before proceeding, note that ét ab N ab π1 (X, x0) = π1 (X, x0) (k) due to [8, Lemma 3.1]. τ Theorem 6.1 (Antei [1, Proposition 3.4]). The commutative torsion group scheme (Pic X)tor N ab is the Cartier dual of the commutative profinite group scheme π1 (X, x0) . More precisely, πN (X, x )ab = lim(Picτ X)[n]∨. 1 0 ←− n>0 ét ab ∨ N ab ∨ If k = C, then π1 (X, x0)tor ' (NS X)tor. We will show that π1 (X, x0)tor ' (NS X)tor for N ab general base fields. Then Theorem 5.10 gives an upper bound on #π1 (X, x0)tor. Lemma 6.2. Let A be a proper commutative group scheme. Let A0 be the identity component 0 of A. Then for every large and divisible m, we have mAtor = Ared,tor. 14 0 0 Proof. Notice that Ared is an abelian variety and A/Ared is a finite group scheme. Suppose 0 that m divides #(A/Ared). Since multiplication by m is surjective on the abelian variety 0 0 0 0 Ared, we get mA = Ared. As a result, mA = Ared, meaning that mAtor = Ared,tor. Lemma 6.3. Let A be a commutative group scheme of finite type. Let B be a subgroup scheme of A. Suppose that B is an abelian variety. Then for every positive integer n, the natural morphism A[n] A ϕ: → [n] B[n] B is an isomorphism. Proof. By the snake lemma,

0 B A A/B 0

×n ×n ×n 0 B A A/B 0

gives the exact sequence A B 0 −→ B[n] −→ A[n] −→ [n] −→ . B nB Since B is an abelian variety, we have B/nB = 0.

Theorem 6.4. Let X be a smooth connected projective variety with base point x0 ∈ X(k). Then N ab ∨ π1 (X, x0)tor ' (NS X)tor. Proof. Keep in mind that limits commute with kernels and colimits commute with cokernels. Keep in mind that the Cartier duality is a contravariant equivalence. Theorem 6.1 implies that ! πN (X, x )ab = lim lim(Picτ X)[n]∨ [m] 1 0 tor −→ ←− m>0 n>0 = lim lim ((Picτ X)[n]∨[m]) −→ ←− m>0 n>0 (Picτ X)[n] ∨ = lim lim −→ ←− τ m>0 n>0 m(Pic X)[n] !∨ (Picτ X)[n] = lim lim −→ −→ τ m>0 n>0 m(Pic X)[n] lim (Picτ X)[n] !∨ = lim −→n>0 −→ lim m(Picτ X)[n] m>0 −→n>0 (Picτ X) ∨ = lim tor −→ τ m>0 m(Pic X)tor 15 !∨ (Picτ X) = lim tor ←− τ m>0 m(Pic X)tor τ ∨ (Pic X)tor = 0 (by Lemma 6.2) (Pic X)red,tor lim (Picτ X)[n] !∨ = −→n>0 lim (Pic0 X) [n] −→n>0 red !∨ (Picτ X)[n] = lim −→ 0 n>0 (Pic X)red[n] !∨ = lim(NS X) [n] (by Lemma 6.3) −→ tor n>0 ∨ = (NS X)tor. N ab Therefore, we obtain an upper bound on #π1 (X, x0)tor. Theorem 6.5. Let X,→ Pr be a smooth connected projective variety defined by homogeneous polynomials of degree ≤ d with base point x0 ∈ X(k). Then N ab #π1 (X, x0)tor ≤ exp2 exp2 exp2 expd exp2(2r + 6 log2 r). N ab Proof. Theorem 6.4 implies that #π1 (X, x0)tor = #(NS X)tor. Therefore, the bound follows from Theorem 5.10. ét ab Suppose that char k = p > 0 and ` 6= p is a prime number. Since π1 (X, x0) = N ab ét ab ∞ ∞ ∨ π1 (X, x0) (k), Theorem 6.4 implies that π1 (X, x0) [` ] = (NS X)[` ] . However, this is not true for p-power torsions. Hence, a bound on #(NS X)tor such as [21, Theorem 4.12] ét ab does not give a bound on #π1 (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 ∈ X(k). Then ét ab #π1 (X, x0)tor ≤ exp2 exp2 exp2 expd exp2(2r + 6 log2 r). ét ab ∨ Proof. By [32, Proposition 69], we have π1 (X, x0)tor = (NS X)tor(k). Therefore, ét ab ∨ #π1 (X, x0)tor = #(NS X)tor(k) ≤ #(NS X)tor, and the bound follows from Theorem 5.10.

