THE F-SIGNATURE FUNCTION ON THE AMPLE CONE SEUNGSU LEE AND SUCHITRA PANDE Abstract. For any fixed globally F-regular projective variety Xover an algebraically

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THE F-SIGNATURE FUNCTION ON THE AMPLE CONE

SEUNGSU LEE AND SUCHITRA PANDE

Abstract. For any ﬁxed globally F-regular projective variety Xover an algebraically

closed ﬁeld of positive characteristic, we study the F-signature of section rings of Xwith

respect to the ample Cartier divisors on X. In particular, we deﬁne an F-signature function

on the ample cone of Xand show that it is locally Lipschitz continuous. We further prove

that the F-signature function extends to the boundary of the ample cone. We also establish

an eﬀective comparison between the F-signature function and the volume function on the

ample cone. As a consequence, we show that for divisors that are nef but not big, the

extension of the F-signature is zero.

1. Introduction

The F-signature is an invariant of the singularities of a Noetherian, F-ﬁnite local ring R

of prime characteristic. First arising implicitly in [SVdB97] and formally deﬁned in [HL02],

this invariant measures the asymptotic growth of the number of Frobenius splittings of R

(See Deﬁnition 2.2). The positivity of the F-signature corresponds exactly to Rbeing a

strongly F-regular singularity [AL03]; only when Ris regular, does the F-signature achieve

its maximum value of 1. This invariant has also attracted attention as a candidate for the

positive characteristic analog of the normalized volume of a Kawamata log-terminal (klt)

singularity, extending the established analogy between strongly F-regular and klt singular-

ities; see [LLX20], [Tay19], [MPST19]. There have been applications of the F-signature to

bounding the sizes of the ´etale fundamental group and the torsion subgroup of the divisor

class group; see [CRST18], [Mar22], [CR22] and [Pol22].

In the global setting, globally F-regular varieties (Deﬁnition 2.9), introduced in [Smi00] are

the positive characteristic analogs of log-Fano type varieties enjoying additional properties

such as satisfying a Kawamata-Viehweg vanishing theorem [SS10]. It was shown in [Smi00]

that a projective variety Xis globally F-regular if and only if the section ring S(X, L)

(Deﬁnition 2.6) is strongly F-regular for some (equivalently, every) ample invertible sheaf L.

Since the F-signature is positive for all strongly F-regular rings, it is natural to ask: How

does the F-signature of the section ring S(X, L)vary with L?The purpose of this paper is

to answer this question. We prove:

Theorem 1.1. Fix any globally F-regular projective variety Xover an algebraically closed

ﬁeld kof positive characteristic p. Assume that the dimension of Xis positive. Then, the

F-signature function L7→ sX(L), assigning to any ample Cartier divisor L, the F-signature

of the section ring of Xwith respect to L, satisﬁes the following properties:

(a) ([VK12],[CR22],Theorem 3.3) The F-signature function sXnaturally extends to a

unique, well-deﬁned, real-valued function

sX: AmpQ(X)−→ R

Lee was supported in part by NSF Grant #2101800 and #1952522.

Pande was partially supported by the NSF grants #1952399, #1801697 and #2101075.

arXiv:2210.00566v2 [math.AG] 28 Feb 2025

2 SEUNGSU LEE AND SUCHITRA PANDE

on the set of rational classes in the ample cone of Xsatisfying:

sX(λL) = 1

λsX(L)for all ample Q-divisors Land all λ∈Q>0.

(b) (Theorem 4.1) The function sXis continuous on the rational ample cone of X, with

respect to the usual topology on the N´eron-Severi space.

cone of X.

Theorem 1.1 provides us with a new tool for the study of globally F-regular varieties.

Such varieties have found various applications, for instance, to the three dimensional min-

imal model program in positive characteristic [HX15] and in the study of Fano type com-

plex varieties [GOST15]. For other investigations regarding globally F-regular varieties, see

[GLP+15], [GT19] and [Kaw21].

Another motivation for considering the F-signature function comes from the volume func-

tion on the big cone of a projective variety. On a projective variety Xover an algebraically

closed ﬁeld, to any Cartier divisor Don X, we can associate a non-negative real number

called the volume of D, measuring the growth of the global sections of multiples of D. A

foundational result in the theory of volumes is that the volume of a big divisor Ddepends

only on its numerical equivalence class. Moreover, it extends suitably to all R-divisors and

varies continuously as Dvaries on the N´eron-Severi space of X. See [Laz04, Section 2.2] and

[LM09] for the details. The study of volumes of divisors has been important in birational ge-

ometry; for example, see [Laz04, Section 2.2], [LM09], [Bou02], [ELM+05], [HM06], [Tak06],

and [K¨

06].

Theorem 1.1 parallels the theory of the F-signature function (Deﬁnition 3.1) on the ample

cone of a globally F-regular variety with the volume function on the big cone. This perspec-

tive was ﬁrst considered in [VK12], where Theorem 1.1 was proved in the special case when

Xis a toric variety.

