A Frobenius version of Tian’s Alpha-Invariant

Swaraj Pande Department of Mathematics
University of Michigan
Ann Arbor, MI 48109-1043
USA [email protected]

Abstract.

For a pair $(X,L)$ consisting of a projective variety $X$ over a perfect field of characteristic $p>0$ and an ample line bundle $L$ on $X$ , we introduce and study a positive characteristic analog of Tian’s $\alpha$ -invariant, which we call the $\alpha_{F}$ -invariant. We utilize the theory of $F$ -singularities in positive characteristics, and our approach is based on replacing klt singularities with the closely related notion of global $F$ -regularity. We show that the $\alpha_{F}$ -invariant of a pair $(X,L)$ can be understood in terms of the global Frobenius splittings of the linear systems $|mL|_{m\geq 0}$ . We establish inequalities relating the $\alpha_{F}$ -invariant with the $F$ -signature, and use that to prove the positivity of the $\alpha_{F}$ -invariant for all globally $F$ -regular projective varieties (with respect to any ample $L$ on $X$ ). When $X$ is a Fano variety and $L=-K_{X}$ , we prove that the $\alpha_{F}$ -invariant of $X$ is always bounded above by $1/2$ and establish tighter comparisons with the $F$ -signature. We also show that for toric Fano varieties, the $\alpha_{F}$ -invariant matches with the usual (complex) $\alpha$ -invariant. Finally, we prove that the $\alpha_{F}$ -invariant is lower semicontinuous in a family of globally $F$ -regular Fano varieties.

This work was partially supported by the NSF grants #1952399, #1801697 and #2101075

1. Introduction

The $\alpha$ -invariant of a complex Fano variety $X$ was introduced by Tian in [Tia87] to provide a sufficient criterion for $K$ -stability of $X$ , a condition that guarantees the existence of a Kähler-Einstein metric on $X$ . Though initially defined analytically, Demailly later reinterpreted the $\alpha$ -invariant in terms of the log canonical threshold [CS08], an algebraic invariant of the singularities of divisors on $X$ . Since then, understanding the $\alpha$ -invariant and $K$ -stability more generally has led to many fundamental advances in our understanding of complex Fano varieties; see [OS12], [Bir21], [Xu21]. The minimal model program (MMP), and the singularities that arise therein have played a key role in these advances.

The purpose of this paper is to study a positive characteristic analog of the $\alpha$ -invariant. To do this, we replace the singularities of the MMP with singularities defined using the Frobenius map (“ $F$ -singularities"). Though $F$ -singularities have fundamentally different definitions than the singularities of the MMP, a dictionary involving many precise relationships between these classes has been established; see [Smi97], [Har98], [MS97], [HW02], [HY03], [Tak04] and [MS18]. Under this dictionary, log canonical (resp. Kawamata log-terminal (klt)) singularities correspond to $F$ -split (resp. strongly $F$ -regular) singularities (2.15). Since the $\alpha$ -invariant of a complex Fano variety $X$ involves the log canonicity of anti-canonical $\mathbb{Q}$ -divisors of $X$ , this inspires our definition of the Frobenius version:

Definition 1.1.

Let $X$ be a globally $F$ -regular Fano variety over a perfect field of positive characteristic. Then, we define the $\alpha_{F}$ -invariant of $X$ as

\alpha_{F}(X):=\sup\{t\geq 0\,|\,(X,t\Delta)\,\text{is globally $F$-split }\forall\,\text{effective $\mathbb{Q}$-divisor }\Delta\sim_{\mathbb{Q}}-K_{X}\}.

Since we intend for the $\alpha_{F}$ -invariant to capture global properties of anti-canonical $\mathbb{Q}$ -divisors on $X$ , we use the notion of global $F$ -splitting (2.15), or equivalently, $F$ -splitting of the cone over the corresponding divisors on $X$ (with respect to $-K_{X}$ ). To do so, we require $X$ to be globally $F$ -regular (2.16), which is equivalent to the cone over $X$ being strongly $F$ -regular. Thus, global $F$ -regularity can be thought of as a Frobenius analog of the klt-condition on complex Fano varieties. Furthermore, we note that while working over $\mathbb{C}$ , simply replacing globally $F$ -regular and globally $F$ -split in 1.1 by klt on the cone, and log canonical on the cone respectively, we obtain the minimum value between the usual $\alpha$ -invariant of $X$ and $1$ (see 4.11). Thus, at least for Fano varieties with $\alpha(X)\leq 1$ , the $\alpha_{F}$ -invariant is a “Frobenius-analog" of Tian’s $\alpha$ -invariant.

Our first theorem proves some surprising properties of the $\alpha_{F}$ -invariant in contrast to the complex version, and establishes connections to the $F$ -signature of $X$ , another important invariant and a Frobenius version of the local volume of singularities:

Theorem 1.2.

Let $X$ be a globally $F$ -regular Fano variety over a perfect field of positive characteristic of positive dimension. Then,

(1)

The $\alpha_{F}$ -invariant of $X$ is at most 1/2 (4.5).
(2)

Assume that $X$ is geometrically connected over the (perfect) base field. Then, we have $\alpha_{F}(X)=1/2$ if and only if the $F$ -signature of $X$ (with respect to $-K_{X}$ ) equals $\frac{\mathrm{vol}(-K_{X})}{2^{d}(d+1)!}$ , where $d$ is the dimension of $X$ (4.8).
(3)

More generally (and still assuming $X$ is geometrically connected), the $F$ -signature of $X$ is at most $\frac{\mathrm{vol}(-K_{X})}{2^{d}(d+1)!}$ (4.8).
(4)

In case $X$ is a toric Fano variety corresponding to a fan $\mathcal{F}$ , then $\alpha_{F}(X)$ is the same as the complex $\alpha$ -invariant of $X_{\mathbb{C}}(\mathcal{F})$ , the complex toric Fano variety corresponding to $\mathcal{F}$ (4.12).

Part (1) of 1.2 is surprising since many complex Fano varieties have $\alpha$ -invariants greater than 1/2 (and less than 1). Parts (1) and (4) of 1.2 together recover, and provide a positive characteristic proof of the well-known fact that the $\alpha$ -invariant of toric Fano varieties is at most 1/2 (see [LZ22, Corollary 3.6]).

The $\alpha_{F}$ -invariant (like the complex $\alpha$ -invariant) can be defined for any pair $(X,L)$ , where $X$ is a globally $F$ -regular projective variety and $L$ is an ample line bundle on $X$ . So, in Section 3, we develop the theory of the $\alpha_{F}$ -invariant in this more general setting. From this perspective, the $\alpha_{F}$ -invariant is an asymptotic invariant of a section ring of a projective variety that shares many properties and relations with the $F$ -signature. In this direction, we prove:

Theorem 1.3.

Let $S$ denote the section ring of a globally $F$ -regular projective variety over a perfect field $k$ , with respect to some ample line bundle over $X$ . Then,

(1)

$\alpha_{F}(S)$ can be calculated as the following limit:

$\alpha_{F}(S)=\lim_{e\to\infty}\frac{m_{e}(S)}{p^{e}}$

where $m_{e}(S)$ denotes that maximum integer $m$ such that for each non-zero homogeneous element $f\in S$ of degree $m$ , the map $S\to F^{e}_{*}S$ sending $1$ to $F^{e}_{*}f$ splits. See 3.8.
(2)

$\alpha_{F}(S)$ is positive (3.10).
(3)

Base-change (3.16): Assume that $S_{0}=k$ and $K$ is any perfect field extension of $k$ . Then,

$\alpha_{F}(S)=\alpha_{F}(S\otimes_{k}K).$

Our third set of results concern the semicontinuity properties of the $\alpha_{F}$ -invariant in families of globally $F$ -regular varieties (5.2), analogous to the results of [BL22] about the complex version. In 5.8, we prove:

Theorem 1.4.

Let $f:\mathcal{X}\to Y$ be family of globally $F$ -regular Fano varieties such that $-K_{\mathcal{X}|Y}$ is $\mathbb{Q}$ -Cartier and $f$ -ample. Assume that $Y$ is regular. Then, the map $Y\to\mathbb{R}_{\geq 0}$ given by

y\mapsto\alpha_{F}(\mathcal{X}_{y^{\infty}})

is lower semicontinuous, where $\mathcal{X}_{y^{\infty}}$ is the perfectified-fiber over $y\in Y$ .

We also prove a weaker version of 1.4 for any polarized family of globally $F$ -regular varieties (see 5.3). This is analogous to the corresponding result for the $F$ -signature proved in [CRST21] and relies on understanding the rate of convergence in Part (1) of 1.3 for the $\alpha_{F}$ -invariant, which may be of independent interest (see 5.4).

Finally, we consider some examples that highlight interesting features of the $\alpha_{F}$ -invariant. For instance, we see that the $\alpha_{F}$ -invariant does not detect regularity of a section ring (6.1). In 6.4, we observe that when viewed through the lens of reduction modulo $p$ , the $\alpha_{F}$ -invariant may depend on the characteristic. Furthermore, while the limit of the $\alpha_{F}$ -invariant as $p\to\infty$ may exist and have an interesting geometric interpretation, the limit is not necessarily the complex $\alpha$ -invariant. This fact raises interesting questions and obstructions to relating log canonical thresholds and $F$ -pure thresholds of divisors on a klt variety. We hope that these observations and questions will lead to other results in the future.

Acknowledgements

I would like to thank my advisor Karen Smith for her guidance, support and many helpful discussions. I would like to thank Harold Blum and Yuchen Liu for their valuable guidance that led to some of the main theorems of this paper. I am thankful for the many useful suggestions given by Devlin Mallory, Mircea Mustaţă, Karl Schwede, Kevin Tucker, and Ziquan Zhuang. I would also like to thank Anna Brosowsky, Seungsu Lee, Linquan Ma, Shravan Patankar, Anurag Singh, David Stapleton, Vijaylaxmi Trivedi and Yueqiao Wu for helpful conversations. I thank Devlin Mallory, Mircea Mustaţă and Austyn Simpson for detailed comments on an earlier draft. Parts of this work were conducted during visits to the University of Utah, the University of Illinois at Chicago, the Tata Institute of Fundamental Research, the Indian Statistical Institute (Bangalore), and the Indian Institute of Science. I thank all these institutes for their facilities and hospitality.

2. Preliminaries

Notation 2.1.

Throughout this paper, all rings are assumed to be Noetherian and commutative with a unit. Unless specified otherwise, $k$ will denote a perfect field of characteristic $p$ . A variety over $k$ is an integral (in particular, connected), separated scheme of finite type over $k$ . For a point $x$ on a scheme $X$ , the residue field $\mathcal{O}_{X,x}/\mathfrak{m}_{x}$ will be denoted by $\kappa(x)$ (where $\mathcal{O}_{X,x}$ is the local ring at $x$ and $\mathfrak{m}_{x}$ is the maximal ideal of the local ring).

Notation 2.2 (Divisors and Pairs).

A prime Weil-divisor on a scheme $X$ is a reduced and irreducible subscheme of $X$ of codimension one. An integral Weil-divisor is a formal $\mathbb{Z}$ -linear combination of prime Weil-divisors. A $\mathbb{Q}$ -divisor is a formal $\mathbb{Q}$ -linear combination of prime Weil-divisors. By a pair $(X,\Delta)$ , we mean that $X$ is a Noetherian, normal scheme and $\Delta$ is an effective $\mathbb{Q}$ -divisor over $X$ . A projective pair is a pair $(X,\Delta)$ where $X$ is a projective variety over $k$ .

2.1. Section Rings and Modules

Definition 2.3.

Let $A$ be a Noetherian ring and $X$ be a projective scheme over $A$ . Given an ample invertible sheaf $\mathcal{L}$ on $X$ and $\mathcal{F}$ a coherent sheaf on $X$ , the $\mathbb{N}$ -graded ring $S$ defined by

S=S(X,\mathcal{L}):=\bigoplus_{n\geq 0}H^{0}(X,\mathcal{L}^{n})

is called the section ring of $X$ with respect to $\mathcal{L}$ . The affine scheme $\operatorname{\text{Spec}}(S)$ is called the (affine) cone over $X$ with respect to $\mathcal{L}$ . The section module of $\mathcal{F}$ with respect to $\mathcal{L}$ is a $\mathbb{Z}$ -graded $S$ -module $M$ defined by

M=M(X,\mathcal{L}):=\bigoplus_{n\in\mathbb{Z}}H^{0}(X,\mathcal{F}\otimes\mathcal{L}^{n}).

Similarly, the sheaf corresponding to $M$ on $\operatorname{\text{Spec}}(S)$ is called the cone over $\mathcal{F}$ with respect to $\mathcal{L}$ .

Let $S$ be a Noetherian, $\mathbb{N}$ -graded domain, and $T$ denote the set of positive degree homogeneous elements of $S$ . For a finitely generated, torsion-free, $\mathbb{Z}$ -graded module $M$ over $S$ , let $M^{\prime}$ denote the localization $M^{\prime}=T^{-1}M$ . Note that $M^{\prime}$ is naturally a $\mathbb{Z}$ -graded module over $T^{-1}S$ . Since $M$ is torsion-free, we can think of $M$ naturally as a subset of $M^{\prime}$ . In this setting, we define the saturation of $M$ to be the $\mathbb{Z}$ -graded module

M^{\text{sat}}=\{m\in M^{\prime}\,|\,\mathfrak{p}^{n}m\in M\quad\text{for some $n>0$}\}

where $\mathfrak{p}$ is the irrelevant ideal $\mathfrak{p}=\bigoplus_{j>0}S_{j}$ . We say $M$ is saturated if $M=M^{\text{sat}}$ .

Lemma 2.4.

Let $A$ be a Noetherian domain, $X$ be an integral, projective scheme over $A$ and $\mathcal{L}$ an ample invertible sheaf over $X$ .

(1)

The section ring $S$ of $X$ with respect to $\mathcal{L}$ is a finitely generated algebra over $A$ and hence, is Noetherian. If $X$ is normal, and $A=k$ , then the section ring is also characterized as the unique normal $\mathbb{N}$ -graded ring $S$ such that $\operatorname{\text{Proj}}(S)$ is isomorphic to $X$ and the corresponding $\mathcal{O}_{X}(1)$ is isomorphic to $\mathcal{L}$ .
(2)

The section module of any torsion-free coherent sheaf over $X$ with respect to $\mathcal{L}$ is finitely generated over $S$ . It is also characterized as the unique saturated, torsion-free, $\mathbb{Z}$ -graded $S$ -module $M$ (with respect to the irrelevant ideal $I=\bigoplus_{j>0}S_{j}$ ) such that the associated coherent sheaf $\tilde{M}$ on $X$ is isomorphic to $\mathcal{F}$ .

(3)

For two torsion-free coherent sheaves $\mathcal{F}$ and $\mathcal{G}$ , we have a natural isomorphism:

\operatorname{Hom}_{\mathcal{O}_{X}}(\mathcal{F},\mathcal{G})\cong\operatorname{Hom}^{\text{gr}}_{S}(M(\mathcal{F},\mathcal{L}),M(\mathcal{G},\mathcal{L}))

where $\operatorname{Hom}^{\text{gr}}_{S}(\,,\,)$ denotes the set of grading preserving $S$ -module maps between two graded $S$ -modules.

Proof.

See [Sta, Tag 0BXF]. ∎

2.2. Affine and Projective Cones

Given an integral, projective scheme $X$ over a noetherian domain $A$ , and an ample invertible sheaf $\mathcal{L}$ over $X$ , let $S$ be the corresponding section ring. Assume that $S_{0}=A$ . Let $\overline{X}$ be the projective $A$ -scheme defined as

\overline{X}=\operatorname{\text{Proj}}(S[z])

where $S[z]$ is the $\mathbb{N}$ -graded ring obtained by adjoining a new variable $z$ to $S$ in degree $1$ . Then, $\overline{X}$ is called the projective cone over $X$ with respect to $\mathcal{L}$ . Denoting by $\mathfrak{m}$ the homogeneous irrelevant ideal $\bigoplus_{j>0}S_{j}$ of $S$ , we have a map of graded $A$ -algebras

S[z]\to(S/\mathfrak{m})[z]\cong A[z]

that induces the “zero-section" map

\sigma:\operatorname{\text{Spec}}(A)\to\overline{X}

over $A$ . We call this map the “vertex of the cone $\overline{X}$ ". We also have natural maps of graded rings

S\subset S[z]\to S[z]/(z)\cong S

which, via the construction in [Har77, II, Exercise 2.14 (b)] induce maps

i_{\infty}:X\hookrightarrow\overline{X}

called the “section at infinity" and

\pi:\overline{X}\setminus\sigma(\operatorname{\text{Spec}}(A))\to X

where $\sigma$ is the zero-section described above. It follows from the Proj construction that $\pi$ is an $\mathbb{A}^{1}$ -bundle over $X$ . The affine cone $Y=\operatorname{\text{Spec}}(S)$ is isomorphic to $\overline{X}\setminus i_{\infty}(X)$ , and the zero section $\sigma$ actually maps into $Y$ . Thus, $\pi$ restricts to a map $Y\setminus\sigma(\operatorname{\text{Spec}}(A))\to X$ that is a $\operatorname{\text{Spec}}(A[t,t^{-1}])$ -bundle over $X$ . See [HS04, Section 2] and [Kol13, Section 3.1] for details.

2.3. Cones over $\mathbb{Q}$ -divisors

We follow the description of the cone over a $\mathbb{Q}$ -divisor as in [SS10, Section 5], where it is explained for a projective variety over a field. Essentially the same description holds in the following more general relative setting: Let $A$ be a normal domain of finite type over $k$ , and $X$ be an integral, normal, projective scheme over $A$ . Assume that $X$ is flat over $A$ and of positive relative dimension. Fix an ample invertible sheaf $\mathcal{L}$ over $X$ and $S$ be the corresponding section ring. Assume further that $S_{0}=A$ . Note that this guarantees that the codimension of the zero section $\sigma$ is at least two in $Y=\operatorname{\text{Spec}}(S)$ . Thus, it follows that $S$ is normal as well. In this situation, given any integral Weil-divisor $D=\sum a_{i}D_{i}$ (for distinct prime Weil divisors $D_{i}$ ) on $X$ , we can construct the corresponding Weil divisor $\tilde{D}$ on $Y$ , the “cone over $D$ ", in three equivalent ways:

(1)

Let $\tilde{D_{i}}$ be the prime Weil divisor on $Y$ corresponding to the height one prime $\mathfrak{p}_{i}\subset S$ corresponding to $D_{i}$ . Then $\tilde{D}=\sum a_{i}\tilde{D_{i}}$ .
(2)

Let $\mathcal{O}_{X}(D)$ be the reflexive sheaf on $X$ corresponding to $D$ . Then, $\tilde{D}$ is the divisor corresponding to the reflexive $S$ -module defined by

$M:=M(D,\mathcal{L})=\bigoplus_{j\in\mathbb{Z}}H^{0}(X,\mathcal{O}_{X}(D)\otimes\mathcal{L}^{j}).$

Note that the fact that $M$ is reflexive can be seen by applying Part (3) of 2.4 to each twist of the graded module $M$ .
(3)

Let $\pi:Y\setminus\sigma(\operatorname{\text{Spec}}(A))\to X$ be the $A[t,t^{-1}]$ -bundle map defined in the previous paragraph. Then, we may define $\tilde{D}$ to be the pull back of $D$ to $Y$ . More precisely, near the generic point of a component of $D$ , if $D$ is given by an equation $f$ , then is $\tilde{D}$ is defined by $\pi^{*}f$ . We then take closures to obtain a Weil-divisor on $Y$ . This defines a unique divisor on $Y$ since $\pi$ is flat and $\operatorname{codim}_{Y}(\sigma(\operatorname{\text{Spec}}(A)))$ is at least $2$ .

Given a principal divisor $D$ on $X$ defined by the rational function $f$ , the cone over $D$ can be seen to be the principal divisor defined by $f$ again. The construction of the cone clearly preserves addition of divisors. This implies that cone construction extends to $\mathbb{Q}$ -divisors and preserves the linear equivalence of Weil-divisors. Furthermore, by taking closures, this construction also extends to the projective cone $\overline{X}$ described in the previous paragraph. Finally, using the third description of the cone, we see that the cone over the canonical (Weil-)divisor $K_{X}$ is the canonical divisor of $K_{Y}$ (recall that $Y$ is also normal).

2.4. $F$ -signature

Let $R$ be any ring of prime characteristic $p$ . Then for any $e\geq 1$ , let $F^{e}:R\to R$ sending $r\mapsto r^{p^{e}}$ be the $e^{\text{th}}$ -iterate of the Frobenius morphism. Since $R$ has characteristic $p$ , $F^{e}$ defines a ring homomorphism, allowing us to define a new $R$ -module for each $e\geq 1$ obtained via restriction of scalars along $F^{e}$ . We denote this new $R$ -module by $F_{*}^{e}R$ and its elements by $F_{*}^{e}r$ (where $r$ is an element of $R$ ). Concretely, $F_{*}^{e}R$ is the same as $R$ as an abelian group, but the $R$ -module action is given by:

r\cdot F_{*}^{e}s:=F_{*}^{e}(r^{p^{e}}s)\textrm{\quad for $r\in R$ and $F_{*}^{e}s\in F_{*}^{e}R$}.