7. Lefschetz Hyperplane Theorem Throughout the section, X,→ Pr is a smooth connected projective variety satisfying dim X ≥ 2, and H is a smooth hyperplane section of X. In this section, we discuss the Lefschetz hyperplane theorem for Picτ X. If k = C, the exponential sequence gives a short exact sequence 1 H (X, Z) ⊗Z R τ 2 0 −→ −→ Pic X −→ H (X, Z)tor −→ 0. H1(X, Z) 16 With the natural analytic topology, the Pontryagin duality gives an exact sequence

τ H1(X, Z) 0 −→ H1(X, Z)tor −→ Homcont(Pic X, R/Z) −→ −→ 0. H1(X, Z)tor The Lefschetz hyperplane theorem implies that the map H1(H, Z) → H1(X, Z) is surjective. Therefore, the lemma below implies that Picτ X → Picτ H is injective. Lemma 7.1. Let f : A → B be a surjective morphism between ﬁnitely generated abelian groups. Consider the morphism between exact sequences

0 0 Ator A A/Ator 0 g f∗ f∗

0 0 Btor B B/Btor 0 where the outer vertical morphisms are induced by f. Then g : A0 → B0 is surjective.

Proof. This follows from the snake lemma. We expect a similar Lefschetz-type theorem for any algebraically closed field k. Over a general base field, we have the Lefschetz hyperplane theorem for étale fundamental groups [11, XII. Corollaire 3.5]. Hence, for a prime number ` 6= char k, (Pic X)[`∞] → (Pic H)[`∞] is injective by Theorem 6.1. However, étale fundamental groups lack the information of (Pic X)[p∞]. Theorem 7.2. Let X be a smooth connected projective variety, and let H be a smooth hyperplane section of X. If dim X ≥ 2, then the kernel of r : Picτ X → Picτ H is a finite commutative group scheme with a connected Cartier dual.

0 Proof. Let ` 6= char k be a prime number. The connected component (ker r)red of (ker r)red 0 is an abelian variety without `-torsion points. Therefore, (ker r)red = 0 and ker r is a finite commutative group scheme. Take a base point x ∈ H, and let M = #(ker r). Then (ker r)∨ = ker(Picτ X → Picτ H)∨ = ker (Pic X)[M] → (Pic H)[M]∨ = coker ((Pic H)[M]∨ → Pic X)[M]∨) πN (H, x)ab πN (X, x)ab = coker 1 → 1 (by Theorem 6.1). N ab N ab Mπ1 (H, x) Mπ1 (X, x) We have πN (X, x)ab πét(X, x)ab 1 (k) = 1 N ab ét ab Mπ1 (X, x) Mπ1 (X, x) by [8, Lemma 3.1]. Moreover, πét(H, x)ab πét(X, x)ab 1 → 1 ét ab ét ab Mπ1 (H, x) Mπ1 (X, x) 17 is surjective by the Lefschetz hyperplane theorem for étale fundamental groups [11, XII. ∨ ∨ Corollaire 3.5]. Thus, (ker r) (k) = 0, meaning that (ker r) is connected.

One may want to show that r : Picτ X → Picτ H is injective. Unfortunately, the Lefschetz hyperplane theorem for Nori’s fundamental group scheme is no longer true [2, Remark 2.4]. Nonetheless, we still have the Lefschetz hyperplane theorem in some cases.

Theorem 7.3. Let d be a suﬃciently large integer. Then for any smooth hyperplane section H0 of the d-uple embedding of X,→ Pr, the natural map

r : Picτ X → Picτ H0 is a closed embedding.

N 0 Proof. Let M = #(ker r). If d is suﬃciently large, then the natural map π1 (H , x) → N π1 (X, x) is faithfully ﬂat by [3, Theorem 1.1]. In this case,

πN (H, x)ab πN (X, x)ab (ker r)∨ = coker 1 → 1 = 0 N ab N ab Mπ1 (H, x) Mπ1 (X, x)

as in the proof of Theorem 7.2. Theorem 7.4 (Langer [22, Theorem 11.3]). Suppose that X has a lifting to a smooth projective scheme over W2(k). Then

r : Picτ X → Picτ H is a closed embedding.