The ample cone is an open cone in the N´eron-Severi space of a projective variety X, and

its closure is represented by the set of nef divisors on X. Hence, it is natural to ask if the

F-signature function sXfrom Theorem 1.1 has a natural extension to the nef cone. We show

that this is indeed true:

Theorem 1.2 (Theorem 5.1).Suppose Xis a globally F-regular projective variety. Then

the F-signature function sXextends continuously to all non-zero classes of the Nef cone of

X. Moreover, if Lis a nef Cartier divisor which is not big, then sX(L)=0.

The proof of Theorem 1.1 and Theorem 1.2 consists of several steps. First, we need

to verify that the F-signature function is well-deﬁned on the rational ample cone of X.

We do this in Section 3. The main result here is that on globally F-regular projective

varieties, numerical equivalence, and Q-linear equivalence coincide. This is reminiscent of

the same fact for log-Fano type varieties over the complex numbers. Once we have this,

we prove the continuity of the F-signature function in Section 4. This needs several ideas

towards analyzing Frobenius splittings of linear systems; a sketch of the proof is presented in

Section 4.2. We further utilize these ideas to extend the F-signature function to all non-zero

nef divisors in Section 5. Lastly, in Theorem 6.1 we prove a local eﬀective upper bound for

the F-signature function.

THE F-SIGNATURE FUNCTION ON THE AMPLE CONE 3

Throughout this paper, unless speciﬁed otherwise, all rings are assumed commutative

with a unit and are of positive characteristic p.kwill denote an algebraically closed ﬁeld

of characteristic p. All varieties are assumed to be integral, separated schemes of ﬁnite type

over k.

2. Preliminaries

2.1. F-signature. Let Rbe any ring of prime characteristic p. Then Ris naturally equipped

with the Frobenius morphism,F:R−→ Rsending r7→ rp. Since Rhas characteristic p,

Fdeﬁnes a ring homomorphism, allowing us to view Ras a new R-module obtained via

restriction of scalars along F. We denote this new R-module by F∗Rand its elements by

F∗r(where ris an element of R). Concretely, F∗Ris the same as Ras an abelian group,

but the R-module action is given by:

r·F∗s:= F∗rpsfor r∈Rand F∗s∈F∗R.

Throughout, we will assume that Ris essentially of ﬁnite type over k, which also makes it

F-ﬁnite, i.e., F∗Ris a ﬁnitely generated R-module. Similarly, for any natural number e≥1,

we have the iterate of the Frobenius, Fe:R−→ Rsending r7→ rpeand the R-module Fe

∗R

obtained by restricting scalars along Fe. We will be interested in invariants of Rdeﬁned by

analyzing the R-module structure of the modules Fe

∗R.

Deﬁnition 2.1 (Free rank).Let Mbe a ﬁnitely generated module over a local ring R.

Consider a decomposition:

M∼

=Ra(M)⊕N

where Nhas no R-summands. Then, since Ris local, the number a(M) is independent of

the decomposition chosen, and is called the free rank of M.

Deﬁnition 2.2 (F-signature).Let (R, m, k) be a local ring, and ae(R) denote the free rank

of Fe

∗R(Deﬁnition 2.1). Then the F-signature of Ris deﬁned to be the limit:

s(R) := lim

e−→∞

ae(R)

ped

where dis the Krull dimension of R. This limit exists by [Tuc12].

The F-signature of Radmits an alternate description as follows: We say a map of R-

modules ϕ:M−→ Nsplits if there exists an R-module map ψ:N−→ Msuch that

ψ◦ϕ= idM. Deﬁne the subset Ie⊆Ras

Ie={x∈R|the map R−→ Fe

∗Rsending 1 7→ Fe

∗xdoes not split}.

Then, we observe that Ieis an ideal of Rand by [Tuc12, Proposition 4.5], the free rank of

∗Requals lR(R/Ie), the length of the R-module R/Ie. Hence, the F-signature of Rcan be

deﬁned as the limit:

s(R) = lim

e−→∞

lR(R/Ie)

ped .

Though the deﬁnition of F-signature is given for a local ring (R, m, k), we may also work

with N-graded rings (S, m, k) i.e. Sis N-graded with S0=kand m=S>0. We next relate

the local and graded situations.

4 SEUNGSU LEE AND SUCHITRA PANDE

Deﬁnition 2.3 (Graded free rank).Let (S, m, k) be an N-graded ring, ﬁnitely generated

over k, with S0=kand Ma ﬁnitely generated Z-graded module over S. Then we can

decompose Mas a graded S-module as:

M∼

=P⊕N

where Pis a graded free S-module (i.e. a direct sum of S(j), the shifted rank 1 free modules,

for various j∈Z) and Nis a graded module with no graded free summands. Then the rank

of Pis independent of the chosen decomposition and we deﬁne it to be the graded free rank

of Mover S(denoted by agr(M)).

Lemma 2.4. Let (S, m, k)and Mbe as above. Then the free rank of Mmover the local ring

Smis the same as the graded free rank of M.

Proof. See [DSPY22, Proposition 5.7] □

Now, we describe the F-signature of N-graded rings, relating it to the (local) F-signature

at the vertex. For similar discussions relating the local and global situations, see [Smi00,

Section 3], [Smi97, Section 4], and [VK12, Section 2.2].