Now let $(R,\mathfrak{m})$ denote a normal local ring and $X$ denote the normal scheme $\operatorname{\text{Spec}}(R)$ . Throughout, we will assume that $R$ is the localization of a finitely generated $k$ -algebra at a maximal ideal, which also makes it $F$ -finite (i.e., $F_{*}^{e}R$ is a finitely generated $R$ -module for any $e\geq 1$ ), with the rank of $F^{e}_{*}R$ over $R$ being $p^{ed}$ , where $d$ is the Krull dimension of $R$ . Let $\Delta$ be an effective $\mathbb{Q}$ -divisor on $X=\operatorname{\text{Spec}}(R)$ . Then, note that since $\Delta$ is effective, for any $e\geq 1$ , we have a natural inclusion $R\subset R(\lceil(p^{e}-1)\Delta\rceil)$ of reflexive $R$ -modules. Here, $R(\lceil(p^{e}-1)\Delta\rceil)$ denotes the $R$ -module corresponding to the reflexive sheaf $\mathcal{O}_{X}(\lceil(p^{e}-1)\Delta\rceil)$ . Thus, applying $\operatorname{Hom}_{R}(\,\textunderscore\,,\,R)$ to the natural inclusion $F^{e}_{*}R\subset F^{e}_{*}(R(\lceil(p^{e}-1)\Delta\rceil))$ , we get

\operatorname{Hom}_{R}\big{(}F^{e}_{*}R(\lceil(p^{e}-1)\Delta\rceil),R\big{)}\subset\operatorname{Hom}_{R}(F^{e}_{*}R,R).

Thus, given any element $\varphi\in\operatorname{Hom}_{R}\big{(}F^{e}_{*}R(\lceil(p^{e}-1)\Delta\rceil),R\big{)}$ , it can be naturally viewed as a map $\varphi:F^{e}_{*}R\to R$ .

Definition 2.5 (Splitting Ideals).

For any $e\geq 1$ , we define the subset $I_{e}^{\Delta}\subseteq R$ as

I_{e}^{\Delta}=\left\{x\in R\mid\varphi(F^{e}_{*}x)\in\mathfrak{m}\,\text{for every map }\varphi\in\operatorname{Hom}_{R}\big{(}F^{e}_{*}R(\lceil(p^{e}-1)\Delta\rceil),R\big{)}\,\right\}.

We observe that $I_{e}^{\Delta}$ is an ideal of finite colength in $R$ and we call

a_{e}^{\Delta}=\ell_{R}(R/I_{e}^{\Delta})

the $\Delta$ -free rank of $F^{e}_{*}R$ , where $\ell_{R}$ denotes the length as an $R$ -module.

Definition 2.6.

[BST11, Theorem 3.11, Proposition 3.5] Let $(R,\Delta)$ be a pair as above, and $a_{e}^{\Delta}(R)$ denote the $\Delta$ -free rank of $F^{e}_{*}R$ (2.5). Then the $F$ -signature of $(R,\Delta)$ is defined to be the limit:

\mathscr{s}(R,\Delta):=\lim_{e\to\infty}\frac{a_{e}^{\Delta}}{p^{ed}}

where $d$ is the Krull dimension of $R$ . This limit exists by [BST11].

2.5. $F$ -signature of cones over projective varieties.

In this subsection, we describe how we can compute the $F$ -signature of cones over projective varieties (and pairs) using global splittings on $X$ . We begin with a useful lemma that relates global Frobenius splitting of a divisor to splitting “on the cone". This is a slight generalization to the relative setting of [Smi00, Theorem 3.10], where it is proved over a field.

Lemma 2.7.

Let $A$ be a regular ring of finite type over $k$ and $X$ be an integral, normal projective scheme over $A$ (with $H^{0}(X,\mathcal{O}_{X})=A$ ). Assume that $X$ is flat over $A$ and of positive relative dimension. Fix an ample invertible sheaf $\mathcal{L}$ and $S$ be the corresponding section ring. Fix an effective Weil divisor $D$ over $X$ and $\tilde{D}$ be the cone over $D$ with respect to $\mathcal{L}$ . Then, for any $e\geq 1$ , the natural map

\mathcal{O}_{X}\to F^{e}_{*}(\mathcal{O}_{X}(D))

splits as a map of $\mathcal{O}_{X}$ -modules if and only if the map on the cones

S\to F^{e}_{*}(S(\tilde{D}))

splits as a map of $S$ -modules.

Proof.

Using the second description of the cone over a $\mathbb{Q}$ -divisor in Section 2.3, and Part (3) of 2.4, the proof is exactly the same as that in [SS10, Proposition 5.3]. ∎

Returning to working over a perfect field $k$ , we next recall a formula to compute the $F$ -signature of the section ring of a projective variety proved in [LP23]. Fix a normal projective variety $X$ over $k$ and $\Delta$ be an effective $\mathbb{Q}$ -divisor over $X$ .

Definition 2.8.

For any Weil-divisor $D$ on $X$ and $e\geq 1$ , define the $k$ -vector subspace $I_{e}^{\Delta}(D)$ of $H^{0}(X,\mathcal{O}_{X}(D))$ as follows:

I_{e}^{\Delta}(D):=\{f\in H^{0}(X,D)\ |\ \varphi(F^{e}_{*}f)=0\text{ for all }\varphi\in\operatorname{Hom}_{\mathcal{O}_{X}}(F^{e}_{*}\mathcal{O}_{X}(\lceil(p^{e}-1)\Delta\rceil+D),\mathcal{O}_{X})\,\}.

Remark 2.9.

Note that the subspace $I_{e}^{\Delta}(D)$ only depends on the sheaf $\mathcal{O}_{X}(D)$ and not on the specific divisor $D$ in its linear equivalence class.

Remark 2.10.

Let $L$ be an ample divisor on $X$ . Then, it follows from 2.7 that $I_{e}^{\Delta}(mL)$ is the degree $m$ component of the $\Delta$ -splitting ideal of the section ring of $S$ with respect to $L$ (2.5).

Lemma 2.11.

[LP23, Lemma 4.7] Let $L$ be an ample Cartier divisor on $X$ and $S$ denote the section ring of $X$ with respect to $L$ . Let $\Delta_{S}$ denote the cone over $\Delta$ with respect to $L$ (Section 2.3). Then, for any $e\geq 1$ , if $a_{e}^{\Delta}(L)$ denotes the $\Delta_{S}$ -free-rank of $F^{e}_{*}S$ (2.5), then $a_{e}^{\Delta}(L)$ is computed by the following formula:

a_{e}^{\Delta}(L)=\frac{1}{[k^{\prime}:k]}\,\sum_{m=0}^{\infty}\dim_{k}\frac{H^{0}(X,mL)}{I_{e}^{\Delta}(mL)}

(2.1)

where $k^{\prime}$ denotes the field $H^{0}(X,\mathcal{O}_{X})$ . Hence, the $F$ -signature of $(X,\Delta)$ with respect to $L$ can be computed as

\mathscr{s}_{(X,\Delta)}(L):=\mathscr{s}(S(X,L),\Delta_{S})=\frac{1}{[k^{\prime}:k]}\,\lim_{e\to\infty}\frac{\sum\limits_{m=0}^{\infty}\dim_{k}\frac{H^{0}(X,mL)}{I_{e}^{\Delta}(mL)}}{p^{e(\dim(X)+1)}}

Proof.

Using 2.7, we have that

I_{e}^{\Delta_{S}}=\bigoplus_{m\geq 0}I_{e}^{\Delta}(mL).

See 2.5 for the definition of $I_{e}^{\Delta_{S}}$ . Therefore, we have

\ell_{S}(S/I_{e}^{\Delta_{S}})=\sum_{m\geq 0}\dim_{k^{\prime}}\frac{H^{0}(mL)}{I_{e}^{\Delta}(mL)}=\frac{1}{[k^{\prime}:k]}\,\sum_{m=0}^{\infty}\dim_{k}\frac{H^{0}(X,mL)}{I_{e}^{\Delta}(mL)}

where $\ell_{S}$ denotes the length as an $S$ -module. This completes the proof of the lemma. ∎

2.6. Duality and the Trace Map

It will be convenient to think of the subspaces $I_{e}^{\Delta}$ (2.8) using a pairing arising out of duality for the Frobenius map. Let $(X,\Delta)$ be a projective pair and $D$ be any Weil divisor on $X$ . We continue to work over any perfect field $k$ of characteristic $p>0$ . But in this subsection, we assume that $H^{0}(X,\mathcal{O}_{X})=k$ , i.e., that $X$ is geometrically connected.

Recall that by applying duality to the Frobenius map, we get the following isomorphism of reflexive $\mathcal{O}_{X}$ -modules:

\mathscr{H}om_{\mathcal{O}_{X}}(F^{e}_{*}\mathcal{O}_{X}(\lceil(p^{e}-1)\Delta\rceil+D),\mathcal{O}_{X})\cong F^{e}_{*}\mathcal{O}_{X}(-(p^{e}-1)K_{X}-\lceil(p^{e}-1)\Delta\rceil-D).

(2.2)

See [SS10, Section 4.1] for a detailed discussion regarding duality for the Frobenius map. Furthermore, when $D=0$ , this gives an isomorphism

\mathscr{H}om_{\mathcal{O}_{X}}(F^{e}_{*}\mathcal{O}_{X}(\lceil(p^{e}-1)\Delta\rceil,\mathcal{O}_{X})\cong F^{e}_{*}\mathcal{O}_{X}(-(p^{e}-1)K_{X}-\lceil(p^{e}-1)\Delta\rceil).

(2.3)

Composing this isomorphism (over the global sections) with the evaluation at $F^{e}_{*}1$ map, we obtain the trace map:

\text{Tr}^{e}_{\Delta}:H^{0}\Big{(}X,F^{e}_{*}\big{(}\mathcal{O}_{X}((1-p^{e})K_{X}-\lceil(p^{e}-1)\Delta\rceil)\big{)}\Big{)}\to k=H^{0}(X,\mathcal{O}_{X}).

(2.4)

Lemma 2.12.

The kernel of the trace map $\text{Tr}^{e}_{\Delta}$ in Equation 2.4 is exactly the subspace $F^{e}_{*}I_{e}^{\Delta}((1-p^{e})K_{X}-\lceil(p^{e}-1)\Delta\rceil)$ . See 2.8 for the definition of the subspace $I_{e}^{\Delta}$ .

Proof.

A section $f\in H^{0}(X,\mathcal{O}_{X}((1-p^{e})K_{X}-\lceil(p^{e}-1)\Delta\rceil))$ is contained in the corresponding $I_{e}^{\Delta}$ -subspace if and only if for every

\varphi\in\operatorname{Hom}_{\mathcal{O}_{X}}(F^{e}_{*}(\mathcal{O}_{X}((1-p^{e})K_{X})),\mathcal{O}_{X})\cong H^{0}(X,F^{e}_{*}\mathcal{O}_{X})\cong F^{e}_{*}k,

we have $\varphi(F^{e}_{*}f)=0$ . But, $\operatorname{Hom}_{\mathcal{O}_{X}}(F^{e}_{*}(\mathcal{O}_{X}((1-p^{e})K_{X})),\mathcal{O}_{X})$ is a one dimensional $k$ -vector space, and is generated by the trace map $\text{Tr}^{e}$ (where $\Delta=0$ ). Now, the map $\text{Tr}^{e}_{\Delta}$ is just the restriction of the trace map $\text{Tr}^{e}$ to the subspace $H^{0}\Big{(}X,F^{e}_{*}\big{(}\mathcal{O}_{X}((1-p^{e})K_{X}-\lceil(p^{e}-1)\Delta\rceil)\big{)}\Big{)}$ . Thus, the lemma follows. ∎

Lemma 2.13.

Let $(X,\Delta)$ be a normal projective pair, and $D$ be any Weil divisor on $X$ . Then, denoting $D_{1}=(1-p^{e})K_{X}-\lceil(p^{e}-1)\Delta\rceil-D$ and $D_{2}=(1-p^{e})K_{X}-\lceil(p^{e}-1)\Delta\rceil$ for any $e\geq 1$ , we have a non-degenerate pairing

\frac{H^{0}(D)}{I_{e}^{\Delta}(D)}\times\frac{H^{0}(D_{1})}{I_{e}^{\Delta}(D_{1})}\to\frac{H^{0}(D_{2})}{I_{e}^{\Delta}(D_{2})}

obtained from multiplication (and reflexifying) global sections. In particular,

\dim_{k}\frac{H^{0}(D)}{I_{e}^{\Delta}(D)}=\dim_{k}\frac{H^{0}(D_{1})}{I_{e}^{\Delta}(D_{1})}.

Proof.

Using Equation 2.2, the natural multiplication map

H^{0}(D)\times H^{0}(D_{1})\to H^{0}(D_{2})

can be identified with the evaluation map

H^{0}\Big{(}F^{e}_{*}\big{(}\mathcal{O}_{X}(D)\big{)}\Big{)}\times\operatorname{Hom}_{\mathcal{O}_{X}}\Big{(}F^{e}_{*}\big{(}\mathcal{O}_{X}(\lceil(p^{e}-1)\Delta\rceil+D)\big{)},\mathcal{O}_{X}\Big{)}\to\operatorname{Hom}_{\mathcal{O}_{X}}(F^{e}_{*}\mathcal{O}_{X}(\lceil(p^{e}-1)\Delta\rceil),\mathcal{O}_{X})

where we identify $\operatorname{Hom}_{\mathcal{O}_{X}}(F^{e}_{*}\mathcal{O}_{X}(\lceil(p^{e}-1)\Delta\rceil+D),\mathcal{O}_{X})$ and $\operatorname{Hom}_{\mathcal{O}_{X}}(F^{e}_{*}\mathcal{O}_{X}(\lceil(p^{e}-1)\Delta\rceil),\mathcal{O}_{X})$ as subspaces of $\operatorname{Hom}_{\mathcal{O}_{X}}(F^{e}_{*}\mathcal{O}_{X}(D),\mathcal{O}_{X})$ and $\operatorname{Hom}_{\mathcal{O}_{X}}(F^{e}_{*}\mathcal{O}_{X},\mathcal{O}_{X})$ respectively, both via the natural inclusion $\mathcal{O}_{X}\to\mathcal{O}_{X}(\lceil(p^{e}-1)\Delta\rceil)$ . Therefore, a section $f\in H^{0}(D)$ is contained in $I_{e}^{\Delta}(D)$ if and only if for all sections $g\in H^{0}(D_{1})$ , the multiplication $f\,g$ is contained in $I_{e}^{\Delta}(D_{2})$ . By symmetry, a section $g\in H^{0}(D_{1})$ is contained in $I_{e}^{\Delta}(D_{1})$ if and only if for all sections $f\in H^{0}(D)$ , $g\,f\in I_{e}^{\Delta}(D_{2})$ . This proves there is a well defined, and non-degenerate pairing as needed.

Finally, we note that since by 2.12, $I_{e}^{\Delta}(D_{2})$ is the kernel of the trace map (Equation 2.4), the vector space $H^{0}(D_{2})/I_{e}^{\Delta}(D_{2})$ is either one-dimensional over $k$ , or equal to $0$ . In either case, the equality of dimensions follows. ∎

The next Proposition allows us to perturb by any divisor while computing the $F$ -signature of a section ring.

Proposition 2.14.

Let $X$ be a normal projective varitey over $k$ and $L$ an ample divisor on $X$ . Assume $H^{0}(X,\mathcal{O}_{X})=k$ . Fix a (not necessarily effective) Weil divisor $D$ on $X$ . Then, there exists a constant $C>0$ (depending only on $D$ and $L$ ) such that

\left|\dim_{k}\frac{H^{0}(mL)}{I_{e}(mL)}-\dim_{k}\frac{H^{0}(mL+D)}{I_{e}(mL+D)}\right|\leq Cp^{e(\dim(X)-1)}

for all $m>0$ and $e>0$ .

Proof.

First we prove the case when $D$ is effective: For any $m\geq 1$ , by using the natural map $\mathcal{O}_{X}(mL)\to\mathcal{O}_{X}(mL+D)$ , we will view $H^{0}(mL)$ as a subspace of $H^{0}(mL+D)$ . Let $J_{e}(mL)$ denote the subspace $H^{0}(mL)\cap I_{e}(mL+D)$ . By [LP23, Lemma 4.12], we see that $I_{e}(mL)\subset J_{e}(mL)$ . Moreover, by Equation 4.12 in [LP23, Proof of Lemma 4.14], we have

\dim_{k}\frac{H^{0}(mL+D)}{I_{e}(mL+D)}=\dim_{k}\frac{H^{0}(mL)}{J_{e}(mL)}+\dim_{k}\frac{H^{0}(mL+D)}{H^{0}(mL)+I_{e}(mL+D)}.

Using this and the triangle inequality, we obtain that

\left|\dim_{k}\frac{H^{0}(mL)}{I_{e}(mL)}-\dim_{k}\frac{H^{0}(mL+D)}{I_{e}(mL+D)}\right|\leq\dim_{k}\frac{H^{0}(mL+D)}{H^{0}(mL)}+\dim_{k}\frac{J_{e}(mL)}{I_{e}(mL)}

(2.5)

for all $m,e>0$ . Next, to compute the second term in the above inequality, fix an $e>0$ and set $\Delta_{e}=\frac{1}{p^{e}-1}D$ . Then, we observe that the subspace $J_{e}(mL)$ is exactly the same as $I_{e}^{\Delta_{e}}(mL)$ (2.8). Moreover, we also similarly have

I_{e}^{\Delta_{e}}((1-p^{e})K_{X}-mL-D)=H^{0}((1-p^{e})K_{X}-mL-D)\cap I_{e}((1-p^{e})K_{X}-mL).

(2.6)

Thus, by 2.13, we have

\dim_{k}\frac{H^{0}(mL)}{I_{e}(mL)}=\dim_{k}\frac{H^{0}((1-p^{e})K_{X}-mL)}{I_{e}((1-p^{e})K_{X}-mL)}

and similarly,

\dim_{k}\frac{H^{0}(mL)}{J_{e}(mL)}=\dim_{k}\frac{H^{0}((1-p^{e})K_{X}-mL-D)}{I_{e}^{\Delta_{e}}((1-p^{e})K_{X}-mL-D)}.

By Equation 2.6, we see the natural map from $H^{0}((1-p^{e})K_{X}-mL-D)$ to $H^{0}((1-p^{e})K_{X}-mL)$ restricts to an injective map

\frac{H^{0}((1-p^{e})K_{X}-mL-D)}{I_{e}^{\Delta_{e}}((1-p^{e})K_{X}-mL-D)}\hookrightarrow\frac{H^{0}((1-p^{e})K_{X}-mL)}{I_{e}((1-p^{e})K_{X}-mL)}.

By considering the cokernel of this map, we get that

\dim_{k}\frac{J_{e}(mL)}{I_{e}(mL)}=\dim_{k}\frac{H^{0}(mL)}{I_{e}(mL)}-\dim_{k}\frac{H^{0}(mL)}{J_{e}(mL)}\leq\dim_{k}\frac{H^{0}((1-p^{e})K_{X}-mL)}{H^{0}((1-p^{e})K_{X}-mL-D)}.

(2.7)

Pick an $M\gg 0$ such that $mL$ admits a global section that doesn’t vanish along $D$ for all $m\geq M$ (this is possible since $L$ is ample). Using the standard exact sequences to restrict to $D$ , we see that

\dim_{k}\frac{H^{0}((1-p^{e})K_{X}-mL)}{H^{0}((1-p^{e})K_{X}-mL-D)}\leq\frac{\mathrm{vol}(-K_{X}|_{D})}{(d-1)!}p^{e(d-1)}+o(p^{e(d-2)})

(2.8)

and

\dim_{k}\frac{H^{0}(mL+D)}{H^{0}(mL)}\leq\frac{\mathrm{vol}(L|_{D})}{(d-1)!}m^{d-1}+o(m^{d-2}).

(2.9)

Finally, by [LP23, Theorem 4.9], we may pick a constant $C_{2}>0$ such that $H^{0}(mL)=I_{e}(mL)$ and $H^{0}(mL+D)=I_{e}(mL+D)$ for $m>C_{2}p^{e}$ . Therefore, to prove the Proposition, it is enough to consider the case when $m\leq C_{2}p^{e}$ . In this case, the Proposition now follows by putting together inequalities in 2.5, 2.7, 2.8 and 2.9. This completes the proof of the Proposition when $D$ is effective.