The author does not know whether or not Picτ X → Picτ H is injective in general. How- ever, the author conjectures that Picτ X → Picτ H can fail to be a closed embedding if char k = p > 0, because of the failure of the Kodaira vanishing theorem. The argument below is a reformulation of the work in [3, Section 2] and [22, Example 10.1].

n Deﬁnition 7.5. Let αpn be the kernel of the p -power Frobenius endomorphism on Ga.

Theorem 7.6. Let D be a smooth ample eﬀective divisor on X. Then the natural map

τ τ r(αp):(Pic X)(αp) → (Pic D)(αp)

1 is injective if and only if H (X, OX (−D)) = 0.

1 Proof. Suppose that H (X, OX (−D)) 6= 0. Because dim X ≥ 2, there is a large integer n 1 n such that H (X, OX (−p D)) = 0. Let FX : X → X be the absolute Frobenius morphism. n ∗ n Then (FX ) OX (−D) = OX (−p D). Thus, we have the diagram with exact rows and columns 18 as below. 0 0

1 h 1 Hfppf (X, αpn ) Hfppf (D, αpn )

1 f 1 g 1 0 H (X, OX (−D)) H (X, OX ) H (D, OD)

n ∗ n ∗ (FX ) (FX )

1 n 1 0 H (X, OX (−p D)) H (X, OX )

1 n ∗ 1 n Take a nonzero s ∈ H (X, OX (−D)). Then (FX ) (f(s)) = 0, because H (X, OX (−p D)) = 1 0. Thus, there is a nonzero t ∈ Hfppf (X, αpn ) such that f(s) = i(t). On the other hand, h(t) = 0, since g(f(s)) = 0. Therefore, h is not injective. By [1, Proposition 3.2], the natural morphism τ τ r(αpn ):(Pic X)(αpn ) → (Pic D)(αpn ) τ is isomorphic to h, so is also not injective. Let ϕ: αpn → Pic X be a nonzero element of ker r(αpn ). Then the image of ϕ is αpm for some m > 0. Furthermore, αpm has a subgroup isomorphic to αp. Thus, we have the commutative diagram below.

αpn ϕ

τ τ αp αpm Pic X Pic D

The second row gives a nonzero element in the kernel of

τ τ r(αp):(Pic X)(αp) → (Pic D)(αp).

1 1 1 Now, suppose that H (X, OX (−D)) = 0. Then H (X, OX ) → H (D, OD) is injective, 1 1 and so is Hfppf (X, αp) → Hfppf (D, αp). By [1, Proposition 3.2],

τ τ r(αp):(Pic X)(αp) → (Pic D)(αp)

is also injective.

1 Raynaud [30] gave an ample eﬀective divisor D such that H (X, OX (−D)) 6= 0. Lau- ritzen [23] [24] showed that the Kodaira vanishing theorem can fail even if D is very ample. However, the author does not know any example of a very ample divisor D such that 1 H (X, OX (−D)) 6= 0. Thus we still do not know the answers to the following.

Question 7.7. Let X be a smooth connected projective variety of dim X ≥ 2, and let H be a smooth hyperplane section. Is the map Picτ X → Picτ H always injective? Is the map Picτ X → Picτ H always a closed embedding? 19 8. The Number of Generators of (NS X)tor Let dim X,→ Pr be a smooth connected projective variety such that dim X ≥ 1. The aim of this section is to prove that (NS X)tor is generated by at most (deg X − 1)(deg X − 2) elements, if X has a lifting over W2(k). In the rest of the section, let C be the intersection of X with dim X −1 general hyperplanes in Pr. Then C is a smooth connected curve, and the genus g(C) of C is at most (deg X − 1)(deg X − 2)/2. Since Picτ C = Jac C, we have a natural map r : Picτ X → Jac C. Theorem 8.1. Let C be a general curve section. Then ker r is a finite commutative group scheme with a connected dual. Proof. By Theorem 7.2 applied repeatedly, r is a composition of homomorphisms with finite connected kernel, so ker r also is finite and connected. Lemma 8.2. If X,→ Pr is the reduction of a smooth connected projective scheme X ,→ r over W (k), then ker r = 0. PW2(k) 2 Proof. Let V,→ r be a general hyperplane, and let H = X ∩ V . Let V ,→ r be a P PW2(k) lifting of V . Then V ∩ X is a smooth projective scheme, and H is its reduction. Therefore, if dim X ≥ 1, then the hypothesis on X is inherited by a general hyperplane section H. We may now apply Theorem 7.4 iteratively to obtain the result.