Let Sbe an N-graded ring. Then Fe

∗Sis also naturally an 1

peN-graded S-module by taking

(Fe

∗S)i

pe=Fe

∗Si.

This gives rise to the N-grading on Fe

∗Sgiven by

(Fe

∗S)n=M

0≤i≤pe−1

(Fe

∗S)i+npe

pe.

Thus, Fe

∗Sdecomposes as

∗S=M

0≤i≤pe−1M

j≥0

∗Si+jpe

as an N-graded S-module.

Deﬁnition 2.5 (F-signature of N-graded rings).Let (S, m, k) be an N-graded, ﬁnitely gen-

erated k-algebra, with S0=k. Then, we deﬁne the F-signature of Sto be the limit:

lim

e−→∞

ae,gr(S)

ped

where ae,gr(S) is the graded free rank of Fe

∗Sand ddenotes the Krull dimension of S. We

note that by Lemma 2.4, the F-signature of Scoincides with the F-signature of Sm, the

localization of Sat the maximal ideal m.

2.2. Section Rings: The N-graded rings we will be interested in arise as the section rings

of projective varieties over kwith respect to some ample divisor.

Deﬁnition 2.6 (Section Rings and Modules).Let Xbe a projective variety over k,Lan

ample invertible sheaf on Xand Fa coherent sheaf on X. Then the N-graded ring Sdeﬁned

S=S(X, L) := M

n≥0

H0(X, Ln)

THE F-SIGNATURE FUNCTION ON THE AMPLE CONE 5

is called the section ring of Xwith respect to L. The aﬃne scheme Spec(S) is called the

cone over Xwith respect to L. The section module of Fwith respect to Lis a Z-graded

S-module Mdeﬁned by

M=M(X, L) := M

n∈Z

H0(X, F⊗Ln).

Similarly, the sheaf corresponding to Mon Spec(S) is called the cone over Fwith respect

to L.

In the next lemma, we record some useful principles concerning direct summands of sheaves

on a proper variety.

Lemma 2.7. On a proper variety Xover k, let L,Mbe invertible sheaves, and F,Gbe

coherent sheaves. Then,

(a) If Lis not a direct summand of Fand G, then Lis also not a summand of F⊕G.

(b) If F∼

=L⊕n⊕Gand Lis not a summand of G, then, nis the maximum number of L

summands of F(in any decomposition).

=M, and both Land Mare summands of F, then L⊕Mis a summand

of F.

Proof. Note that an OX-summand (i.e., a summand isomorphic to OX) of a coherent sheaf

Fis equivalent to a non-zero global section s∈H0(X, F) and a map φ∈HomOX(F,OX)

such that φ(s)̸= 0.

(a) By twisting by L−1, we may assume that L=OX. An OX-summand of F⊕Gis given

by a global section

s= (s1, s2)∈H0(X, F⊕G) = H0(X, F)⊕H0(X, G)

and a map

φ= (φ1, φ2)∈HomOX(F⊕G,OX) = HomOX(F,OX)⊕HomOX(G,OX)

such that φ(s)̸= 0. However, φ(s) = φ1(s1) + φ2(s2). So, if φ(s)̸= 0, then φi(si)̸= 0

for some i= 1,2, giving an OX-summand of either For G, which is a contradiction.

(b) Again, twisting by L−1, we may reduce to the case when L=OX. Suppose that there

is another decomposition F∼

=O⊕(n+m)

X⊕G′for some m > 0. Let φ:O⊕(n+m)

XLG′−→

O⊕n

XLGbe an isomorphism. Now, consider the map ψ:H0(X, O⊕(n+m)

X)−→ H0(X, O⊕n

induced by the inclusion of O⊕(n+m)

Xinto F, the isomorphism φand the projection

onto O⊕n

X. Since mis positive, there exists a non-zero section s∈O⊕(n+m)

Xsuch that

ψ(s) = 0.

Now write φ(s, 0) = (0, g) for some g∈H0(X, G). Note that (s, 0) gives an OX-

summand of F. Hence, gmust be an OX-summand of G, which is a contradiction,

since Gwas assumed to have no OX-summands.

=L⊕G. Now, by

part (a), if Mis a direct summand of L⊕G, then Mis direct summand of either L

or G. However, since M̸∼

=L,Mmust be a direct summand of G. Hence, L⊕Mis a

direct summand of F.□

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THEF-SIGNATUREFUNCTIONONTHEAMPLECONESEUNGSULEEANDSUCHITRAPANDEAbstract.ForanyfixedgloballyF-regularprojectivevarietyXoveranalgebraicallyclosedfieldofpositivecharacteristic,westudytheF-signatureofsectionringsofXwithrespecttotheampleCartierdivisorsonX.Inparticular,wedefineanF-signaturefunctionontheamp...

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THE F-SIGNATURE FUNCTION ON THE AMPLE CONE SEUNGSU LEE AND SUCHITRA PANDE Abstract. For any fixed globally F-regular projective variety Xover an algebraically

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