More generally, we first pick an $r\gg 0$ such that $rL$ and $D+rL$ are both effective. Then, for any $e\geq 1$ and $m>r$ , we have

\begin{split}\left|\dim_{k}\frac{H^{0}(mL)}{I_{e}(mL)}-\dim_{k}\frac{H^{0}(mL+D)}{I_{e}(mL+D)}\right|\leq&\left|\dim_{k}\frac{H^{0}(mL)}{I_{e}(mL)}-\dim_{k}\frac{H^{0}((m-r)L)}{I_{e}((m-r)L)}\right|\\ +&\left|\dim_{k}\frac{H^{0}((m-r)L)}{I_{e}((m-r)L)}-\dim_{k}\frac{H^{0}((m-r)L+rL+D)}{I_{e}((m-r)L+rL+D)}\right|.\end{split}

Now, we may apply the previous case of the Proposition (since both $D+rL$ and $rL$ are effective) to each of the two terms in the above inequality. Since $r$ was independent of $e$ , this completes the proof of the Proposition . ∎

2.7. $F$ -regularity:

Definition 2.15 (Sharp $F$ -splitting).

[SS10, Definition 3.1] Let $X$ be a normal variety over $k$ and $\Delta\geq 0$ be an effective $\mathbb{Q}$ -divisor. The pair $(X,\Delta)$ is said to be globally sharply $F$ -split (resp. locally sharply $F$ -split) if there exists an integer $e\gg 0$ , such that, the natural map

\mathcal{O}_{X}\to F^{e}_{*}\mathcal{O}_{X}(\lceil(p^{e}-1)\Delta\rceil)

splits (resp. splits locally) as a map of $\mathcal{O}_{X}$ -modules. A normal variety $X$ is said to globally $F$ -split if the pair $(X,0)$ is globally sharply $F$ -split.

Definition 2.16 ( $F$ -regularity).

[SS10, Definition 3.1] Let $X$ be a normal variety over $k$ and $\Delta\geq 0$ be an effective $\mathbb{Q}$ -divisor. The pair $(X,\Delta)$ is said to be globally $F$ -regular (resp. locally strongly $F$ -regular) if for any effective Weil divisor $D$ on $X$ , there exists an integer $e\gg 0$ , such that, the natural map

\mathcal{O}_{X}\to F^{e}_{*}\mathcal{O}_{X}(\lceil(p^{e}-1)\Delta\rceil+D)

splits (resp. splits locally) as a map of $\mathcal{O}_{X}$ -modules. A normal variety $X$ is said to globally $F$ -regular if the pair $(X,0)$ is globally $F$ -regular. Similarly, a ring $R$ is called strongly $F$ -regular if the pair $(\operatorname{\text{Spec}}(R),0)$ is locally (equivalently, globally) strongly $F$ -regular.

Remark 2.17.

When $X=\operatorname{\text{Spec}}(R)$ is an affine variety and $\Delta$ is an effective $\mathbb{Q}$ -divisor, the pair $(X,\Delta)$ being globally $F$ -regular (resp. globally sharply $F$ -split) is equivalent to the pair $(R,\Delta)$ being locally strongly $F$ -regular (resp. locally sharply $F$ -split) [SS10].

Remark 2.18.

A local domain $R$ is strongly $F$ -regular if and only if its $F$ -signature $\mathscr{s}(R)$ is positive [AL03]. More generally, a pair $(R,\Delta)$ (where $R$ is normal, local) is strongly $F$ -regular if and only if the $F$ -signature $s(R,\Delta)$ (2.6) is positive [BST11, Theorem 3.18].

Theorem 2.19.

[Smi00, Theorem 3.10] Let $X$ be a projective variety over $k$ . Then, $X$ is globally $F$ -regular if and only if the section ring $S(X,L)$ (2.3) with respect to some (equivalently, every) ample invertible sheaf $L$ is strongly $F$ -regular.

Remark 2.20.

(Locally) Strongly $F$ -regular varieties are normal and Cohen-Macaulay. Similarly, globally $F$ -regular varieties enjoy many of nice properties such as:

•

As proved in [SS10, Theorem 4.3], they are log-Fano type. More precisely, there exists an effective divisor $\Delta\geq 0$ such that the pair $(X,\Delta)$ is globally $F$ -regular and $-K_{X}-\Delta$ is ample.
•

A version of the Kawamata-Viehweg vanishing theorem holds on all globally $F$ -regular varieties [SS10, Theorem 6.8].

Theorem 2.21 ([Smi00], Corollary 4.3).

Let $X$ be a projective, globally $F$ -regular variety over $k$ . Suppose $L$ is a nef invertible sheaf over $X$ . Then,

H^{i}(X,L)=0\quad\text{for all $i>0$}.

We need a slight variation of 2.21 for $\mathbb{Q}$ -ample divisors that we prove here for completeness.

Proposition 2.22.

Let $X$ be a globally $F$ -split normal variety and $L$ be a $\mathbb{Q}$ -ample Weil divisor i.e., $L$ is an integral Weil divisor such that $rL$ is an ample Cartier divisor for some integer $r>0$ . Then,

H^{i}(X,\mathcal{O}_{X}(L))=0\quad\text{for $i>0$.}

Proof.

Let $r$ be an integer such that $rL$ is Cartier. Write $r=p^{e_{0}}s$ such that $s$ is coprime to $p$ . Pick an $e>0$ such that $s$ divides $p^{e}-1$ . Then, since $p^{e_{0}}(p^{ne}-1)$ is a multiple of $r$ for all $n>0$ , using Serre vanishing theorem, we have

H^{i}(X,\mathcal{O}_{X}(p^{ne+e_{0}}L))=H^{i}(X,\mathcal{O}_{X}(p^{e_{0}}L+(p^{ne}-1)p^{e_{0}}L)=0\quad\text{for all $i>0$ and $n\gg 0$.}

(2.10)

Since the map

\mathcal{O}_{X}\to F_{*}^{ne+e_{0}}\mathcal{O}_{X}

is split, twisting by $\mathcal{O}_{X}(L)$ and reflexifying, we get that

\mathcal{O}_{X}(L)\to F_{*}^{ne+e_{0}}\mathcal{O}_{X}(p^{ne+e_{0}}L)

is split as well. Now the Proposition follows from the vanishing in (2.10). ∎

The next Proposition is a technical result that helps us to restrict $\mathbb{Q}$ -Cartier divisors to normal, locally complete intersection subvarieties. This is very close to [PS12, Corollary 3.3], but we will need it in the form stated below.

Proposition 2.23.

Let $X=\operatorname{\text{Spec}}(R)$ , where $(R,\mathfrak{m})$ is an $F$ -finite, strongly $F$ -regular, local ring and $D$ be an integral Weil-divisor on $X$ such that $rD$ is Cartier for some integer $r$ . Then, for each $m\geq 0$ ,

(1)

There exists an $e\gg 0$ such that the module $R(mD)=H^{0}(X,\mathcal{O}_{X}(mD))$ is isomorphic to an $R$ -module summand of $F^{e}_{*}R$ . In particular, $R(mD)$ is a Cohen-Macaulay module over $R$ .
(2)

Suppose that $x_{1},\dots,x_{t}$ is a regular sequence on $R$ such that the ring $R/(x_{1},\dots,x_{t})R$ is normal. Then, the sheaf $\mathcal{O}_{X}(mD)\otimes_{R}\mathcal{O}_{Y}$ is reflexive on $Y=\operatorname{\text{Spec}}(R/(x_{1},\dots,x_{t})R)$ . Furthermore, if we assume that the support of $D$ does not contain the subscheme $Y$ , then natural map

$\mathcal{O}_{X}(mD)\otimes_{R}\mathcal{O}_{Y}\to\mathcal{O}_{Y}(mD_{Y})$ (2.11)

is an isomorphism, where $D_{Y}$ denotes the restriction of $D$ to $Y$ (see the description in the proof below).

Proof.

By adding a principal divisor if necessary (which leaves the module $R(mD)$ isomorphic), we assume that $D$ is effective.

(1)

Since $R$ is strongly $F$ -regular, there exists an $e\gg 0$ such that the map $\mathcal{O}_{X}\to F^{e}_{*}(\mathcal{O}_{X}(sD))$ splits for each $0\leq s\leq r$ . Once we have such an $e$ , choose $s\leq r$ such that $p^{e}+s$ is divisible by $r$ . Thus, we may assume that the map (got by twisting by $D$ and reflexifying):

$\mathcal{O}_{X}(D)\to F^{e}_{*}\mathcal{O}_{X}((p^{e}+s)D)$

is split. Since $r$ divides $p^{e}+s$ , $\mathcal{O}_{X}((p^{e}+s)D)$ is isomorphic to $\mathcal{O}_{X}$ since $R$ is local and $rD$ is Cartier. Therefore, taking global sections, we have that the map

$R(mD)\to F^{e}_{*}R$ (2.12)

is split. Note that the $e$ obtained is independent of $m$ . The Cohen-Macaulayness of $R(mD)$ follows because $F^{e}_{*}R$ is a Cohen-Macaulay module over $R$ , since $R$ itself is Cohen-Macaulay.
(2)

Firstly, we may assume $m=1$ since the discussion holds for an arbitrary Weil divisor and is compatible with addition of Weil-divisors. Now, since $Y$ is a normal, complete intersection subscheme of $X$ , we may “restrict" the rank one reflexive sheaf $\mathcal{F}:=\mathcal{O}_{X}(D)$ on $X$ to a reflexive sheaf $\mathcal{F}_{Y}$ on $Y$ as follows: Let $U$ be the regular locus of $Y$ . Then there is an open subset $V\subset X_{\text{reg}}$ (where $X_{\text{reg}}$ denotes the regular locus of $X$ ) such that $V\cap Y=U$ . This is possible because $Y$ is a complete intersection in $X$ . Therefore, we may restrict $\mathcal{F}$ to $V$ and then to an invertible sheaf on $U$ , since $\mathcal{F}|_{V}$ is invertible. Define $\mathcal{F}_{Y}$ to be

$\mathcal{F}_{Y}:=i_{*}(\mathcal{F}|_{U})$

where $i:U\to Y$ is the inclusion. Then, $\mathcal{F}_{Y}$ is a rank one reflexive sheaf on $Y$ because $Y$ is normal and $U$ contains all the codimension one points of $Y$ . Thus, we can write $\mathcal{F}_{Y}$ as $\mathcal{O}_{Y}(D_{Y})$ for some Weil-divisor $D_{Y}$ on $Y$ . Furthermore, if $\text{Supp}(D)$ does not contain $Y$ , then since $Y$ is normal, hence integral, $D$ naturally restricts to a Cartier divisor $D_{U}$ on $U$ (given by restricting the equation for $D$ ) and we may take $D_{Y}$ to be the closure of $D_{U}$ . It is also clear from the description of restriction that it commutes with addition of Weil-divisors (since the restriction of Cartier divisors on the regular locus commutes with addition).

Now, since $\mathcal{F}|_{U}$ is the restriction of the sheaf $\mathcal{F}$ (i.e., isomorphic to $\mathcal{F}\otimes_{\mathcal{O}_{X}}\mathcal{O}_{U}$ ), there is a natual map

$\mathcal{F}\otimes_{\mathcal{O}_{X}}\mathcal{O}_{Y}\to\mathcal{F}_{Y}$

which is an isomorphism if and only if $\mathcal{F}\otimes_{\mathcal{O}_{X}}\mathcal{O}_{Y}$ is reflexive. Therefore, it is sufficient to show that the module $R(D)\otimes_{R}R/(x_{1},\dots,x_{t})R$ satisfies the S₂ condition on $R/(x_{1},\dots,x_{t})R$ (since $R/(x_{1},\dots,x_{t})R$ is normal). But since $R(mD)$ is Cohen-Macaulay by Part (1) (and clearly full dimensional), and $x_{1},\dots,x_{t}$ is a regular sequence on $R$ , we get that $R(mD)\otimes R/(x_{1},\dots,x_{t})R$ is Cohen-Macaulay as well. This completes the proof of the Proposition. ∎

3. The $\alpha_{F}$ -invariant of section rings

In analogy with Tian’s $\alpha$ -invariant in complex geometry, we define the “Frobenius- $\alpha$ " invariant (denoted by $\alpha_{F}$ ) for any pair $(X,L)$ where $X$ is a globally $F$ -regular projective variety (2.16) and $L$ is an ample Cartier divisor on $L$ .

Throughout this section, by a section ring $S$ , we mean that $S$ is the section ring $S(X,L)$ of some projective variety $X\,(\cong\operatorname{\text{Proj}}(S))$ with respect to some ample line bundle $L\,(\cong\mathcal{O}_{X}(1))$ on $X$ (see 2.3).

3.1. Definitions

Definition 3.1.

Let $(R,\mathfrak{m})$ be an $F$ -finite normal local ring. Assume that $R$ is strongly $F$ -regular (2.16). Then, we define the $F$ -pure threshold of an effective $\mathbb{Q}$ -divisor $D\geq 0$ on $X$ to be:

\text{fpt}_{\mathfrak{m}}(R,D):=\sup\{\,\lambda\,|\,(R,\lambda D)\text{ is sharply $F$-split}\,\}.

Moreover, by [SS10, Lemma 4.9], it follows that the $F$ -pure threshold of $(R,D)$ is equivalently characterized as the supremum of the set $\{\,\lambda\,|\,(R,\lambda D)\text{ is strongly $F$-regular}\}$ .

If $D$ is the principal divisor corresponding to a function $f\in R$ , we write $f$ instead of $D$ in the notation for the $F$ -pure threshold.

Definition 3.2.

Let $(S,\mathfrak{m})$ be a strongly $F$ -regular section ring of a projective variety. Then, we define

\alpha_{F}(S)=\inf\{\,\text{fpt}_{\mathfrak{m}}(S,f)\,\deg(f)\,|\,0\neq f\in S\text{ homogeneous element}\,\}.

If $S$ is the section ring of a projective variety $X$ with respect to an ample divisor $L$ , we may also use $\alpha_{F}(X,L)$ to denote $\alpha_{F}(S)$ .

We have the following equivalent ways of characterizing the $\alpha_{F}$ -invariant of a section ring.

Lemma 3.3.

Let $X$ be a globally $F$ -regular projective variety and $L$ be an ample Cartier divisor on $X$ . Let $S=S(X,L)$ be the section ring of $X$ with respect to $L$ . Then, $\alpha_{F}(S)$ from 3.2 is equal to the supremum of any of the following sets:

(1)

The set of $\lambda\geq 0$ such that the pair $(S,\frac{\lambda}{n}D)$ is sharply $F$ -split for every $n\in\mathbb{N}$ and every effective divisor $D\sim nL$ .
(2)

The set of $\lambda\geq 0$ such that the pair $(S,\frac{\lambda}{n}D)$ is strongly $F$ -regular for every $n\in\mathbb{N}$ and every effective divisor $D\sim nL$ .
(3)

The set of $\lambda\geq 0$ such that the pair $(X,\frac{\lambda}{n}D)$ is globally sharply $F$ -split for every $n\in\mathbb{N}$ and every effective divisor $D\sim nL$ .
(4)

The set of $\lambda\geq 0$ such that the pair $(X,\frac{\lambda}{n}D)$ is globally $F$ -regular for every $n\in\mathbb{N}$ and every effective divisor $D\sim nL$ .
(5)

The set of $\lambda\geq 0$ such that the pair $(X,\lambda D)$ is globally $F$ -regular for every effective $\mathbb{Q}$ -divisor $D\sim_{\mathbb{Q}}L$ .

Proof.

Statements (1) and (2) follow immediately from the definition of the $\alpha_{F}$ -invariant and the definition of the $F$ -pure threshold (3.1). Statements (3) and (4) follow from (1) and (2) by using 2.7. Part (5) is just a reformulation of (4) since every effective $\mathbb{Q}$ -divisor $D\sim_{\mathbb{Q}}L$ is of the form $\frac{1}{n}nD$ for some effective Cartier divisor $nD\sim nL$ . ∎

Next, we explain a more precise connection between the $\alpha_{F}$ -invariant and Frobenius splittings in $S$ .

Proposition 3.4.

Let $S$ be a strongly $F$ -regular section ring and $\alpha^{\prime}(S)$ denote the supremum of the following set:

\mathscr{A}(S):=\{\lambda\in\mathbb{R}_{\geq 0}\,|\,\text{ for any integers $e\geq 1$ and $m\leq\lambda(p^{e}-1)$, we have }I_{e}(m)=0\}.

Then, $\alpha^{\prime}(S)=\alpha_{F}(S)$ . Moreover, $\alpha_{F}(S)$ belongs to the set $\mathscr{A}(S)$ .

Remark 3.5.

Note that in the above Proposition, it is unclear if the set $\mathscr{A}(S)$ contains any non-zero element. This is equivalent to the positivity of $\alpha_{F}(S)$ and will be addressed in 3.10.

The proof of the 3.4 is based on the following lemma which is a slight generalization of a result of Hernández:

Lemma 3.6.

Let $(S,\mathfrak{m})$ be an $F$ -finite, strongly $F$ -regular local ring. Fix an effective Weil-divisor $D$ on $X=\textrm{Spec}(S)$ . Then, for any fixed $e_{0}>0$ , let $\psi_{e_{0}}$ denote the natural map

\psi_{e_{0}}:\mathcal{O}_{X}\to F_{*}^{e_{0}}(\mathcal{O}_{X}(D)).

Then, the following are equivalent:

(1)

The map $\psi_{e_{0}}$ splits as a map of $\mathcal{O}_{X}$ -modules.
(2)

The pair $(X,\frac{1}{p^{e_{0}}-1}D)$ is sharply $F$ -split (2.15).
(3)

The $F$ -pure threshold of $(X,D)$ is at least $\frac{1}{p^{e_{0}}-1}$ (3.1).

Proof.

It follows immediately from the definitions that (1) implies (2), and (2) implies (3). Hence, it remains to show that (3) implies (1).

Following [Her12, Thoerem 4.9], if $\text{fpt}_{\mathfrak{m}}(X,D)\geq\frac{1}{p^{e_{0}}-1}$ , we must must have that the pair $(X,\frac{1}{p^{e_{0}}}D)$ is sharply $F$ -split. Thus, there is an $e>0$ such that the natural map

\psi_{e}(D):\mathcal{O}_{X}\to F^{e}_{*}(\mathcal{O}_{X}(\lceil\frac{p^{e}-1}{p^{e_{0}}}\rceil D)

(3.1)

splits. Since the same holds for $\psi_{ne}(D)$ for any natural number $n\geq 1$ (see [Sch08, Proposition 3.3] for the proof), we get the map:

\psi_{ee_{0}}(D):\mathcal{O}_{X}\to F_{*}^{ee_{0}}(\mathcal{O}_{X}(\lceil\frac{p^{ee_{0}}-1}{p^{e_{0}}}\rceil D))=F_{*}^{ee_{0}}(\mathcal{O}_{X}(p^{(e-1)e_{0}}D))

splits. Note that $\psi_{e_{0}}$ as defined in Equation 3.1 matches with the map considered in the statement of the Lemma.

Let $U\subset X$ denote the regular locus of $X$ . Since $\phi_{e_{0}}$ is a map between reflexive sheaves, to show that it splits, it sufficient to show that its restriction of $U$ splits. Over $U$ , we may construct the map $\psi_{ee_{0}}$ as follows: First consider the map

\phi_{(e-1)e_{0}}:\mathcal{O}_{U}(D)\to F_{*}^{(e-1)e_{0}}\mathcal{O}_{U}(p^{(e-1)e_{0}}D)

obtained by twisting the $(e-1)e_{0}^{\text{th}}$ -iterate of the Frobenius map by the invertible sheaf $\mathcal{O}_{U}(D)$ . If $f$ denotes the local equation of $D$ , then $\phi_{(e-1)e_{0}}$ is defined by sending

f\mapsto F_{*}^{(e-1)e_{0}}f^{p^{(e-1)e_{0}}}.

Then, after restricting to $U$ , we have that

\psi_{ee_{0}}=F_{*}^{e_{0}}\phi_{(e-1)e_{0}}\circ(\psi_{e_{0}}|_{U})

where the right hand side is the composition

F_{*}^{e_{0}}\phi_{(e-1)e_{0}}\circ(\psi_{e_{0}}|_{U}):\mathcal{O}_{U}\to F_{*}^{e_{0}}\mathcal{O}_{U}(D)\to F_{*}^{ee_{0}}(\mathcal{O}_{X}(p^{(e-1)e_{0}}D)).

Therefore, if $\psi_{ee_{0}}$ splits then so does $\psi_{e_{0}}$ . This proves that part (3) implies part (1), completing the proof of the lemma. ∎

Proof of 3.4.

Set $\alpha=\alpha_{F}(S)$ , and $\alpha^{\prime}=\alpha^{\prime}(S)$ (which is defined in the satement of the Proposition). First we will prove that $\alpha\leq\alpha^{\prime}$ . This is clear if $\alpha=0$ , so we assume that $\alpha$ is positive. For any non-zero element $f\in S_{m}$ , by definition of $\alpha$ , we must have $\text{fpt}_{\mathfrak{m}}(S,f)\geq\frac{\alpha}{m}$ . So, if $m\leq\alpha(p^{e}-1)$ , we have

\text{fpt}_{\mathfrak{m}}(S,f)\geq\frac{\alpha}{m}\geq\frac{\alpha}{\alpha(p^{e}-1)}=\frac{1}{p^{e}-1}.