Theorem 8.3. If ker r is connected, then (NS X)tor is generated by (deg X − 1)(deg X − 2) elements.

Proof. Let N = #(NS X)tor. Then (NS X)tor is a quotient group of (Pic X)[N]. Since ker r is connected, Picτ X → (Jac C)(k) is injective. Thus, (Picτ X)[N] → (Jac C)(k)[N] is also injective. The group (Jac C)(k)[N] is generated by 2g(C) elements, and 2g(C) ≤ (deg X−1)(deg X− 2). Thus, its subquotient (NS X)tor is also generated by ≤ (deg X − 1)(deg X − 2) elements.

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 Proof. This follows from Lemma 8.2 and Theorem 8.3. Furthermore, the p-power torsion subgroups have smaller upper bounds. Theorem 8.5. Suppose that char k = p > 0. Then (NS X)[p∞]∨(k) is generated by at most (deg X − 1)(deg X − 2)/2 elements. If ker r is connected, then (NS X)[p∞] is also generated by at most (deg X − 1)(deg X − 2)/2 elements. 20 Proof. Take a suﬃciently large integer N. Let (Jac C)∨ be the dual abelian variety of (Jac C). Since (Jac C)∨[pN ] = (Jac C)[pN ]∨, (Jac C)∨[pN ](k) → (Pic X)[pN ]∨(k) is surjective by Theorem 8.1. Because N is large, (NS X)[p∞]∨ is a subgroup scheme of (Pic X)[pN ]∨. Since (Jac C)∨(k)[pN ] is generated by ≤ (deg X − 1)(deg X − 2)/2 elements, so is (NS X)[p∞]∨(k). Now, suppose that ker r is connected. Then we have an injection (Picτ X)[pN ] → (Jac C)(k)[pN ]. Because N is large, (NS X)[p∞] is a quotient of (Picτ X)[pN ]. Since (Jac C)(k)[pN ] is gener- ∞ ated by ≤ (deg X − 1)(deg X − 2)/2 elements, so is (NS X)[p ]. Acknowledgement The author thanks his advisor Bjorn Poonen for his careful advice. The author thanks János Kollár for answering questions regarding Lefschetz-type theorems. The author thanks Barry Mazur for suggesting Nori’s fundamental group schemes. The author also thanks Chenyang Xu, Steven Kleiman and Davesh Maulik for many helpful conversations.