Thus, by 3.6, the map $R\to F^{e}_{*}R$ sending $1$ to $F^{e}_{*}f$ splits. Since $f$ was an arbitrary non-zero element of degree $m$ , this shows that $I_{e}(m)=0$ whenever $m\leq p^{e}-1$ . Therefore, $\alpha$ belongs to the set $\mathscr{A}(S)$ , which proves that $\alpha\geq\alpha^{\prime}$ .

Next we prove that $\alpha\geq\alpha^{\prime}$ . Fix any $\lambda<\alpha^{\prime}$ . Then, whenever $m\leq\lambda(p^{e}-1)$ and $f\in S_{m}$ is any non-zero element, we know that $f$ is not contained in $I_{e}(m)$ . By 3.6, we have $\text{fpt}_{\mathfrak{m}}(S,f)\geq\frac{1}{p^{e}-1}$ . In other words, if $e$ is the smallest integer such that $m\leq\lambda(p^{e}-1)$ , (equivalently, $e=\lceil\log_{p}(\frac{m}{\lambda})\rceil$ ), then $\text{fpt}_{\mathfrak{m}}(S,f)\geq\frac{1}{p^{e}-1}.$ Now combining this with the fact that $\text{fpt}_{\mathfrak{m}}(S,f^{a})=\frac{\text{fpt}_{\mathfrak{m}}(S,f)}{a}$ for any integer $a$ to get:

\text{fpt}_{\mathfrak{m}}(S,f)\geq\sup_{a\geq 1}\frac{a}{p^{\lceil\log_{p}(\frac{am}{\lambda})\rceil}-1}\geq\frac{\lambda}{m}.

(3.2)

To see the right inequality, we make the following observations: Fixing $m$ and $\lambda$ and for any $a$ , write

\lceil\log_{p}(a)+\log_{p}(\frac{m}{\lambda})\rceil=\log_{p}(a)+\log_{p}(\frac{m}{\lambda})+\varepsilon(a)

for some non-negative real number $\varepsilon(a)$ . Then, we have

\inf_{a\geq 1}\frac{p^{\log_{p}(a)+\log_{p}(\frac{m}{\lambda})+\varepsilon(a)}-1}{a}=\inf_{a\geq 1}\frac{m}{\lambda}p^{\varepsilon(a)}-\frac{1}{a}\leq\inf_{a\geq 1}\frac{m}{\lambda}p^{\varepsilon(a)}

for each $a\geq 1$ . So it is sufficient to show that

\inf_{a\geq 1}p^{\varepsilon(a)}=1.

This is true because given any real number $\gamma$ , the infimum of the set $\{\lceil\gamma+\log_{p}(a)\rceil-\gamma-\log_{p}(a)\,|\,a\in\mathbb{N}\}$ is zero. This proves the inequality in (3.2) .

Since $f$ was an arbitrary non-zero homogeneous element of degree $m$ , it follows from Equation 3.2 that $\alpha\geq\lambda$ . Since $\lambda$ was an arbitrary number smaller than $\alpha^{\prime}$ , we must have $\alpha\geq\alpha^{\prime}$ as well. This completes the proof that $\alpha=\alpha^{\prime}$ . ∎

3.2. Finite-degree approximations

Now we will define finite-degree approximations to the $\alpha_{F}$ -invariant. This establishes a limit formula for the $\alpha_{F}$ -invariant that is analogous to the $F$ -signature (see 2.6 and the classical definition in [Tuc12]).

Definition 3.7.

Let $S$ be an $\mathbb{N}$ -graded section ring over $k$ . For each integer $e\geq 1$ , we define

m_{e}(S):=\max\{m\geq 0\,|\,I_{e}(m)=0\}

and define

\alpha_{e}(S):=\frac{m_{e}(S)}{p^{e}}.

Theorem 3.8.

Let $S$ be a strongly $F$ -regular $\mathbb{N}$ -graded section ring over $k$ . Then, we have

\lim_{e\to\infty}\alpha_{e}(S)=\alpha_{F}(S).

In particular, the limit exists. See 3.2 for the definition of $\alpha_{F}(S)$ .

Lemma 3.9.

Let $S$ be a strongly $F$ -regular $\mathbb{N}$ -graded section ring over $k$ . For any $e\geq 1$ , we have

\alpha_{e}(S)+\frac{1}{p^{e}}\geq\alpha_{e+1}(S)+\frac{1}{p^{e+1}}.

Proof.

First note that since $S$ is strongly $F$ -regular, $S$ is a normal domain. For any $e\geq 1$ , let $0\neq f$ be an element of $I_{e}(m_{e}+1)$ . Then, we have that $f^{p}$ is a non-zero element of $I_{e+1}(p(m_{e}+1))$ (see [Tuc12, Lemma 4.4]). This proves that

p(m_{e}+1)\geq m_{e+1}+1.

Dividing both sides by $p^{e+1}$ , we obtain the required inequality. ∎

Proof of 3.8.

The sequence $\{\alpha_{e}+\frac{1}{p^{e}}\}_{e\geq 1}$ is decreasing, by 3.9. Since it is a decreasing sequence of non-negative real numbers, the sequence converges to its infimum. Moreover, since the sequence $\frac{1}{p^{e}}$ converges to zero, the sequence $\{\alpha_{e}\}_{e\geq 1}$ also converges and

\lim_{e\to\infty}\alpha_{e}=\inf_{e\geq 1}\{\alpha_{e}+\frac{1}{p^{e}}\}.

It remains to show that the limit is equal to $\alpha_{F}(S)$ . Using the definition of $\alpha_{e}$ , we have that

\alpha_{F}(S)\leq\frac{p^{e}}{p^{e}-1}(\alpha_{e}+\frac{1}{p^{e}})=\frac{m_{e}+1}{p^{e}-1}

for each $e\geq 1$ . This is because we know that $\alpha_{F}(S)$ belongs to the set $\mathscr{A}(S)$ from 3.4. Taking a limit over $e$ , we obtain

\alpha_{F}(S)\leq\lim_{e\to\infty}\alpha_{e}.

For the reverse inequality, setting $\alpha:=\lim\alpha_{e}$ , we note that

\alpha(p^{e}-1)\leq(\alpha_{e}+\frac{1}{p^{e}})(p^{e}-1)<p^{e}\alpha_{e}+1.

By the definition of $m_{e}=p^{e}\alpha_{e}$ , the subspace $I_{e}(m)$ is equal to zero for each $m\leq\alpha(p^{e}-1)\leq m_{e}$ . Thus, $\alpha$ belongs to the set $\mathscr{A}(S)$ defined in 3.4. Since $\alpha_{F}(S)$ is the supremum of $\mathscr{A}(S)$ , we get that $\alpha_{F}(S)\geq\alpha$ . This completes the proof of 3.8. ∎

3.3. Positivity and comparison to the $F$ -signature.

Next we will show that the $\alpha_{F}$ -invariant is positive by comparing it to the $F$ -signature (2.6). Recall that for a section ring $S$ of any globally $F$ -regular projective variety, there exists a positive constant $C$ such that for any $e>0$ and any $m\leq C\,p^{e}$ , we have $I_{e}(m)=S_{m}.$ This follows from [LP23, Theorem 4.9].

Theorem 3.10.

The $\alpha_{F}$ -invariant of a strongly $F$ -regular section ring is positive. Moreover, setting $\alpha=\alpha_{F}(S)$ and fixing a constant $C$ as discussed above (so that for any $e>0$ and any $m\leq C\,p^{e}$ , we have $I_{e}(m)=S_{m}$ ), we have the following comparisons:

\frac{e(S)\,\alpha^{\dim(S)}}{\dim(S)!}\,\leq\,\mathscr{s}(S)\,\leq\frac{e(S)}{\dim(S)!}\,\big{(}C^{\dim(S)}-(C-\alpha)^{\dim(S)}\big{)}

(3.3)

where $e(S)$ denotes the Hilbert-Samuel multiplicity of $S$ .

Lemma 3.11.

Given a non-zero homogeneous element $f$ in $S$ of degree $n$ , let $\lambda>n\,\text{fpt}_{\mathfrak{m}}(S,f)$ be a real number. Then,

\mathscr{s}(S)\leq\frac{e(S)}{\dim(S)!}\,\big{(}C^{\dim(S)}-(C-\lambda)^{\dim(S)}\big{)}.

(3.4)

Proof.

Since we have assumed that $\lambda>n\,\text{fpt}_{\mathfrak{m}}(S,f)$ , there exist integers $a,e_{0}>0$ such that

\text{fpt}_{\mathfrak{m}}(S,f)<\frac{a}{p^{e_{0}}-1}<\frac{\lambda}{d}.

Replacing $f$ by $f^{a}$ , by 3.6 we may assume that the map $S\to F_{*}^{e_{0}}S$ defined by $1\to F_{*}^{e_{0}}f$ does not split (since $\text{fpt}_{\mathfrak{m}}(S,f^{a})<\frac{1}{p^{e_{0}}-1})$ . We may also assume that $n\leq\lambda\,(p^{e_{0}}-1)$ . Now, since $f$ belongs to the ideal $I_{e_{0}}$ , we have $S_{m-n}\cdot f\subset I_{e_{0}}(m)$ for any $m\geq d$ yielding the inequality

\dim_{k}\frac{S_{m}}{I_{e_{0}}(m)}\leq\dim_{k}S_{m}-\dim_{k}S_{m-n}.

(3.5)

for all $m\geq n$ . Further, setting $v_{r}=\frac{p^{re_{0}}-1}{p^{e_{0}}-1}$ for any integer $r$ , we have that $\text{fpt}_{\mathfrak{m}}(S,f^{v_{r}})<\frac{1}{p^{re_{0}}-1}$ , and so $f^{v_{r}}$ belongs to $I_{re_{0}}(nv_{r})$ . Therefore, we similarly have

\dim_{k}\frac{S_{m}}{I_{ne_{0}}(m)}\leq\dim_{k}S_{m}-\dim_{k}S_{m-nv_{r}}

(3.6)

for all $m\geq nv_{r}$ . Then, using the 2.11 we may compute the $F$ -signature $\mathscr{s}(S)$ as follows:

\mathscr{s}(S)=\frac{1}{[k^{\prime}:k]}\,\lim_{r\to\infty}\frac{\sum\limits_{m=0}^{m=Cp^{re_{0}}}\dim_{k}\frac{S_{m}}{I_{re_{0}}(m)}}{p^{re_{0}\dim(S)}}\leq\frac{1}{[k^{\prime}:k]}\,\Bigg{(}\lim_{r\to\infty}\frac{\sum\limits_{m=0}^{m=Cp^{re_{0}}}\dim_{k}S_{m}}{p^{re_{0}\dim(S)}}-\frac{\sum\limits_{m=dv_{r}}^{m=Cp^{re_{0}}}\dim_{k}S_{m-nv_{r}}}{p^{re_{0}\dim(S)}}\Bigg{)},

where $k^{\prime}$ is the field $S_{0}$ . Here we have used Equation 3.6 and the defining property of the constant $C$ . Finally, calculating the dimensions in the above inequality using the formula

\dim_{k}S_{m}=[k^{\prime}:k]\,\frac{e(S)}{(\dim S-1)!}m^{\dim S-1}+o(m^{\dim S-2}),

we obtain

\mathscr{s}(S)\leq\frac{e(S)}{\dim(S)!}\big{(}C^{\dim(S)}-(C-\frac{n}{p^{e_{0}}-1})^{\dim(S)}\big{)}.

The proof of the lemma is now complete by using the fact that $\lambda\geq\frac{n}{p^{e_{0}}-1}$ . ∎

Proof of 3.10.

We note that if $\alpha_{F}(S)=0$ , then the rightmost inequality of Equation 3.3 implies that the $F$ -signature $\mathscr{s}(S)$ is zero. But this is a contradiction since $S$ was assumed to be strongly $F$ -regular (see [AL03, Theorem 0.2]). So the positivity of $\alpha_{F}(S)$ follows from Equation 3.3, which we will now prove.

The rightmost inequality follows from 3.11 by taking a limit as $\lambda\to\alpha_{F}(S)$ , since the Lemma applies to each $\lambda$ such that $\lambda>\text{fpt}_{\mathfrak{m}}(S,f)\,\deg(f)$ for some non-zero $f$ .

To prove the leftmost inequality, we use 2.11 again to compute the $F$ -signature of $S$ and we observe that for any $e\geq 1$ ,

\mathscr{s}(S)=\lim_{e\to\infty}\frac{\sum\limits_{m=0}^{m=Cp^{e}}\dim_{k}\frac{S_{m}}{I_{e}(m)}}{p^{e\dim(S)}}\geq\lim_{e\to\infty}\frac{\sum\limits_{m=0}^{m=m_{e}}\dim_{k}S_{m}}{p^{e\dim(S)}}.

(3.7)

Recall that for any $e\geq 1$ , $m_{e}$ is the largest $m$ such that $I_{e}(m)=0$ , which justifies Equation 3.7. But the right hand side is equal to

\lim_{e\to\infty}\frac{e(S)}{\dim(S)!}\big{(}\frac{m_{e}}{p^{e}}\big{)}^{\dim(S)}.

We conclude the proof by using 3.8, which says that $\lim_{e\to\infty}\frac{m_{e}}{p^{e}}=\alpha_{F}(S)$ . This concludes the proof of Equation 3.3 and thus of 3.10. ∎

Remark 3.12.

The positivity of the $\alpha_{F}$ -invariant proved in 3.10 can also be deduced from the main theorem of [Sat18].

3.4. Behaviour under certain ring extensions.

In this subsection, we record some useful results on the behaviour of the $\alpha_{F}$ -invariant under suitably nice extensions of section rings.

Proposition 3.13.

Let $S$ and $S^{\prime}$ be two $\mathbb{N}$ -graded section rings (of possibly different varieties) and $S\subset S^{\prime}$ be an inclusion such that for a fixed integer $n\geq 1$ and any other $m$ , all degree $m$ elements of $S$ are mapped to degree $nm$ elements of $S^{\prime}$ . Further, assume that the inclusion $S\subset S^{\prime}$ splits as a map of $S$ -modules. Then, we have

\alpha_{F}(S)\geq\frac{\alpha_{F}(S^{\prime})}{n}.

(3.8)

Moreover, equality holds in (3.8) if $S$ is the $n^{\text{th}}$ -Veronese subring of $S^{\prime}$ .

Proof.

The first part follows immediately from 3.8 and the fact that a homogeneous element $f$ of $S$ splits from $F^{e}_{*}S$ if it splits from $F^{e}_{*}S^{\prime}$ . In other words, we have

I_{e}(S_{m})\subset I_{e}(S^{\prime}_{mn})

for any $e$ and $m$ . For the statement about Veronese subrings, the key observation is that since $S^{\prime}$ is a section ring (of $(X,L)$ , say), then

I_{e}(S^{\prime}_{mn})=I_{e}(S_{m})=I_{e}(X,L^{mn}).

Thus, the equality again follows from 3.8. ∎

Remark 3.14.

The $F$ -signature of section rings also transforms in a similar manner to the $\alpha_{F}$ -invariant above. Indeed, see [VK12, Theorem 2.6.2] and its generalization in [CR17, Theorem 4.8]. A simple proof of this transformation rule for section rings can also be obtained by using 2.14.

Proposition 3.15.

Let $(S,\mathfrak{m})\subset(S^{\prime},\mathfrak{n})$ be a degree preserving map of $\mathbb{N}$ -graded, strongly $F$ -regular section rings (of possibly different varieties and over possibly different perfect fields). Assume that both $S$ and $S^{\prime}$ are generated in degree one , $S^{\prime}$ is flat over $S$ , and that the ring $S^{\prime}/\mathfrak{m}S^{\prime}$ is regular. Then, we have

\alpha_{F}(S^{\prime})=\alpha_{F}(S).

Proof.

First note that since $S$ and $S^{\prime}$ are generated in degree $1$ , the $\alpha_{F}$ -invariant must be at most $1$ for both of them. Next, for any $e\geq 1$ and $m\leq p^{e}-1$ , we may apply Claim 3.4 in [CRST21, Proof of Theorem 3.1] to get that

I_{e}(S,m)\,S^{\prime}=I_{e}(S^{\prime},m).

This implies that $m_{e}(S)=m_{e}(S^{\prime})$ for any $e\geq 1$ . Now, the Proposition follows immediately by using 3.8. ∎

Recall that $k$ was assumed to be a perfect field of characteristic $p>0$ .

Corollary 3.16.

Let $S$ be a strongly $F$ -regular section ring over $k$ and $K$ be an arbitrary perfect field extension of $k$ . Then, the base-change $S\otimes_{k}K$ is isomorphic to a product of strongly $F$ -regular section rings $S^{i}$ over finite extensions of $K$ and for each $i$ , we have

\alpha_{F}(S)=\alpha_{F}(S^{i}).

Proof.

Firstly, we may assume that $S$ is generated in degree one by using 3.13. Set $S^{\prime}=S\otimes_{k}K$ . Since $k$ is perfect and $S/\mathfrak{m}$ is a finite separable extension of $k$ , we see that

S^{\prime}/\mathfrak{m}S^{\prime}\cong S/\mathfrak{m}\otimes_{k}K\cong\prod_{i}L_{i}

is a finite product of perfect fields $L_{i}$ . Thus, if $S$ was the section ring of $X$ , then $S^{\prime}\cong\prod_{i}S^{i}$ where $S^{i}$ is the section ring of $X\times_{S/\mathfrak{m}}L_{i}$ . Note that $S^{i}$ is isomorphic to $S\otimes_{S/\mathfrak{m}}L_{i}$ , and hence each $S^{i}$ is strongly $F$ -regular by [CRST21, Theorem 3.6]. The corollary now follows from 3.15 by applying it to each inclusion $S\subset S^{i}$ . ∎

Remark 3.17.

In the setting of 3.16, the same results also holds for the $F$ -signature of $S$ instead of the $\alpha_{F}$ -invariant by using [Yao06, Theorem 5.6] in place of 3.15. This allows to reduce computing both the $F$ -signature and the $\alpha_{F}$ -invariant of a section ring to the geometrically connected case, i.e., when $S_{0}=k$ .

Remark 3.18.

The results of this section naturally extend to the more general setting of globally $F$ -regular pairs $(X,\Delta)$ such that $(p^{e}-1)\Delta$ is an integral Weil-divisor. This will be addressed in a future version of the paper.

4. The $\alpha_{F}$ -invariant of globally $F$ -regular Fano varieties.

In this section, we specialize the study of the $\alpha_{F}$ -invariant to the case of globally $F$ -regular Fano varieties (and when the ample divisor is a multiple of $-K_{X}$ ). We begin by defining what we mean by a $\mathbb{Q}$ -Fano variety in positive characteristic. Recall that $k$ denotes a perfect field of characteristic $p>0$ .

Definition 4.1.

A $\mathbb{Q}$ -Fano variety $X$ is a projective variety over $k$ such that

(1)

$X$ is locally strongly $F$ -regular (2.16).
(2)

$K_{X}$ is a $\mathbb{Q}$ -Cartier divisor.
(3)

$-K_{X}$ is ample.

Note that since $X$ has only strongly $F$ -regular singularities, $X$ is automatically normal and Cohen-Macaulay. In particular, we may define the canonical Weil-divisor $K_{X}$ by extending a canonical divisor from the smooth locus. In fact, $\omega_{X}=\mathcal{O}_{X}(K_{X})$ is a dualizing sheaf over $X$ . In particular, we have

H^{d}(X,\omega_{X})\cong k

(4.1)

where $d$ is the dimension of $X$ . Moreover, the second and third conditions in Definition 4.1 guarantee that there is a positive integer $r$ such that $rK_{X}$ is Cartier and $-rK_{X}$ is ample. The smallest such $r$ is called the index of $K_{X}$ .

Definition 4.2.

Let $X$ be a globally $F$ -regular $\mathbb{Q}$ -Fano variety over $k$ and $r$ be a positive integer divisible by the index of $K_{X}$ . Let

S:=S(X,-rK_{X})=\bigoplus_{m\geq 0}H^{0}(X,\mathcal{O}_{X}(-mrK_{X}))

denote the section ring of $X$ with respect to $-rK_{X}$ . Then, the $\alpha_{F}$ -invariant of $X$ is defined to be

\alpha_{F}(X):=r\,\alpha_{F}(S)

where $\alpha_{F}(S)$ denotes the $\alpha_{F}$ -invariant of the strongly $F$ -regular ring $S$ , as defined in Defintion 3.2.

By taking the affine cone over a $\mathbb{Q}$ -Fano variety, we also define the global $F$ -signature of the a $\mathbb{Q}$ -Fano variety.

Definition 4.3 ( $F$ -signature).

Let $X$ be a $\mathbb{Q}$ -Fano variety over $k$ and $r$ denote a positive integer divisible by the index of $K_{X}$ . Let

S:=S(X,-rK_{X})=\bigoplus_{m\geq 0}H^{0}(X,\mathcal{O}_{X}(-mrK_{X}))

denote the section ring of $X$ with respect to $-rK_{X}$ . Then, the $F$ -signature of $X$ is defined to be

\mathscr{s}(X):=r\,\mathscr{s}(S)

where $\mathscr{s}(S)$ denotes the $F$ -signature of $S$ , as defined in Defintion 2.6.

Remark 4.4.

Though the definitions of the $\alpha_{F}$ -invariant and the $F$ -signature involve making a choice of a multiple of the index of $K_{X}$ , both invariants are well-defined thanks to 3.13 (for the $\alpha_{F}$ -invariant) and [VK12, Theorem 2.6.2] (for the $F$ -signature).

Theorem 4.5.

Let $X$ be a globally $F$ -regular $\mathbb{Q}$ -Fano variety of positive dimension. Then, $\alpha_{F}(X)$ is at most $1/2$ .

Proof of 4.5:.

Let $d$ denote the dimension of $X$ , and $r\gg 0$ be an integer divisible by the index of $K_{X}$ and such that $H^{0}(X,\mathcal{O}_{X}(-mK_{X}))\neq 0$ for all $m\geq r$ . First, we claim that there is an integer $n>0$ such that

\dim_{k}H^{0}(X,\mathcal{O}_{X}(-mK_{X}))<\dim_{k}H^{0}(X,\mathcal{O}_{X}(-(m+n)K_{X}))

for all $m\gg 0$ . This is clear if $-K_{X}$ is a Cartier divisor (and we may take $n=1$ in this case), since $\dim_{k}H^{0}(X,\mathcal{O}_{X}(-mK_{X}))$ is a polynomial in $m$ of degree $d>0$ and a positive leading term (because $-K_{X}$ is ample). More generally, by the asymptotic Riemann-Roch formula ([Laz04, Example 1.2.19]), for each $0\leq i\leq r-1$ , there exists polynomials $P_{i}$ of degree $d$ such that for all $m\gg 0$

\dim_{k}H^{0}(X,\mathcal{O}_{X}(-(i+mr)K_{X}))=P_{i}(m).

Moreover, setting $V=(-rK_{X})^{d}$ , each $P_{i}$ has the form

P_{i}(m)=\frac{V}{d!}m^{d}+Q_{i}(m)

for polynomials $Q_{i}$ of degree at most $d-1$ . In this situation, the existence of an integer $n$ as required is guaranteed by 4.6 stated and proved below.

Assume, for the sake of contradiction, that $\alpha_{F}(X)>\frac{1}{2}+\varepsilon$ for some small $\varepsilon>0$ . Then, note that by 3.4, for all $e\gg 0$ , we have $I_{e}(-mrK_{X})=0$ for all $m\leq\frac{p^{e}-1}{2r}+\frac{\varepsilon}{r}(p^{e}-1)$ .

Now, for $e\gg 0$ , we can find an integer $m$ satisfying the following properties:

•

$mr<\frac{p^{e}-1}{2}$ ,
•

$p^{e}-1-mr+2r<\frac{p^{e}-1}{2}+\varepsilon(p^{e}-1)$ , and,
•

$n\leq p^{e}-1-2mr$ .

This is equivalent to finding an integer $m$ such that

\frac{p^{e}-1}{2r}+2-\frac{\varepsilon}{r}(p^{e}-1)\leq m\leq\frac{p^{e}-1}{2r}-\frac{n}{2r},

which is possible since $n$ is fixed and $\frac{\varepsilon}{2}(p^{e}-1)\to\infty$ as $e\to\infty$ . The third condition on $m$ guarantees that

\dim_{k}H^{0}(X,-mrK_{X})<\dim_{k}H^{0}(X,-(p^{e}-1-mr)K_{X}).

(4.2)

The second condition guarantees that there exists a non-zero effective Weil divisor $E\geq 0$ that induces an injective map

\mathcal{O}_{X}(-(p^{e}-1-mr)K_{X})\hookrightarrow\mathcal{O}_{X}(-m^{\prime}rK_{X})

for some $m<m^{\prime}\leq\frac{p^{e}-1}{2r}+\frac{\varepsilon}{r}(p^{e}-1)$ . Therefore, we know that $I_{e}(-mrK_{X})=I_{e}(-m^{\prime}rK_{X})=0$ as noted above. By [LP23, Lemma 4.12], this implies that $I_{e}(-(p^{e}-1-mr)K_{X})=0$ as well. Finally, note that 2.13 applied to $D=-mrK_{X}$ tells us that

\dim_{k}\frac{H^{0}(X,\mathcal{O}_{X}(-mrK_{X}))}{I_{e}(-mrK_{X})}=\dim_{k}\frac{H^{0}(X,\mathcal{O}_{X}(-(p^{e}-1-mr)K_{X}))}{I_{e}(-(p^{e}-1-mr)K_{X})}

But since both the $I_{e}$ -subspaces in the above equation are zero, this is in contradiction to Equation 4.2. This proves that $\alpha_{F}(X)$ is at most 1/2. ∎

The following lemma was used in the proof of 4.5.

Lemma 4.6.

For each $0\leq i<r$ , let $P_{i}$ be a polynomial with real coefficients. Moreover, assume that all the $P_{i}$ ’s have the same positive degree and the same positive leading term. In other words, for each $0\leq i\leq r-1$ , there exists a real polynomial $Q_{i}$ of degree at most $d-1$ such that

P_{i}(m)=a_{d}m^{d}+Q_{i}(m)

for some real number $a_{d}>0$ (independent of $i$ ). Then, there is an integer $n\gg 0$ such that for all integers $m\gg 0$ , and all pairs $0\leq i,j\leq r-1$ , we have

P_{i}(m)\leq P_{j}(m+n).

Proof.

Since each $Q_{i}$ is a real polynomial of degree at most $d-1$ , we may find positive constants $C_{1},C_{2}$ such that

-C_{2}m^{d-1}\leq Q_{i}(m)\leq C_{1}m^{d-1}

for each $m\geq 0$ and each $0\leq i\leq r-1$ . Now, it is sufficient to find a constant $n$ such that

a_{d}m^{d}+C_{1}m^{d-1}\leq a_{d}(m+n)^{d}-C_{2}(m+n)^{d-1}

(4.3)

for all $m\gg 0$ . For this, expanding $(m+n)^{d}$ using the binomial theorem, Equation 4.3 is equivalent to

C_{1}m^{d-1}\leq a_{d}(dnm^{d-1}+\dots+dmn^{d-1}+n^{d})-C_{2}(m+n)^{d-1}

First, we note that we may choose $n\gg 0$ such that $\frac{a_{d}dn}{2}$ is larger than both $C_{1}$ and $C_{2}$ . This implies that $\frac{a_{d}dn}{2}m^{d-1}>C_{1}m^{d-1}$ for all $m\geq 0$ . Furthermore, having chosen the $n\gg 0$ as before, we have $\frac{a_{d}dn}{2}m^{d-1}\geq C_{2}(m+n)^{d-1}$ for all $m\gg 0$ . This proves the lemma. ∎

For the rest of this section, we assume that $H^{0}(X,\mathcal{O}_{X})=k$ , i.e., that $X$ is geometrically connected. However, see 3.17 for ways to extend the results to more general cases. In the case of Fano varieties, we have a stronger version of comparison of the $\alpha_{F}$ -invariant to the $F$ -signature than the formula in 3.10:

Theorem 4.7.

Let $X$ be a $d$ -dimensional globally $F$ -regular $\mathbb{Q}$ -Fano variety over $k$ . Assume that $H^{0}(X,\mathcal{O}_{X})=k$ and that $d$ is positive. Set $\alpha=\alpha_{F}(X)$ . Then, we have the following inequalities relating the $F$ -signature and the $\alpha_{F}$ -invariant:

\frac{2\,\alpha^{d+1}\,\mathrm{vol}(X)}{(d+1)!}\leq\mathscr{s}(X)\leq\frac{2\,\big{(}(\frac{1}{2})^{d+1}-(\frac{1}{2}-\alpha)^{d+1}\big{)}\,\mathrm{vol}(X)}{(d+1)!}

(4.4)

Here, $\mathrm{vol}(X)$ denotes the volume of the $\mathbb{Q}$ -Cartier divisor $-K_{X}$ .

Corollary 4.8.

Let $X$ be a globally $F$ -regular $\mathbb{Q}$ -Fano variety of dimension $d>0$ . Then,

(1)

We have

$\mathscr{s}(X)\leq\frac{\mathrm{vol}(X)}{2^{d}(d+1)!}.$
(2)

Moreover, $\alpha_{F}(X)$ is equal to $1/2$ if and only if the value of the $F$ -signature $\mathscr{s}(X)$ is equal to

$\frac{\mathrm{vol}(X)}{2^{d}(d+1)!}.$

For $\mathbb{Q}$ -Fano varieties, the $F$ -signature has a more refined formula than 2.11, which we prove next.

Proposition 4.9.

Let $X$ be a globally $F$ -regular $\mathbb{Q}$ -Fano variety over $k$ . Assume that $H^{0}(X,\mathcal{O}_{X})=k$ . Let $r$ be an integer such that $rK_{X}$ is Cartier. Then, the $F$ -signature of $X$ can be computed as

\mathscr{s}(X)=\lim_{e\to\infty}\frac{2r\,\sum\limits_{m=0}^{\lfloor\frac{p^{e}-1}{2r}\rfloor}\dim_{k}\frac{H^{0}(-mrK_{X})}{I_{e}(-mrK_{X})}}{p^{e(\dim(X)+1)}}

Lemma 4.10.

Let $X$ be a globally $F$ -regular $\mathbb{Q}$ -Fano variety that is geometrically connected over $k$ . Then, for any $r>0$ , we have $H^{0}(-mrK_{X})=I_{e}(-mrK_{X})$ whenever $m>\frac{p^{e}-1}{r}$ .

Proof.

We will prove this lemma in two different ways, since both ideas may be useful in other situations.

Proof 1:

Let $m>\frac{p^{e}-1}{r}$ . By definition of the subspace $I_{e}$ (2.8), it is sufficient to show that there are no non-zero maps $\phi:F^{e}_{*}\mathcal{O}_{X}(-mrK_{X})\to\mathcal{O}_{X}$ . We have

\operatorname{Hom}_{\mathcal{O}_{X}}(F^{e}_{*}\mathcal{O}_{X}(-mrK_{X},\mathcal{O}_{X})\cong H^{0}(X,(1-p^{e}+mr)K_{X}).

By assumption, we have $1-p^{e}+mr>0$ . Since $-rK_{X}$ is ample, this means that $H^{0}(X,(1-p^{e}+mr)K_{X})=0$ . Hence, there are no non-zero maps $\phi:F^{e}_{*}\mathcal{O}_{X}(-mrK_{X})\to\mathcal{O}_{X}$ , which proves the lemma.

Proof 2:

Let $m>\frac{p^{e}-1}{r}$ and suppose that there is a non-zero global section $f\in H^{0}(-mrK_{X})$ that is not in $I_{e}(-mrK_{X})$ . Then, by definition of $I_{e}$ , we have a map

\phi\in\operatorname{Hom}_{\mathcal{O}_{X}}(F^{e}_{*}\mathcal{O}_{X}(-mrK_{X}),\mathcal{O}_{X})

such that $\phi(F^{e}_{*}f)=1$ . Thus, we have the splitting

\mathcal{O}_{X}\hookrightarrow F^{e}_{*}\mathcal{O}_{X}(-mrK_{X})\twoheadrightarrow\mathcal{O}_{X}

(4.5)

where the first map is got by sending $1\to F^{e}_{*}f$ and the second map is $\phi$ . Twisting equation 4.5 by $\mathcal{O}_{X}(K_{X})$ and reflexifying, we obtain a splitting:

\mathcal{O}_{X}(K_{X})\hookrightarrow F^{e}_{*}\mathcal{O}_{X}(-(mr-p^{e})K_{X})\twoheadrightarrow\mathcal{O}_{X}(K_{X})

(4.6)

By assumption, $mr-p^{e}$ is non-negative. Hence, $H^{d}(X,F^{e}_{*}\mathcal{O}_{X}(-(mr-p^{e})K_{X}))=0$ by 2.22, in turn implying that $H^{d}(X,\mathcal{O}_{X}(K_{X}))=0$ (using the splitting in equation 4.6). This is a contradiction, since $\mathcal{O}_{X}(K_{X})$ is the canonical sheaf of $X$ . This completes the proof of 4.10. ∎

Proof of 4.9.

Let $S$ denote the section ring of $X$ with respect to $-rK_{X}$ . Fix an $e>0$ and let $a_{e}$ denote the free rank of $F^{e}_{*}S$ as an $S$ -module (2.5). Recall that by 2.11, we have

a_{e}=\sum_{m=0}^{\infty}\dim_{k}\frac{H^{0}(-mrK_{X})}{I_{e}(-mrK_{X})}

(4.7)

so that

\mathscr{s}(X)=r\,\lim_{e\to\infty}\frac{a_{e}}{p^{e(\dim(X)+1)}}.

Then, 4.10 shows that the terms of the sum in Equation 4.7 are zero for $m>\frac{p^{e}-1}{r}$ . Furthermore, using 2.13, we have

\sum_{m=\lceil\frac{p^{e}-1}{2r}\rceil}^{\lfloor\frac{p^{e}-1}{r}\rfloor}\dim_{k}\frac{H^{0}(-mrK_{X})}{I_{e}(-mrK_{X})}=\sum_{m=\lceil\frac{p^{e}-1}{2r}\rceil}^{\lfloor\frac{p^{e}-1}{r}\rfloor}\dim_{k}\frac{H^{0}(-(p^{e}-1-mr)K_{X})}{I_{e}(-(p^{e}-1-mr)K_{X})}.

Let $a$ be an integer between $0$ and $r$ such that $p^{e}-1\equiv a\mod r$ . Hence, we have

\sum_{m=\lceil\frac{p^{e}-1}{2r}\rceil}^{\lfloor\frac{p^{e}-1}{r}\rfloor}\dim_{k}\frac{H^{0}(-(p^{e}-1-mr)K_{X})}{I_{e}(-(p^{e}-1-mr)K_{X})}=\sum_{m=0}^{\lfloor\frac{p^{e}-1}{2r}\rfloor}\dim_{k}\frac{H^{0}(-(a+mr)K_{X})}{I_{e}(-(a+mr)K_{X})}.

Thus, we have

a_{e}=\sum_{m=0}^{\lfloor\frac{p^{e}-1}{2r}\rfloor}\dim_{k}\frac{H^{0}(-mrK_{X})}{I_{e}(-mrK_{X})}+\sum_{m=0}^{\lfloor\frac{p^{e}-1}{2r}\rfloor}\dim_{k}\frac{H^{0}(-(a+mr)K_{X})}{I_{e}(-(a+mr)K_{X})}.

Moreover, using 2.14, we see that there is a constant $C>0$ such that

\left|\frac{H^{0}(-(a+mr)K_{X})}{I_{e}(-(a+mr)K_{X})}-\frac{H^{0}(-mrK_{X})}{I_{e}(-mrK_{X})}\right|<Cp^{e(\dim(X)-1)}.

Thus, we have that

\left|a_{e}-2\sum\limits_{m=0}^{\lfloor\frac{p^{e}-1}{2r}\rfloor}\dim_{k}\frac{H^{0}(-mrK_{X})}{I_{e}(-mrK_{X})}\right|<Cp^{e\dim(X)}.

The proof is now complete since the right hand side limits to zero when divided by $p^{e(\dim(X)+1)}$ and as $e\to\infty$ . ∎

Proof of 4.7.

The proof of this Theorem is exactly the same proof as the proof of Equation 3.3 in 3.10 (see the proof of 3.11), once we replace the formula from 2.11 with the formula from 4.9 to compute the $F$ -signature of $S=S(X,-rK_{X})$ . ∎

Proof of 4.8.

Part (1) follows immediately from the right-hand inequality in 4.7, since we know that $\alpha_{F}(X)\leq\frac{1}{2}$ by 4.5. We also see that if $\alpha_{F}(X)<\frac{1}{2}$ , we must have

\mathscr{s}(X)<\frac{\mathrm{vol}(X)}{2^{d}(d+1)!}.

Thus, Part (2) also follows from Equation 4.4 once we note that when $\alpha_{F}(X)=\frac{1}{2}$ , both sides of the inequality in Equation 4.4 are equal to $\frac{\mathrm{vol}(X)}{2^{d}(d+1)!}$ . ∎

Remark 4.11.

Let $X$ be a $\mathbb{Q}$ -Fano variety over $\mathbb{C}$ . This means that $X$ is a normal variety, $-K_{X}$ is $\mathbb{Q}$ -Cartier and ample and $X$ has only klt singularities. Let $r$ be such that $-rK_{X}$ is Cartier, and $S=S(X,-rK_{X})$ . Then, for any effective $\mathbb{Q}$ -divisor $\Delta$ on $X$ with $\Delta\sim_{\mathbb{Q}}-rK_{X}$ , we have

\text{lct}_{\mathfrak{m}}(S,\Delta_{S})=\min\,\{\,\text{lct}(X,\Delta),\,\frac{1}{r}\,\}

where lct denotes the log canonical threshold and $\Delta_{S}$ denotes the cone over $\Delta$ . This follows from [Kol13, Lemma 3.1]. Thus, if we let

\tilde{\alpha}(X)=r\,\inf\,\{\text{lct}_{\mathfrak{m}}(S,\Delta_{S})\,|\,\Delta\geq 0\,\text{is a $\mathbb{Q}$-divisor on $X$ such that $\Delta\sim_{\mathbb{Q}}-K_{X}$}\,\},

then, we have that

\tilde{\alpha}(X)=\min\,\{\alpha(X),\,1\}.

Therefore, for any $\mathbb{Q}$ -Fano variety with $\alpha(X)\leq 1$ , the $\alpha_{F}$ -invariant from 4.2 is a “Frobenius analog" of the complex $\alpha$ -invariant.

The $\alpha_{F}$ -invariant of toric Fano varieties

Let $k$ denote an algebraically closed field of prime characteristic $p>0$ . Fix a lattice $N\cong\mathbb{Z}^{d}$ and let $M$ be the dual lattice (where $d$ is some positive integer).

Theorem 4.12.

Let $X_{p}$ be a $\mathbb{Q}$ -Fano toric variety over $k$ defined by a fan $\mathcal{F}$ in $N$ . Let $X_{\mathbb{C}}$ be the corresponding complex toric variety (which is also automatically $\mathbb{Q}$ -Fano). Then, we have

\alpha_{F}(X_{p})=\alpha(X_{\mathbb{C}}).

Proof.

Let $v_{1},\dots,v_{n}$ denote the primitive generators for the one dimensional cones in $\mathcal{F}$ and write $-K_{X}=\sum_{i}b_{i}v_{i}$ for rational numbers $b_{i}$ .

First we choose an $r>0$ such that $rb_{i}\in\mathbb{Z}$ for each $i$ and the section ring $S(X,-rK_{X})$ is generated in degree one. Let $P\subset M$ denote the polytope associated to $-rK_{X}$ , and defined by:

P=\{u\in M_{\mathbb{R}}=M\otimes_{\mathbb{Z}}\mathbb{R}\,|\,\langle u,v_{i}\rangle\geq-b_{i}\,\,\text{for all }1\leq i\leq n\}.

Since we are assuming that $S(X,-rK_{X})$ is generated in degree $1$ , the vertices of $P$ are lattice points of $M$ . For any $u\in P\cap M$ , let $D_{u}$ be the corresponding effective divisor in the linear system $|-rK_{X}|$ . By [BJ20, Corollary 7.16], we have that

\alpha(X_{\mathbb{C}})=\min_{u\in P\cap M}\,r\,\text{lct}(X_{\mathbb{C}},D_{u})

(4.8)

where $\text{lct}(X_{\mathbb{C}},-)$ denotes the log canonical threshold of a divisor on $X_{\mathbb{C}}$ . Note that since the vertices of $P$ are lattice points, just the vertices are sufficient to compute $\alpha(X_{\mathbb{C}})$ .

Let $\tilde{P}$ denote the polytope $P\times\{1\}\subset M\times\mathbb{Z}$ . Then, the section ring $S(X,-rK_{X})$ is the semigroup ring associated to the cone over $\tilde{P}$ in $(M\times\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{R}$ . Note that $S$ is $\mathbb{Q}$ -Gorenstein. Therefore, by [Bli04, Theorem 3], we see that for any $\tilde{u}\in\tilde{P}\cap(M\times\mathbb{Z})$ , we have

\text{fpt}_{\mathfrak{m}}(S(X_{p},-K_{X_{p}}),D_{\tilde{u}})=\text{lct}_{\mathfrak{m}}(S(X_{\mathbb{C}},-K_{X_{\mathbb{C}}}),D_{\tilde{u}}).