References [1] Marco Antei. On the abelian fundamental group scheme of a family of varieties. Israel J. Math., 186:427– 446, 2011. [2] Indranil Biswas and Yogish I. Holla. Comparison of fundamental group schemes of a projective variety and an ample hypersurface. J. Algebraic Geom., 16(3):547–597, 2007. [3] Indranil Biswas and Yogish I. Holla. Comparison of fundamental group schemes of a projective variety and an ample hypersurface. J. Algebraic Geom., 16(3):547–597, 2007. [4] Fabrizio Catanese. Chow varieties, Hilbert schemes and moduli spaces of surfaces of general type. J. Algebraic Geom., 1(4):561–595, 1992. [5] M. Demazure, A. Grothendieck, and M. Artin. Schémas en groupes (SGA 3): Propriétés générales des schémas en groupes. Documents mathématiques. Société mathematique de France, 2011. [6] Thomas W. Dubé. The structure of polynomial ideals and Gröbner bases. SIAM J. Comput., 19(4):750– 775, 1990. [7] David Eisenbud. Commutative Algebra: with a view toward algebraic geometry, volume 150. Springer Science & Business Media, 2013. [8] Hélène Esnault, Phùng Hô Hai, and Xiaotao Sun. On Nori’s fundamental group scheme. In Geometry and dynamics of groups and spaces, volume 265 of Progr. Math., pages 377–398. Birkhäuser, Basel, 2008. [9] Gerd Gotzmann. Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Ringes. Math. Z., 158(1):61–70, 1978. [10] Alexander Grothendieck. Éléments de géométrie algébrique: IV. étude locale des schémas et des mor- phismes de schémas, seconde partie. Publications Mathématiques de l’IHÉS, 24:5–231, 1965. [11] Alexander Grothendieck. Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2). North-Holland Publishing Co., Amsterdam; Masson & Cie, Éditeur, Paris, 1968. Augmenté d’un exposé par Michèle Raynaud, Séminaire de Géométrie Algébrique du Bois-Marie, 1962, Advanced Studies in Pure Mathematics, Vol. 2. [12] Joe Harris. Algebraic geometry, volume 133 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1992. A first course. [13] Robin Hartshorne. Algebraic geometry. Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. [14] Joos Heintz. Definability and fast quantifier elimination in algebraically closed fields. Theoret. Comput. Sci., 24(3):239–277, 1983. 21 [15] Jürgen Herzog and Takayuki Hibi. Monomial ideals, volume 260 of Graduate Texts in Mathematics. Springer-Verlag London, Ltd., London, 2011. [16] Lê Tuân Hoa. Finiteness of Hilbert functions and bounds for Castelnuovo-Mumford regularity of initial ideals. Trans. Amer. Math. Soc., 360(9):4519–4540, 2008. [17] Anthony Iarrobino and Vassil Kanev. Power sums, Gorenstein algebras, and determinantal loci, volume 1721 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1999. Appendix C by Iarrobino and Steven L. Kleiman. [18] Jun-ichi Igusa. On some problems in abstract algebraic geometry. Proc. Nat. Acad. Sci. U.S.A., 41:964– 967, 1955. [19] Steven L. Kleiman. The Picard scheme. In Fundamental algebraic geometry, volume 123 of Math. Surveys Monogr., pages 235–321. Amer. Math. Soc., Providence, RI, 2005. [20] János Kollár. Rational curves on algebraic varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 1996. [21] Hyuk Jun Kweon. Bounds on the torsion subgroups of néron-severi groups, 2019. https://arxiv.org/ pdf/1902.02753.pdf. [22] Adrian Langer. On the S-fundamental group scheme. Ann. Inst. Fourier (Grenoble), 61(5):2077–2119 (2012), 2011. [23] N. Lauritzen and A. P. Rao. Elementary counterexamples to Kodaira vanishing in prime characteristic. Proc. Indian Acad. Sci. Math. Sci., 107(1):21–25, 1997. [24] Niels Lauritzen. Embeddings of homogeneous spaces in prime characteristics. Amer. J. Math., 118(2):377–387, 1996. [25] Ernst W. Mayr and Albert R. Meyer. The complexity of the word problems for commutative semigroups and polynomial ideals. Adv. in Math., 46(3):305–329, 1982. [26] David Mumford. Lectures on curves on an algebraic surface. With a section by G. M. Bergman. Annals of Mathematics Studies, No. 59. Princeton University Press, Princeton, N.J., 1966. [27] André Néron. Problèmes arithmétiques et géométriques rattachés à la notion de rang d’une courbe algébrique dans un corps. Bull. Soc. Math. France, 80:101–166, 1952. [28] Madhav V. Nori. The fundamental group-scheme. Proc. Indian Acad. Sci. Math. Sci., 91(2):73–122, 1982. [29] Bjorn Poonen, Damiano Testa, and Ronald van Luijk. Computing Néron-Severi groups and cycle class groups. Compos. Math., 151(4):713–734, 2015. [30] M. Raynaud. Contre-exemple au “vanishing theorem” en caractéristique p > 0. In C. P. Ramanujam—a tribute, volume 8 of Tata Inst. Fund. Res. Studies in Math., pages 273–278. Springer, Berlin-New York, 1978. [31] Francesco Severi. La base per le varietà algebriche di dimensione qualunque contenuta in una data, e la teoria generate delle corrispondénze fra i punti di due superficie algebriche. Mem. R. Accad. Italia, 5:239–83, 1933. [32] Jakob Stix. Rational points and arithmetic of fundamental groups, volume 2054 of Lecture Notes in Mathematics. Springer, Heidelberg, 2013. Evidence for the section conjecture.

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA Email address: [email protected] URL: https://kweon7182.github.io/