(4.9)

Next, note that since $X$ is a normal toric variety, it is automatically globally $F$ -regular. Now we prove that the $\alpha_{F}$ -invariant of $X_{p}$ can also be computed by only considering the torus invariant divisors. To see this, let $S=S(X_{p},-rK_{X_{p}})$ and let $f\in S$ be a non-zero homogeneous element. Then, following the discussion in [BJ20, Section 7.4] and [Eis95, Theorem 15.17], there exists an integral weight vector $\mu=(\mu_{1},\dots,\mu_{d+1})$ with $\mu_{i}\in\mathbb{Z}_{>0}$ such that $\text{in}_{>_{\mu}}(f)=\text{in}_{>}(f)$ . Here $>_{\mu}$ denotes the weight monomial order with respect to $\mu$ and $>$ denotes the graded lexicographic monomial order on $S$ . Then, we have a flat degeneration of $f$ to its initial term. In other words, if $f=\sum_{u}\beta_{u}\chi^{u}$ for monomials $\chi^{u}\in S$ , then setting $w=\max\{\langle\mu,u\rangle\,|\,\beta_{u}\neq 0\}$ , the element

\tilde{f}=t^{w}\,\sum_{u}\beta_{u}t^{-\langle\mu,u\rangle}\chi^{u}\in S[t]

satisfies the following properties:

•

Viewing $S[t]$ as a $k[t]$ -algebra, the ring $S[t]/(\tilde{f})$ is a flat $k[t]$ -module.
•

The image of $\tilde{f}$ modulo $t$ is equal to $\text{in}_{>}(f)$ , the initial term of $f$ with respect to the graded lex monomial order on $S$ .
•

For any point $0\neq\lambda\in k$ , the image $f_{\lambda}$ of $\tilde{f}$ in $S[t]/(t-\lambda)$ satisfies

$\text{fpt}_{\mathfrak{m}}(S,f_{\lambda})=\text{fpt}_{\mathfrak{m}}(S,f).$

With this construction in place, we conclude the proof of the theorem with the following lemma:

Lemma 4.13.

For any non-zero homogeneous element $f$ of $S$ , we have

\text{fpt}_{\mathfrak{m}}(S,f)\geq\text{fpt}_{\mathfrak{m}}(S,\text{in}_{>}(f)).

Assuming this lemma for a moment, we see that

\alpha_{F}(X_{p})=\inf_{\tilde{u}\in\tilde{P}\cap(M\times\mathbb{Z})}\text{fpt}_{\mathfrak{m}}(S,D_{\tilde{u}}).

Furthermore, by Equation 4.9, we have

\alpha_{F}(X_{p})=\inf_{\tilde{u}\in\tilde{P}\cap(M\times\mathbb{Z})}\text{lct}_{\mathfrak{m}}(S(X_{\mathbb{C}},-rK_{X_{\mathbb{C}}}),D_{\tilde{u}}).

(4.10)

Since by 4.5 we have $\alpha_{F}(X_{p})\leq 1/2$ , we must have $\text{lct}(S(X_{\mathbb{C}},-rK_{X_{\mathbb{C}}}),D_{\tilde{u}})<\frac{1}{nr}$ for some $\tilde{u}=(u,n)\in\tilde{P}\cap(M\times\mathbb{Z})$ . Note that $D_{u}$ corresponds to a torus-invariant divisor on $X_{\mathbb{C}}$ linearly equivalent to $-nrK_{X_{\mathbb{C}}}$ . Therefore, by 4.11, we have

\text{lct}(X_{\mathbb{C}},D_{u})=\text{lct}_{\mathfrak{m}}(S(X_{\mathbb{C}},-rK_{X_{\mathbb{C}}}),D_{\tilde{u}})

for any $\tilde{u}=(u,n)$ such that $\text{lct}(S(X_{\mathbb{C}},-rK_{X_{\mathbb{C}}}),D_{\tilde{u}})<\frac{1}{nr}$ . Putting this together with Equation 4.10 and Equation 4.8, we get that

\alpha_{F}(X_{p})=\alpha(X_{\mathbb{C}})

as required. ∎

Finally, it remains to prove 4.13.

Proof of 4.13.

By 3.6, it is sufficient to show that for all rational numbers of the form $\frac{a}{p^{e}-1}$ such that

\frac{a}{p^{e}-1}<\text{fpt}_{\mathfrak{m}}(S,\text{in}_{>}(f)),

the map $S\to F^{e}_{*}S$ sending $1$ to $F^{e}_{*}f^{a}$ splits. Equivalently, for all such $\frac{a}{p^{e}-1}$ , it suffices to show that $f^{a}\notin I_{e}(S)$ . Since the pair $(S,\text{in}_{>}(f)^{\frac{a}{p^{e}-1}})$ is strongly $F$ -regular, in particular it is sharply $F$ -split. By 3.6 again, we know that $\text{in}_{>}(f^{a})=(\text{in}_{>}(f))^{a}\notin I_{e}(S)$ . Now, since $S$ is a toric ring, $I_{e}(S)$ is a monomial ideal of $S$ . Therefore, if $\text{in}_{>}(f^{a})\notin I_{e}(S)$ , we also have $f^{a}\notin I_{e}(S)$ as required. ∎

Remark 4.14.

Combining 4.12 with 4.5, we recover the well-known fact that the $\alpha$ -invariant of a toric Fano variety is at most 1/2 (see [LZ22, Corollary 3.6]).

Remark 4.15.

The results of this section also extend naturally to the case of globally $F$ -regular log-Fano pairs $(X,\Delta)$ such that $(p^{e}-1)\Delta$ is an integral Weil-divisor. This will be addressed in a future version of the paper.

5. Semicontinuity properties.

In this section we will examine the behaviour of the $\alpha_{F}$ -invariant in geometric families, analogous to the case of the $F$ -signature which was proved in [CRST21] and the case of complex $\alpha$ -invariant discussed in [BL22]. Throughout this section, we fix $k$ to be an algebraically closed field of characteristic $p>0$ .

5.1. Weak semicontinuity

First, we will prove a result for a family of arbitrary globally $F$ -regular varieties.

Notation 5.1.

Recall that the perfection of a field $K$ (of positive characteristic) is the union

K^{\infty}:=\bigcup_{e=1}^{\infty}K^{1/p^{e}}

of all the $p^{e}$ -th roots of elements of $K$ . For a map $f:\mathcal{X}\to Y$ of varieties over $k$ , and a point $y\in Y$ (not necessarily closed), we denote the fiber of $f$ over $y$ as $\mathcal{X}_{y}$ and denote the perfectified fiber over $y$ as

\mathcal{X}_{y^{\infty}}:=\mathcal{X}\times_{Y}\operatorname{\text{Spec}}(\kappa(y)^{\infty}).

Similar for a coherent sheaf $\mathcal{F}$ on $\mathcal{X}$ , we have the fiber $\mathcal{F}_{y}$ and the perfectified fiber $\mathcal{F}_{y^{\infty}}$ . Furthermore , if $Y=\operatorname{\text{Spec}}(A)$ and $S$ is a finitely generated $A$ -algebra, then for any $y\in Y$ , we denote the perfectified fiber $S\otimes_{A}\kappa(y)^{\infty}$ by $S_{y^{\infty}}$ .

Notation 5.2.

By a family of globally $F$ -regular varieties, we mean that

(1)

We have a flat and projective morphism $f:\mathcal{X}\to Y$ where $\mathcal{X}$ is normal and $Y$ is smooth over $k$ .
(2)

We assume additionally that $f$ has connected fibers (i.e., $f_{*}\mathcal{O}_{\mathcal{X}}=\mathcal{O}_{Y}$ ).
(3)

For each point $y\in Y$ (not necessarily closed), the fiber $X_{y^{\infty}}$ (5.1) is globally $F$ -regular (2.16).

Recall that as in 3.2, we can consider the $\alpha_{F}$ -invariant of a pair $(X,L)$ where $X$ is a globally $F$ -regular projective variety (over a perfect field) and $L$ is an ample line bundle over $X$ .

Theorem 5.3.

Let $f:(\mathcal{X},\mathcal{L})\to Y$ be a flat family of globally $F$ -regular varieties, where $\mathcal{L}$ is an ample line bundle over $\mathcal{X}$ . Let $K$ denote the fraction field of $Y$ and $\alpha_{\text{gen}}$ denote the $\alpha_{F}$ -invariant of $(X_{K^{\infty}},\mathcal{L}|_{X_{K^{\infty}}})$ , the perfectified generic fiber. Then, for each real number $0<\alpha<\alpha_{\text{gen}}$ , there exists a dense open subset $U_{\alpha}\subset Y$ such that

\alpha_{F}(X_{y^{\infty}},\mathcal{L}|_{X_{y^{\infty}}})>\alpha\quad\text{for every point $y$ in $U_{\alpha}$.}

Recall that in 3.8, we defined the sequence $(\alpha_{e})_{e\geq 1}$ that converges to the $\alpha_{F}$ -invariant. To prove 5.3, we need to understand the rate of convergence in 3.8. For this we will use “degree-lowering operators" as below.

Theorem 5.4.

Let $X$ be a projective globally $F$ -regular variety over $k$ and $L$ be an ample invertible sheaf over $X$ . Suppose we have integers $N\geq 1$ and $e>0$ such that the sheaf

\mathscr{H}om_{\mathcal{O}_{X}}\big{(}\,(F^{e}_{*}(L^{m}),L^{N}\,\big{)}

is generically globally generated for each $0\leq m\leq p^{e}-1$ . Then, if $S$ denotes the section ring $S(X,L)$ , we have

\left|\alpha_{F}(S)-\alpha_{e}(S)\right|\leq\frac{N}{p^{e}-1}.

Proof.

Set $d=\dim(X)$ . First, we claim that for each $0\leq m\leq p^{e}-1$ , we can find an injective map of $\mathcal{O}_{X}$ -modules

\iota_{e,m}:F^{e}_{*}(L^{m})\hookrightarrow(L^{N})^{\oplus p^{ed}}.

(5.1)

To see this, let $\eta$ denote the generic point of $X$ and consider the following restriction map to the generic stalk:

\operatorname{Hom}_{\mathcal{O}_{X}}(F^{e}_{*}(L^{m}),L^{N})\to\mathscr{H}om_{\mathcal{O}_{X}}(F^{e}_{*}(L^{m}),L^{N})_{\eta}\cong\operatorname{Hom}_{\mathcal{O}_{X,\eta}}(F^{e}_{*}(L^{m}_{\eta}),L^{N}_{\eta}).

(5.2)

The assumption that $\mathscr{H}om_{\mathcal{O}_{X}}\big{(}\,(F^{e}_{*}(L^{m}),L^{N}\,\big{)}$ is generically globally generated means that the image of the map in Equation 5.2 generates $\operatorname{Hom}_{\mathcal{O}_{X,\eta}}(F^{e}_{*}(L^{m}_{\eta}),L^{N}_{\eta})$ as an $\mathcal{O}_{X,\eta}$ -module. Recall that $\mathcal{O}_{X,\eta}$ is just the fraction field of $X$ . Since $F^{e}_{*}(L^{m}_{\eta})$ is a free $\mathcal{O}_{X,\eta}$ -vector space of rank $p^{ed}$ , we can choose $p^{ed}$ maps $\phi_{1},\dots,\phi_{p^{ed}}$ in $\operatorname{Hom}_{\mathcal{O}_{X}}(F^{e}_{*}(L^{m}),L^{N})$ such that their images under the map in Equation 5.2 forms a basis of $\operatorname{Hom}_{\mathcal{O}_{X,\eta}}(F^{e}_{*}(L^{m}_{\eta}),L^{N}_{\eta})$ over $\mathcal{O}_{X,\eta}$ . Thus, defining $\iota_{e,m}$ to be the product map

\iota_{e,m}:=\phi_{1}\times\dots\times\phi_{p^{ed}}:F^{e}_{*}(L^{m})\rightarrow(L^{N})^{\oplus p^{ed}},

we see that $\iota_{e,m}$ is generically an isomorphism by construction. Furthermore, since $F^{e}_{*}(L^{m})$ is a torsion-free sheaf of rank $p^{ed}$ over $\mathcal{O}_{X}$ , $\iota_{e,m}$ is injective since it is generically an isomorphism. This completes that proof of the claim that maps as in Equation 5.1 exist.

Now, fix a map $\iota_{e,m}$ for each $0\leq m\leq p^{e}-1$ as in Equation 5.1. Then, by taking section modules with respect to $L$ (2.3), we get a corresponding map of graded $S$ -modules

\iota_{e,m}:(F^{e}_{*}S)_{m\bmod p^{e}}\hookrightarrow S(N)^{\oplus p^{ed}}

and let

\iota_{e}:F^{e}_{*}S\hookrightarrow S(N)^{\oplus p^{e(d+1)}}

denote the direct sum of the $\iota_{e,m}$ ’s. Here, for any $m\in\mathbb{Z}$ , $(F^{e}_{*}S)_{m\bmod p^{e}}$ denotes the $S$ -module

(F^{e}_{*}S)_{m\bmod p^{e}}:=\bigoplus_{j\in\mathbb{Z}}F^{e}_{*}(S_{m+jp^{e}}),

and we naturally have an $\mathbb{N}$ -graded $S$ -module decomposition

F^{e}_{*}S\cong\bigoplus_{0\leq m\leq p^{e}-1}(F^{e}_{*}S)_{m\bmod p^{e}}.

Note that $\iota_{e}$ is injective because the $\iota_{e,m}$ ’s were injective. Furthermore, the key property of $\iota_{e}$ that we will use is the following: for each non-zero homogeneous element $f$ of degree $m$ in $S$ , $\iota_{e}(F^{e}_{*}(f))$ is a non-zero homogeneous element of degree

\lfloor\frac{m}{p^{e}}\rfloor+N.

(5.3)

Thus, we may use this map as a “degree-lowering operator".

From 3.8, recall that $\alpha_{e}:=\alpha_{e}(S)=\frac{m_{e}}{p^{e}}$ where $m_{e}$ is defined to be the number $\max\{m\,|\,I_{e}(m)=0\}$ . The condition that $I_{e}(m)=0$ means that for all non-zero elements $f\in S_{m}$ , there exists a splitting $F^{e}_{*}S\to S$ that sends $F^{e}_{*}f$ to $1$ .

Claim: For any $\ell\geq 1$ , let $e^{\prime}=\ell e$ , and $m$ be an integer such that $m\leq(\alpha_{e}-\sum_{t=1}^{\ell-1}\frac{N}{p^{te}})p^{\ell e}$ . Then $I_{e^{\prime}}(m)=0$ .

We will prove the claim by induction on $\ell$ . If $\ell=1$ , note that the sum in the claim is empty, and hence we have $m\leq m_{e}=\alpha_{e}p^{e}$ . In this case, $I_{e}(m)=0$ by the definition of $m_{e}$ .

Now let $\ell>1$ and $f$ be any non-zero element of $S_{m}$ . We need to show that $F^{e}_{*}f$ splits from $F^{e}_{*}S$ . To see this, note that by Equation 5.3, $\iota_{e}(F^{e}_{*}(f))$ is a non-zero element of degree at most

\lfloor\frac{1}{p^{e}}\,(\alpha_{e}-\sum_{t=1}^{\ell-1}\frac{N}{p^{te}})\,p^{\ell e}\rfloor+N=m_{e}\,p^{(\ell-2)e}-N\,p^{(\ell-2)e}-\dots-N+N=(\alpha_{e}-\sum_{t=1}^{\ell-2}\frac{N}{p^{te}})\,p^{(\ell-1)e}.

Thus, the inductive hypothesis applies to $\iota_{e}(F^{e}_{*}(f))$ , implying that $\iota_{e}(F^{e}_{*}(f))$ is not contained in $I_{e^{\prime}-e}$ . Let $\varphi:F_{*}^{e^{\prime}-e}S\to S$ denote a splitting of $\iota_{e}(F^{e}_{*}(f))$ . Then, we see that (forgetting the degrees) $\varphi\circ F_{*}^{e^{\prime}-e}\iota_{e}:F_{*}^{e^{\prime}}S\to S$ defines a splitting of $F_{*}^{e^{\prime}}(f)$ , as required. This completes the proof of the claim.

To complete the proof of the Theorem, we note that the claim above implies that for any $\ell\geq 1$ ,

\alpha_{\ell e}=\frac{m_{\ell e}}{p^{\ell e}}\geq\alpha_{e}-\sum_{t=1}^{\ell-1}\frac{N}{p^{te}}.

Letting $\ell\to\infty$ and using 3.8, we get

\alpha_{e}(S)-\alpha_{F}(S)\leq\frac{N}{p^{e}-1}.

(5.4)

Lastly, note that by 3.9, we already have

\alpha_{F}(S)-\alpha_{e}(S)\leq\frac{1}{p^{e}},

which, together with Equation 5.4 completes the proof of the theorem. ∎

Lemma 5.5.

Let $X$ be a globally $F$ -regular variety of dimension $d$ and $L$ be an ample and globally generated invertible sheaf. Suppose $e_{0}>0$ is such that $(1-p^{e})K_{X}$ is linearly equivalent to an effective Weil divisor for all $e\geq e_{0}$ . Then, for all $e\geq e_{0}$ , and each $0\leq m\leq p^{e}-1$ , the sheaf

\mathscr{H}om_{\mathcal{O}_{X}}(F^{e}_{*}(L^{m}),L^{d+1})

is generically globally generated.

Proof.

First, by an application of Castelnuovo-Mumford regularity, we prove the following statement:

Claim: For all $e\geq 1$ and all $0\leq n\leq p^{e}-1$ , the sheaf $F^{e}_{*}(L^{n+dp^{e}})$ is $0$ -regular with respect to $L$ , and hence globally generated.

To see this, we check that

H^{i}(X,F^{e}_{*}(L^{n+dp^{e}})\otimes L^{-i})=H^{i}(X,L^{n+(d-i)p^{e}})=0\quad\text{for all $i>0$.}

Here, we have used the projection formula to see that $F^{e}_{*}(L^{n+dp^{e}})\otimes L^{-i}\cong F^{e}_{*}(L^{n+(d-i)p^{e}})$ and the cohomology vanishing follows from (2.21), since $L^{n+(d-i)p^{e}}$ is nef for all $i\leq d$ . Thus, $F^{e}_{*}(L^{n+dp^{e}})$ is $0$ -regular with respect to $L$ , and hence is globally generated (see [Laz04, Theorem 1.8.5] for the details regarding Castelnuovo-Mumford regularity).

Next, for any $e\geq e_{0}$ and $n\geq 0$ , write $(1-p^{e})K_{X}\sim D_{e}$ where $D_{e}$ is an effective Weil-divisor. Note that it is always possible to find such an $e_{0}$ thanks to [SS10, Theorem 4.3]. Let $\phi_{e}(L^{n})$ denote the map obtained by twisting the defining map for $D_{e}$ by $L^{n}$ and pushing forward under $F^{e}$ :

\phi_{e}(L^{n}):F^{e}_{*}(L^{n})\hookrightarrow F^{e}_{*}(\mathcal{O}_{X}(D_{e})\otimes L^{n}).

Note that for any point $x\notin\text{Supp}(D_{e})$ , $\phi_{e}$ restricts to an isomorphism in an open neighbourhood around $x$ . For any sheaf $\mathcal{F}$ , let $\mathcal{F}_{x}$ denote the stalk of $\mathcal{F}$ at $x$ .

Applying duality for the Frobenius map (Equation 2.2), we have (for any $m\in\mathbb{Z}$ ):

\mathscr{H}om_{\mathcal{O}_{X}}(F^{e}_{*}(L^{m}),L^{d+1})\cong F^{e}_{*}(\mathcal{O}_{X}(D_{e})\otimes L^{dp^{e}+p^{e}-m}).

Therefore, for any $e\geq e_{0}$ and $0\leq m\leq p^{e}-1$ , set $n=dp^{e}+p^{e}-m$ and consider the diagram

where $x$ is any point not contained in $\text{Supp}(D_{e})$ . Since the horizontal arrows are injective and the bottom horizontal arrow is an isomorphism, any set of global sections generating $F^{e}_{*}(L^{n})_{x}$ , viewed as global sections of $F^{e}_{*}(\mathcal{O}_{X}(D_{e})\otimes L^{n})$ , will also generate $F^{e}_{*}(\mathcal{O}_{X}(D_{e})\otimes L^{n})_{x}$ . Since by the claim above $F^{e}_{*}(L^{n})$ is globally generated, we have that $\mathscr{H}om_{\mathcal{O}_{X}}(F^{e}_{*}(L^{m}),L^{d+1})\cong F^{e}_{*}(\mathcal{O}_{X}(D_{e})\otimes L^{n})$ is globally generated at any $x\notin\text{Supp}(D_{e})$ (and hence generically globally generated). This completes the proof of the lemma. ∎

Lemma 5.6.

Let $f:\mathcal{X}\to Y$ be a flat family of globally $F$ -regular varieties over $k$ where $Y$ is regular. Let $\mathcal{D}$ be an integral Weil-divisor such that $\mathcal{L}=\mathcal{O}_{\mathcal{X}}(r\mathcal{D})$ is Cartier for some integer $r\geq 1$ . We also assume that $\mathcal{L}$ is ample.

(1)

Then, for any integer $m\geq 0$ , the sheaf $f_{*}(\mathcal{L}^{m})$ is locally free on $Y$ and for any point $y\in Y$ (not necessarily closed), the natural “restriction to the fiber" map

f_{*}(\mathcal{L}^{m})\otimes_{\mathcal{O}_{Y}}\kappa(y)\to H^{0}(\mathcal{X}_{y},\mathcal{L}^{m}|_{\mathcal{X}_{y}})

is an isomorphism. Moreover, for any $y\in Y$ , there exists an affine open neighbourhood $\operatorname{\text{Spec}}(B)=U\subset Y$ of $y$ such that if the closure of $y$ is defined by a regular sequence $b_{1},\dots b_{t}\in B$ and for any $m\geq 0$ , the natural map

H^{0}(f^{-1}(U),\mathcal{L}^{m})\otimes_{B}B/\mathfrak{p}B\to H^{0}(\mathcal{X}_{B/\mathfrak{p}B},\mathcal{L}^{m}|_{\mathcal{X}_{B/\mathfrak{p}B}})

is an isomorphism where $\mathfrak{p}$ is the ideal $(b_{1},\dots,b_{t})$ .

(2)

Suppose, in addition that $\mathcal{X}$ is a locally strongly $F$ -regular variety (2.16). Then, for any $m\geq 0$ , setting $\mathcal{F}:=\mathcal{O}_{\mathcal{X}}(m\mathcal{D})$ , we have

(a)

$\mathcal{F}$ is flat over $Y$ .
(b)

For any $y\in Y$ , the restriction $\mathcal{F}_{y}:=\mathcal{F}\otimes_{\mathcal{O}_{\mathcal{X}}}\mathcal{O}_{\mathcal{X}_{y}}$ is reflexive.
(c)

$f_{*}\mathcal{F}$ is locally free on $Y$ and for any $y\in Y$ , the natural map

$f_{*}(\mathcal{F})\otimes_{\mathcal{O}_{Y}}\kappa(y)\to H^{0}(\mathcal{X}_{y},\mathcal{F}_{y})$

is an isomorphism.

(d)

Assume that $\text{Supp}(\mathcal{D})$ does not contain any fiber of $f$ . Then,

f_{*}(\mathcal{O}_{\mathcal{X}}(m\mathcal{D}))\otimes_{\mathcal{O}_{Y}}\kappa(y)\to H^{0}(\mathcal{X}_{y},\mathcal{O}_{\mathcal{X}_{y}}(m\mathcal{D}|_{\mathcal{X}_{y}}))

is an isomorphism as well. Here, we restrict the Weil-divisor $m\mathcal{D}$ to $Y$ as explained in 2.23.

Proof.

Note that since $\mathcal{L}^{m}$ is an invertible sheaf on $\mathcal{X}$ , it is flat over $Y$ for any $m$ .

(1)

Since $\mathcal{O}_{\mathcal{X}}(m\mathcal{L})$ is flat over $Y$ , the claim follows from Grauert’s Theorem [Har77, Chapter III, Corollary 12.9] once we note that the function

y\mapsto\dim_{\kappa(y)}H^{0}(X_{y},\mathcal{L}_{y})

is constant on $Y$ . To see this, we note that if $\chi$ denotes the Euler-characteristic for sheaf-cohomology, we have

\dim_{\kappa(y)}H^{0}(X_{y},\mathcal{L}_{y}^{m})=\chi(\mathcal{L}^{m}_{y})=\chi(\mathcal{L}^{m}_{\eta})=H^{0}(X_{\eta},\mathcal{L}^{m}_{\eta})

where $\eta$ denotes the generic point of $Y$ . Here, we are using the higher cohomology vanishing (2.21) for nef invertible sheaves on the fibers (which are globally $F$ -regular varieties by assumption), and the fact that the Euler-characteristic is constant for all fibers of a flat map [Har77, III, Theorem 9.9]. Since the restriction of $\mathcal{L}^{m}$ to each fiber of $f$ has vanishing higher cohomology, Grauert’s Theorem also implies that $R^{i}f_{*}(\mathcal{L}^{m})$ is zero for any $m\geq 0$ .

Fix a $y\in Y$ and choose an open neighbourhood $\operatorname{\text{Spec}}(B)=U\subset Y$ of $y$ such that $\mathfrak{p}$ is generated by a regular sequence $(b_{1},\dots,b_{t})$ and $B/\mathfrak{p}B$ is regular (where $\mathfrak{p}$ is the prime ideal of $B$ corresponding to $y$ ). This is possible because $Y$ is regular by assumption. Then, note that the following exact sequence of sheaves on $f^{-1}(U)$ :

remains exact after applying $H^{0}(f^{-1}(U),\,-\,)$ since $H^{1}(f^{-1}(U),\mathcal{L}^{m})=0$ . This tells us that

H^{0}(f^{-1}(U),\mathcal{L}^{m})\otimes_{B}B/b_{1}B\cong H^{0}(\mathcal{X}_{B/x_{1}B},\mathcal{L}^{m}|_{\mathcal{X}_{B/x_{1}B}}).

Now, since $B/x_{1}B$ is also regular, we may proceed inductively to complete the proof of Part (1).

(2)
Fix any $y\in Y$ and let $A$ be the local ring at $y\in Y$ and $R$ be the local ring of any point $x\in X$ mapping to $y$ .
1. (a)
  
  Since $R$ is strongly $F$ -regular by assumption, we may apply Part (1) of 2.23 to conclude that $R(m\mathcal{D})$ is isomorphic to a summand of $F^{e}_{*}R$ . Since $A$ is regular, we have $F^{e}_{*}A$ is flat over $A$ and by assumption $F^{e}_{*}R$ is flat over $F^{e}_{*}A$ . Thus, we see that $F^{e}_{*}R$ is flat over $A$ and consequently, $R(m\mathcal{D})$ is flat over $A$ because it is a direct summand of $F^{e}_{*}R$ .
2. (b)
  
  Fix a regular sequence $b_{1},\dots,b_{t}$ on $A$ generating the maximal ideal (this is possible because $A$ is regular). Because $R$ is flat over $A$ , $b_{1},\dots,b_{t}$ is also a regular sequence on $R$ . Now, it is sufficient to show that $R(m\mathcal{D})\otimes_{R}R^{\prime}$ is reflexive over $R^{\prime}=R/(b_{1},\dots,b_{t})R$ . Note that $R^{\prime}$ is normal by [Mat89, Theorem 23.9] since all the fibers of $f$ are assumed to be globally $F$ -regular, which in particular implies that they are normal. Now the fact that $R(m\mathcal{D})\otimes_{R}R^{\prime}$ is reflexive over $R^{\prime}$ is guaranteed by Part (2) of 2.23.
3. (c)
  
  Since we have proved that $\mathcal{F}$ is flat over $Y$ in Part (a), we can repeat the argument from Part (1) of the Lemma using Grauert’s Theorem by replacing the vanishing theorem for $\mathbb{Q}$ -Cartier ample divisors proved in 2.22 instead of 2.21.
4. (d)
  
  This is immediate by combining Part (c) with the last part of 2.23. ∎

Remark 5.7.

Though 5.6 is stated for restriction to the fibers of $f$ , all parts of the lemma hold if we further base-change to the perfectified-fibers by flat base-change [Har77, III, Proposition 9.3] and using the fact that the perfectified-fibers of $f$ are also normal by assumption (since they are globally $F$ -regular).

Proof of 5.3.

We divide the proof into several steps, since some of the steps will be used again in the next subsection.

Step 1:

By 3.13, we may replace $\mathcal{L}$ by a multiple of $\mathcal{L}$ if necessary and assume that $\mathcal{L}$ is globally generated on $\mathcal{X}$ . In that case, the restriction of $\mathcal{L}$ to each perfectified-fiber of $f$ is also automatically globally generated.

Step 2:

Recall the sequence $\alpha_{e}$ converging to the $\alpha_{F}$ -invariant introduced in 3.8. Putting together 5.5 and 5.4, we get that for each $y\in Y$ , and each $e\geq 1$ ,

|\alpha_{e}(S_{y^{\infty}})-\alpha_{F}(S_{y^{\infty}})|\leq\frac{d+1}{p^{e}}.

(5.5)

Here, $S_{y^{\infty}}$ denotes the section ring of the perfect fiber of $f$ over $y$ with respect to the restriction of $\mathcal{L}$ , and $d$ denotes the dimension of every fiber of $f$ (which is well-defined because $f$ is flat).

Now, given any $\alpha<\alpha_{\text{gen}}$ , let $\varepsilon=\alpha_{\text{gen}}-\alpha$ . Fix an $e\gg 0$ such that $\frac{d+1}{p^{e}}<\varepsilon/2$ . Then, applying Equation 5.5 to the (perfectified) generic fiber of $f$ , we have

\alpha_{\text{gen}}-\varepsilon/2<\alpha_{e}(S_{K^{\infty}}).

(5.6)

Claim:

There exists a dense open set $U_{\alpha}\subset Y$ (depending on $e$ ) such that

\alpha_{e}(S_{y^{\infty}})\geq\alpha_{e}(S_{K^{\infty}})

for each $y\in U_{\alpha}$ . Assuming the claim, Equation 5.5, and Equation 5.6 together imply that

\alpha_{F}(S_{y^{\infty}})>\alpha_{e}(S_{y^{\infty}})-\varepsilon/2\geq\alpha_{e}(S_{K^{\infty}})-\varepsilon/2>\alpha_{\text{gen}}-\varepsilon=\alpha

for every $y\in U_{\alpha}$ as required.

Step 3:

Fix an $e\gg 0$ as in Step 2. We now proceed to prove the claim used above, i.e., that there exists a dense open set $U_{\alpha}\subset Y$ (depending on the choice of $e>0$ ) such that $\alpha_{e}(S_{y^{\infty}})\geq\alpha_{e}(S_{K^{\infty}})$ for every $y\in U_{\alpha}$ . Working locally, we may assume that $Y=\operatorname{\text{Spec}}(A)$ and let $S_{A}$ denote the section ring $S(\mathcal{X},\mathcal{L})$ . Set $m_{e}=m_{e}(S_{K^{\infty}})$ . By 5.6, $H^{0}(\mathcal{X},\mathcal{L}^{m})$ is a locally free $A$ -module for any $m\geq 0$ . So by shrinking $Y$ around any point if necessary, we may assume that $H^{0}(\mathcal{X},\mathcal{L}^{m})$ is a free $A$ -module with a basis $\mathscr{B}_{m}$ for each $0\leq m\leq m_{e}$ . By 5.6 again, $\mathscr{B}_{m}$ restricts to a basis of $H^{0}(X_{y},\mathcal{L}_{y})$ for any $y\in Y$ . Let $\mathscr{B}=\sqcup_{m=0}^{m_{e}}\mathscr{B}_{m}$ . Note that $m_{e}$ is at most $p^{e}-1$ since for any non-zero section $f\in H^{0}(X_{K^{\infty}},\mathcal{L}_{K^{\infty}})$ (which exists because $\mathcal{L}$ is assumed to be globally generated), we have $0\neq f^{p^{e}}\in I_{e}(p^{e})$ .

Step 4:

By definition of $m_{e}$ (3.7), for any $m\leq m_{e}$ and any non-zero $f\in H^{0}(X_{K^{\infty}},\mathcal{L}^{m}_{K^{\infty}})$ , the map $S_{K^{\infty}}\to F^{e}_{*}S_{K^{\infty}}$ sending $1$ to $F^{e}_{*}f$ splits. Thus, using [LP23, Lemma 2.7 (a)] repeatedly on the set $\mathscr{B}_{m}$ (fixed in Step 3), we can construct a surjective map $\psi_{m}:F^{e}_{*}(\mathcal{L}_{K^{\infty}}^{m})\to\mathcal{O}_{X_{K^{\infty}}}^{\oplus\mathscr{B}_{m}}$ such that if $f\in\mathscr{B}_{m}$ is a basis element, then $\psi_{m}(F^{e}_{*}f)=\mathbf{1}_{f}$ , where $\mathbf{1}_{f}$ is the standard basis element corresponding to $f$ of $\mathcal{O}_{X_{K^{\infty}}}^{\oplus\mathscr{B}_{m}}$ . Considering the induced map on the section modules and putting together the maps $\psi_{m}$ for all $0\leq m\leq m_{e}$ , along with the zero map for $m_{e}<m\leq p^{e}-1$ , we get a surjective $S_{K^{\infty}}$ -module map

\psi_{K^{\infty}}:F^{e}_{*}S_{K^{\infty}}\to S_{K^{\infty}}^{\oplus\mathscr{B}}

(5.7)

which satisfies the following property: if $0\neq f\in H^{0}(X_{K^{\infty}},\mathcal{L}^{m}_{K^{\infty}})$ and $m\leq m_{e}$ , then $\psi(F^{e}_{*}f)$ is a basis element of $S^{\oplus\mathscr{B}}_{K^{\infty}}$ . In other words, $\psi_{K^{\infty}}$ simultaneously splits all the non-zero sections of degree at most $m_{e}$ on $X_{K^{\infty}}$ . By [CRST21, Lemma 4.8], there is an integer $d_{e}>0$ , a non-zero element $g\in A$ such that if $B=A[g^{-1}]$ , we have a map

\psi_{B^{1/p^{e+d_{e}}}}:S_{B^{1/p^{e+d_{e}}}}^{1/p^{e}}\to S_{B^{1/p^{e+d_{e}}}}^{\oplus\mathscr{B}}

which satisfies $\psi_{B^{1/p^{e+d_{e}}}}\otimes_{B^{1/p^{e+d_{e}}}}K^{\infty}=\psi_{K^{\infty}}$ . Here, $S_{B^{1/p^{e+d_{e}}}}^{1/p^{e}}$ denotes the $e^{\text{th}}$ -relative Frobenius over the base change $S_{B^{1/p^{e+d_{e}}}}:=S_{A}\otimes_{A}B^{1/p^{e+d_{e}}}$ . Note also that after base-changing to $K^{\infty}$ , we are identifying the relative and absolute $e^{\text{th}}$ -Frobenius over $K^{\infty}$ . Since every $f\in\mathscr{B}$ is mapped to a basis element of $S_{B^{1/p^{e+d_{e}}}}^{\oplus\mathscr{B}}$ after tensoring to $K^{\infty}$ , we may assume that the same is true for $\psi_{B^{1/p^{e+d_{e}}}}$ after inverting another element of $A$ if necessary. Finally, for any $y\in\operatorname{\text{Spec}}(B)$ , base changing to $\kappa(y)^{\infty}$ , we see that

\psi_{\kappa(y)^{\infty}}=\big{(}\psi_{B^{1/p^{e+d_{e}}}}\otimes_{B^{1/p^{e+d_{e}}}}\kappa(y)^{1/p^{e+d_{e}}}\big{)}\otimes_{\kappa(y)^{1/p^{e+d_{e}}}}\kappa(y)^{\infty}.

simultaneously splits each non-zero element $0\neq f\in H^{0}(X_{y^{\infty}},\mathcal{L}^{m}_{y^{\infty}})$ . Note that we are again identifying the absolute and relative Frobenius over the perfect field $\kappa(y)^{\infty}$ . This shows that $\alpha_{e}(S_{y^{\infty}})\geq\alpha_{e}(S_{K^{\infty}})$ . This completes the proof of 5.3. ∎

5.2. Semicontinuity for a family of $\mathbb{Q}$ -Fano varieties

In this subsection, we will prove that the $\alpha_{F}$ -invariant is lower semicontinuous in a family of globally $F$ -regular $\mathbb{Q}$ -Fano varieties (see 4.1 for the definition of $\mathbb{Q}$ -Fano varieties):

Theorem 5.8.

Let $f:\mathcal{X}\to Y$ be flat family of globally $F$ -regular $\mathbb{Q}$ -Fano varieties over $k$ , i.e., $f$ is a family of globally $F$ -regular varieties (with $Y$ regular) such that $-K_{\mathcal{X}|Y}$ is $\mathbb{Q}$ -Cartier and $f$ -ample. Then, the map from $Y\to\mathbb{R}_{\geq 0}$ given by

y\mapsto\alpha_{F}(\mathcal{X}_{y^{\infty}})

is lower semicontinuous, where $\mathcal{X}_{y^{\infty}}$ is the perfectified-fiber over $y\in Y$ . See 4.2 for the definition of the $\alpha_{F}$ -invariant of a Fano variety.

Remark 5.9.

For related semicontinuity results for the $F$ -signature and the Hilbert-Kunz multiplicity, see [Pol18], [PT18], [Smi16] and [Smi20]. Similarly, for the corresponding lower semicontinuity result for the complex $\alpha$ -invariant, see [BL22].

Idea of the proof:

Roughly, the proof of 5.8 involves combining 5.3 with the inversion of adjunction for strong $F$ -regularity as proved in [PSZ18]. The main technical difficulty arises when $p$ divides the index of $K_{\mathcal{X}|Y}$ . In this situation, we use a standard perturbation trick similar to [Pat14, Lemma 3.15] and [HX15, Lemma 2.13]. But to do this in a family, we need to be able to restrict $\mathbb{Q}$ -Cartier, ample Weil-divisors to the fibers of the family. So we begin by observing that this can indeed be done in our situation.

Setup and Notation:

Let $A$ be a regular $k$ -algebra of finite type. Suppose we have a flat family $f:\mathcal{X}\to Y=\operatorname{\text{Spec}}(A)$ of globally $F$ -regular Fano varieties (see 5.2) such that $\mathcal{L}=\mathcal{O}_{\mathcal{X}}(-rK_{\mathcal{X}|Y})$ is Cartier and ample for some integer $r>0$ . Then, we can form the section ring $S_{A}:=S(\mathcal{X},\mathcal{L})$ as described in 2.3. Since $\mathcal{X}$ is normal and $\mathcal{L}$ is ample, $S_{A}$ is a normal, finitely generated , $\mathbb{N}$ -graded algebra over $A$ . For any $A$ -algebra $B$ , let $S_{B}$ denote the section ring $S(\mathcal{X}_{B},\mathcal{L}|_{\mathcal{X}_{B}})$ , where $\mathcal{X}_{B}$ denotes the base change $\mathcal{X}\times_{A}\operatorname{\text{Spec}}(B)$ .

Lemma 5.10.

With notation as above, the construction of $S_{A}$ satisfies the following properties:

(1)

$S_{A}$ is flat over $A$ and for any prime ideal $\mathfrak{p}\subset A$ we have that

$S_{A}\otimes_{A}\kappa(\mathfrak{p})\cong S_{\kappa(\mathfrak{p})}.$
(2)

For each prime $\mathfrak{p}\in\operatorname{\text{Spec}}(A)$ , there is an affine open neighbourhood $\operatorname{\text{Spec}}(B)=U\subset Y$ containing $\mathfrak{p}$ such that the restriction of $K_{\mathcal{X}}$ to $\operatorname{\text{Spec}}(B/\mathfrak{p}B)$ (as explained in 2.23) is linearly equivalent to $K_{\mathcal{X}_{B/\mathfrak{p}B}}$ . In particular, for any $\mathfrak{p}\in\operatorname{\text{Spec}}(A)$ , $-rK_{\mathcal{X}_{\mathfrak{p}}}$ is Cartier. Moreover, $\mathcal{X}$ and $S_{A}$ are both $\mathbb{Q}$ -Gorenstein.
(3)

For each $\mathfrak{p}\in\operatorname{\text{Spec}}(A)$ , there is an affine open neighbourhood $\operatorname{\text{Spec}}(B)=U\subset Y$ containing $\mathfrak{p}$ and a regular sequence $b_{1},\dots,b_{t}$ on $B$ generating $\mathfrak{p}$ such that $S_{B}$ , and $S_{B/(b_{1},\dots,b_{i})B}$ is strongly $F$ -regular for each $1\leq i\leq t$ . In particular, $\mathcal{X}$ and $S_{A}$ are both globally $F$ -regular.

(4)

For any Weil-divisor $\mathcal{D}$ on $\mathcal{X}$ such that $r\mathcal{D}$ is Cartier and ample for some $r>0$ , the section module

M_{A}(\mathcal{D})=\bigoplus_{m\geq 0}H^{0}(\mathcal{X},\mathcal{O}_{X}(\mathcal{D})\otimes\mathcal{L}^{m})

is flat over $A$ , and compatible with base change to fibers. In other words, for any $\mathfrak{p}\in\operatorname{\text{Spec}}(A)$ , the natural map

M_{A}(\mathcal{D})\otimes_{A}\kappa(\mathfrak{p})\to M_{\kappa(\mathfrak{p})}(\mathcal{O}_{\mathcal{X}}(\mathcal{D})|_{\mathcal{X}_{\mathfrak{p}}})

is an isomorphism. Moreover, if $\text{Supp}(\mathcal{D})$ does not contain any fiber of $f$ then the natural map

M_{A}(\mathcal{D})\otimes_{A}\kappa(\mathfrak{p})\to M_{\kappa(\mathfrak{p})}(\mathcal{D}|_{\mathfrak{p}})

is an isomorphism as well.

Proof.

Since all parts of the lemma can be proved locally on $Y$ , we may shrink $Y$ if necessary to assume that $\omega_{Y}$ is a free $A$ -module.

(1)

This is immediate from Part (1) of 5.6.

(2)

By inverting an element $g\in A\setminus\mathfrak{p}$ , and setting $B=A[g^{-1}]$ , we may assume that $\mathfrak{p}$ is generated by a regular sequence $b_{1},\dots,b_{t}$ on $B$ (this is possible since $A_{\mathfrak{p}}$ is regular). Fix any $1\leq i\leq t$ and let $B_{i}=B/(b_{1},\dots,b_{i})B$ . By [Mat89, Theorem 23.9], since all fibers of $f$ are normal, we in particular know that each $\mathcal{X}_{B_{i}}$ (and hence, $S_{B_{i}}$ ) is normal. By shrinking $U$ further if necessary, using Part (1) of 5.6, we may also assume $S_{B_{i}}\cong S_{B}\otimes_{B}B_{i}$ .

Now we will show that $K_{\mathcal{X}}$ restricted to ${\mathcal{X}_{B_{i}}}$ is linearly equivalent to the divisor $K_{\mathcal{X}_{B_{i}}}$ . Let $V^{\prime}\subset\mathcal{X}_{B_{i}}$ denote the smooth locus of $\mathcal{X}_{B_{i}}$ , and $V\subset\mathcal{X}_{\text{sm}}$ be an open set such that $V\cap\mathcal{X}_{B_{i}}=V^{\prime}$ . This is possible because $\mathcal{X}_{B_{i}}$ is a complete intersection in $\mathcal{X}_{B}$ . Then, applying the adjunction formula for the complete intersection $V^{\prime}\subset V$ , we have $K_{V^{\prime}}\sim K_{V}|_{V^{\prime}}$ . Since $V^{\prime}\subset\mathcal{X}_{B_{i}}$ is the smooth locus and $\mathcal{X}_{B_{i}}$ is normal, it contains all the codimension one points of $\mathcal{X}_{B_{i}}$ . Thus, taking closures we get that

-rK_{\mathcal{X}}|_{\mathcal{X}_{B_{i}}}=\overline{-rK_{\mathcal{X}}|_{V^{\prime}}}=\overline{-rK_{V}|_{V^{\prime}}}\sim-rK_{\mathcal{X}_{B_{i}}}.

This proves that $\mathcal{L}$ restricted to $\mathcal{X}_{B_{i}}$ is linearly equivalent to $-rK_{\mathcal{X}_{B_{i}}}$ . In particular $-rK_{\mathcal{X}_{B_{i}}}$ is Cartier. Furthermore, it follows from the discussion in [SS10, Section 5.2]) that the canonical divisor $K_{S_{B_{i}}}$ is the cone over the canonical divisor $K_{\mathcal{X}_{B_{i}}}$ (as Weil-divisor). Thus, we get that $\mathcal{O}_{S_{B_{i}}}(-rK_{S_{B_{i}}})\cong\mathcal{O}_{S_{B_{i}}}(1)$ as a graded module over $S_{B_{i}}$ . Also note that for any maximal ideal $\mathfrak{m}\subset A$ , the fiber $S_{\kappa(\mathfrak{m})}$ is strongly $F$ -regular. In particular, all fibers over closed points of $\operatorname{\text{Spec}}(A)$ are Cohen-Macaulay, we see that $S_{A}$ is Cohen-Macaulay as well. Therefore, $S_{A}$ (and similarly $S_{B_{i}}$ ) is $\mathbb{Q}$ -Gorenstein.

(3)

From part (2), we may assume that $\mathfrak{p}$ is generated by a regular sequence $a_{1},\dots a_{t}$ on $A$ and $-rK_{\mathcal{X}_{A_{i}}}$ is Cartier for each $1\leq i\leq t$ , where $A_{i}=A/(a_{1},\dots,a_{i})A$ . Next, note that by assumption (and Part (1)), we have $S_{\kappa(\mathfrak{p})}$ is strongly $F$ -regular. So, there exists a non-zero element $g\in A\setminus\mathfrak{p}$ such that $S_{B_{t}}$ is strongly $F$ -regular where $B_{t}=A_{t}[g^{-1}]$ . Here, we are using the fact that the non-strongly $F$ -regular locus of $S_{B_{t}}$ is closed, compatible with localization and homogeneous with respect to the $\mathbb{N}$ -grading on $S_{B_{t}}$ . Let $B_{t-1}=A_{t-1}\,[g^{-1}]$ . Then, since $-K_{S_{B_{t-1}}}$ is $\mathbb{Q}$ -Cartier and $a_{t}$ is a non-zero divisor on $B_{t-1}$ , by [Das15, Theorem A], we conclude that $S_{B_{t-1}}$ is strongly $F$ -regular in a neighbourhood of $\mathbb{V}(b_{t})$ . Since the locus of points where $S_{B_{t-1}}$ is not strongly $F$ -regular is defined by a homogeneous ideal (and is disjoint from $\mathbb{V}(a_{t})$ ), we may pick a non-zero element $h\in A[g^{-1}]\setminus(a_{1},\dots,a_{t})$ such that if we set $B_{t-1}^{\prime}=B_{t-1}[h^{-1}]$ , $S_{B^{\prime}_{t-1}}$ is strongly $F$ -regular. Replacing $\mathfrak{p}$ with $(a_{1},\dots,a_{t-1})$ , we may proceed inductively to get a localization $B$ of $A$ (at finitely many elements), and an open neighbourhood $U=\operatorname{\text{Spec}}(B)\subset\operatorname{\text{Spec}}(A)$ of $\mathfrak{p}$ such that $S_{B}$ and $S_{B/(a_{1},\dots,a_{i})B}$ is strongly $F$ -regular for each $1\leq i\leq t$ , as required.

Applying this to maximal ideals in $A$ , we see that $S_{A}$ is strongly $F$ -regular. Moreover, let $\mathcal{D}\sim\mathcal{L}^{n}$ for some $n\gg 0$ be an effective divisor and $f$ denote the corresponding degree $n$ element of $S_{A}$ . Then $\operatorname{\text{Spec}}(S_{A}[\frac{1}{f}])$ is isomorphic to the product of $\operatorname{\text{Spec}}(A[t,t^{-1}])$ and $\mathcal{X}\setminus\text{Supp}(\mathcal{D})$ . Since $\mathcal{X}\setminus\text{Supp}(\mathcal{D})$ is affine and strongly $F$ -regular (because we have shown that $S_{A}$ is strongly $F$ -regular), to show that $\mathcal{X}$ is globally $F$ -regular, using [SS10, Theorem 3.9], it is sufficient to show that for some $e\gg 0$ , the map $\mathcal{O}_{\mathcal{X}}\to F^{e}_{*}\mathcal{O}_{\mathcal{X}}(\mathcal{D})$ splits. But this again follows from the fact that $S_{A}$ is strongly $F$ -regular by using 2.7. Hence, $\mathcal{X}$ is globally $F$ -regular.
(4)

Using Part (3) above, $\mathcal{X}$ is in particular, locally strongly $F$ -regular. Now the claim is an immediate consequence of Part (2) of 5.6. ∎

Lemma 5.11.

Let $f:\mathcal{X}\to Y=\operatorname{\text{Spec}}(A)$ be as above and assume $A$ is regular. Suppose $\mathcal{D}$ is a $\mathbb{Q}$ -divisor on $\mathcal{X}$ satisfying the following two properties:

(1)

there is some $e>0$ for which $(p^{e}-1)\mathcal{D}$ is an integral Weil divisor linearly equivalent to $(1-p^{e})K_{\mathcal{X}|Y}$ as Weil divisors, and
(2)

$\text{Supp}(\mathcal{D})$ does not contain any fiber of $f$ .

Then, if $y_{0}\in Y$ is a point such that the pair $(\mathcal{X}_{y_{0}^{\infty}},\lambda\,\mathcal{D}_{y_{0}^{\infty}})$ is globally $F$ -regular for some $\lambda\in\mathbb{Q}_{\geq 0}$ , then there is an open neighbourhood $U\subset Y$ of $y_{0}$ such that $(\mathcal{X}_{y^{\infty}},\lambda\mathcal{D}_{y^{\infty}})$ is also globally $F$ -regular for all $y\in U$ .

Proof.

The proof is divided into several steps, but the strategy is to apply [PSZ18, Corollary 4.19] carefully. See Section 2.3 for a detailed discussion of the process of taking cones over divisors in family.

Step 1:

By shrinking $Y$ to a neighbourhood of $y_{0}$ if necessary, we may also assume that $\omega_{Y}$ is isomorphic to $A$ . Fix an $r>0$ such that $-rK_{\mathcal{X}}$ is an ample Cartier divisor, and set $\mathcal{L}:=-rK_{\mathcal{X}}$ and $S_{A}=S(\mathcal{X},\mathcal{L})$ be the corresponding section ring and $\operatorname{\text{Spec}}(S_{A})$ is the cone over $\mathcal{X}$ . By 5.6, taking the cone over $\mathcal{X}$ commutes with base change to fibers of $f$ . For any integral Weil divisor $\mathcal{E}$ , let $M_{A}(\mathcal{E})=\bigoplus_{m\geq 0}H^{0}(\mathcal{X},\mathcal{O}_{X}(\mathcal{E})\otimes\mathcal{L}^{m})$ denote the corresponding section module over $S_{A}$ .

Step 2:

For any $e$ sufficiently divisible, let $\mathcal{D}_{e}=(p^{e}-1)\mathcal{D}$ , which we may assume is an integral Weil-divisor. Since $-K_{\mathcal{X}}$ is $\mathbb{Q}$ -Cartier, so is $\mathcal{D}_{e}$ . Therefore, by Part (4) of 5.10, the section module $M_{A}(\mathcal{D}_{e})$ is compatible with base change to fibers. Thus, we may consider the cone over $\mathcal{D}_{e}$ as a Weil-divisor on $\operatorname{\text{Spec}}(S_{A})$ defined by the reflexive sheaf $M_{A}(\mathcal{D}_{e})$ . The compatibility with base changing to fibers guarantees that taking the cone over $\mathcal{D}_{e}$ commutes with restricting $\mathcal{D}_{e}$ to the fibers of $f$ .

Step 3:

Since the pair $(\mathcal{X}_{\kappa(y_{0})^{\infty}},\mu\,\mathcal{D}_{\kappa(y_{0})^{\infty}})$ is globally $F$ -regular for $\mu=\lambda$ , it remains so for all $\mu<\lambda+\varepsilon$ for some small $\varepsilon>0$ . Now we set $\mu:=\frac{\ell_{1}}{\ell_{2}}$ to be a rational number such that $\lambda\leq\mu<\lambda+\varepsilon$ for positive integers $\ell_{1}$ and $\ell_{2}$ with the following properties:

(1)

$\ell_{2}-\ell_{1}$ is divisible by $r$ (the Cartier index of $K_{\mathcal{X}}$ ).
(2)

$\ell_{2}$ is not divisible by $p$ .

Furthermore, choose $e_{0}>0$ such that $\ell_{2}$ divides $p^{e_{0}}-1$ and $(p^{e_{0}}-1)\mathcal{D}$ and $-(p^{e_{0}}-1)K_{\mathcal{X}}$ are both integral Weil-divisors. Such an $e_{0}$ exists by our assumptions on $\mathcal{D}$ . Lastly, set $\Delta=\mu\,\mathcal{D}$ . With this notation, we note that the following properties are satisfied:

•

$\Delta=\frac{\ell_{1}}{(p^{e_{0}}-1)\ell_{2}}\,(p^{e_{0}}-1)\mathcal{D}$ , where $(p^{e_{0}}-1)\mathcal{D}$ is a Weil-divisor and $p$ does not divide $(p^{e_{0}}-1)\ell_{2}$ .

•

$(p^{e_{0}}-1)^{2}\,(K_{\mathcal{X}}+\Delta)$ is linearly equivalent to

(p^{e_{0}}-1)^{2}\,(K_{\mathcal{X}}+\Delta)\sim(p^{e_{0}}-1)^{2}(1-\frac{\ell_{1}}{\ell_{2}})\,K_{\mathcal{X}}=\frac{(p^{e_{0}}-1)^{2}(\ell_{2}-\ell_{1})}{\ell_{2}}\,K_{\mathcal{X}}.

(5.8)

And since $r$ divides $\ell_{2}-\ell_{1}$ , we get that $(p^{e_{0}}-1)^{2}(K_{\mathcal{X}}+\Delta)$ is an integral Cartier divisor. Also note that $(p^{e_{0}}-1)^{2}$ clearly divides $(p^{e_{0}}-1)\ell_{2}$ and is not divisible by $p$ .

•

The fibers of $f$ are geometrically normal since the perfect fibers are globally $F$ -regular. They are also geometrically connected by assumption. Thus, our assumption that $\text{Supp}(\mathcal{D})$ does not contain any fibers of $f$ guarantees that $\text{Supp}(\mathcal{D})$ does not contain any generic point of any geometric fiber of $f$ .

Step 4:

In this context, we may use [PSZ18, Corollary 4.19] applied to the projective cone of $f$ with respect to $\mathcal{L}$ to conclude the proof (see Section 2.2 for details about the projective cone construction). More precisely, consider the map

\overline{f}:\overline{\mathcal{X}}:=\operatorname{\text{Proj}}(S_{A}[z])\to\operatorname{\text{Spec}}(A)

where $z$ is just another variable adjoined to $S_{A}$ in degree $1$ . Note that $-rK_{\overline{\mathcal{X}}}$ is Cartier on $\overline{\mathcal{X}}$ (since this is true at the zero-section by construction, and away from the zero section we know that $\overline{\mathcal{X}}$ is an $\mathbb{A}^{1}$ -bundle over $\mathcal{X}$ ).

By 5.10, the construction of the projective cone with respect to $\mathcal{L}$ is compatible with base change to fibers. In other words, for any $y\in Y$ , the fiber $\overline{f}_{y}:\overline{\mathcal{X}}_{y}\to\operatorname{\text{Spec}}(\kappa(y))$ is the map

\overline{f}_{y}:\overline{\mathcal{X}}_{y}=\operatorname{\text{Proj}}(S_{\kappa(y)}[z])\to\operatorname{\text{Spec}}(\kappa(y)).

Let $\overline{\Delta}$ denote the $\mathbb{Q}$ -divisor obtained as the closure in $\overline{\mathcal{X}}$ of the cone over $\Delta$ . For any $r>0$ such that $r\Delta$ is integral, the section module corresponding to $r\overline{\Delta}$ is $M_{A}(r\Delta)[z]$ by construction. Thus, by Step 2,the construction of the projective cone over $\Delta$ is compatible with restricting to fibers. Thus, by Equation 5.8 and the fact that away from the zero-section, $\overline{\mathcal{X}}$ is an $\mathbb{A}^{1}$ -bundle over $\mathcal{X}$ , we have that $K_{\overline{\mathcal{X}}}+\overline{\Delta}$ is $\mathbb{Q}$ -Cartier with index not divisible by $p$ .

Step 5:

Finally, we observe that for any $y\in Y$ , the local strong $F$ -regularity of $(\overline{\mathcal{X}}_{\kappa(y)^{\infty}},\overline{\Delta}_{\kappa(y)^{\infty}})$ is equivalent to the global $F$ -regularity of $(\mathcal{X}_{y^{\infty}},\Delta_{y^{\infty}})$ , since both correspond to the strong $F$ -regularity of the pair $(S_{A},\Delta)$ . With these observations in place, [PSZ18, Corollary 4.19] gives us an open neighbourhood $U\subset Y$ of $y_{0}$ such that $(\mathcal{X}_{y^{\infty}},\Delta_{y^{\infty}})$ is globally $F$ -regular for all $y\in U$ . Since, $\Delta=\mu\mathcal{D}$ and $\lambda\leq\mu$ , the same is true for $(\mathcal{X}_{y^{\infty}},\lambda\mathcal{D}_{y^{\infty}})$ as required. ∎

Proof of 5.8.

Recall that to prove that the given map is lower semicontinuous, we need to show that given any point $y_{0}\in Y$ and $\alpha>0$ such that $\alpha_{F}(\mathcal{X}_{y_{0}^{\infty}})>\alpha$ , there exists an open neighbourhood $U(y_{0},\alpha)\subset Y$ such that $\alpha_{F}(X_{y^{\infty}})>\alpha$ for all $y\in U(y_{0},\alpha)$ . The idea of the proof is similar to the proof of 5.3, but we need a slight variation since we need an open neighbourhood of $y_{0}$ instead of just any open subset of $Y$ .

Firstly, by shrinking $Y$ to a neighbourhood of $y_{0}$ , we assume $Y=\operatorname{\text{Spec}}(A)$ is affine and $\omega_{Y}$ is a free $A$ -module. Next, using 3.13, it is sufficient to prove the lower semicontinuity of the function

y\mapsto\alpha_{F}(S(\mathcal{X}_{y^{\infty}},-rK_{\mathcal{X}_{y^{\infty}}}))

for any $r\gg 0$ . So we pick an $r\gg 0$ such that $-rK_{\mathcal{X}}$ is a globally generated ample divisor on $\mathcal{X}$ . In particular, $-rK_{\mathcal{X}}$ is Cartier. Therefore, Part (2) of 5.10, we have that $-rK_{\mathcal{X}_{y^{\infty}}}$ is a globally generated ample Cartier divisor on $\mathcal{X}_{y^{\infty}}$ for any $y\in Y$ . Additionally, fix an integer $t>0$ such that $H^{0}(\mathcal{X},\mathcal{O}_{\mathcal{X}}(-mK_{\mathcal{X}}))\neq 0$ for all $m\geq t$ .

Let $d$ be the relative dimension of $f$ , let $\alpha_{0}$ denote $\alpha_{F}(S_{y_{0}^{\infty}})$ , and $\varepsilon:=\alpha_{0}-\alpha$ . Recall that for any $A$ -algebra $B$ , $S_{B}$ denotes the section ring $S(\mathcal{X}_{B},\mathcal{L}|_{\mathcal{X}_{B}})$ , where $\mathcal{X}_{B}$ denotes the base change $\mathcal{X}\times_{A}\operatorname{\text{Spec}}(B)$ and $\mathcal{L}=\mathcal{O}_{X}(-rK_{\mathcal{X}})$ . By the argument in Step 2 of the proof of 5.3 (replacing the generic point with $y_{0}$ ), it is sufficient to show that there exists an $e\gg 0$ such that $\frac{d+1}{p^{e}}<\varepsilon/2$ and an open neighbourhood $U(y_{0},\alpha)$ of $y_{0}$ such that

\alpha_{e}(S_{y^{\infty}})\geq\alpha_{0}-\frac{\varepsilon}{2}

for all $y\in U(y_{0},\alpha)$ . To prove this, choose an $e\gg 0$ such that $\frac{d+1}{p^{e}}<\varepsilon/2$ , and

\alpha_{0}\,(p^{e}-1)>p^{e}\,(\alpha_{0}-\varepsilon/2)+t.

(5.9)

Let $n$ be an integer such that $r(p^{e}(\alpha_{0}-\varepsilon/2)+t)<n<r\,\alpha_{0}\,(p^{e}-1)$ be an integer that is not divisible by $p$ . By Part 2c of 5.6, we may assume (by shrinking $Y$ if necessary) that $H^{0}(\mathcal{X},\mathcal{O}_{\mathcal{X}}(-mK_{\mathcal{X}}))$ is a free $A$ -module for each $0\leq m\leq n$ with a basis $\mathscr{B}_{m}$ . Let $\mathcal{D}$ be any element of $\mathscr{B}_{n}$ and $\frac{n}{p^{e}-1}<\lambda<r\,\alpha_{0}$ be any rational number. Since $\mathcal{D}$ is a basis element using Part 2c of 5.6 again, we see that $\mathcal{D}$ does not contain any fibers of $f$ , since $\mathcal{D}$ restricts to a non-zero global section on each fiber. Then, we apply 5.11, to the $\mathbb{Q}$ -divisor $\frac{1}{n}\mathcal{D}$ to get an open neighbourhood $U=U(y_{0},\alpha)$ such that for each $y\in U$ , the pair $(\mathcal{X}_{y^{\infty}},\frac{\lambda}{n}\mathcal{D}|_{\mathcal{X}_{y^{\infty}}})$ is globally $F$ -regular. This is because since $\lambda<r\alpha_{0}$ , the pair $(\mathcal{X}_{y_{0}^{\infty}},\frac{\lambda}{n}\mathcal{D}|_{y_{0}^{\infty}})$ is globally $F$ -regular by the definition of $\alpha_{0}$ . Since $\frac{\lambda}{n}>\frac{1}{p^{e}-1}$ by construction, 3.6 tells us that for each $y\in Y$ , and each $\mathcal{D}$ in $\mathscr{B}_{n}$ , the map

S_{y^{\infty}}\to F^{e}_{*}(S_{y^{\infty}}(\mathcal{D}_{y^{\infty}}))

(5.10)

splits. Furthermore, we may pick an integer $m\leq n$ satisfying: $r$ divides $m$ , $n-m\geq t$ , and $m>rp^{e}\,(\alpha_{0}-\varepsilon/2)$ . This is possible by our choice of $n$ . Since $n-m\geq t$ , we can pick a non-zero $\mathcal{E}\in\mathscr{B}_{n-m}$ that restricts to a non-zero Weil-divisor on each fiber (by base-change). Thus, for any element $\mathcal{D}$ in $\mathscr{B}_{m}$ , since the corresponding map for $(\mathcal{D}+\mathcal{E})_{y^{\infty}}$ splits for each $y\in U$ by Equation 5.10, the corresponding map for $\mathcal{D}_{y^{\infty}}$ also splits for each $y\in U$ . Finally, we apply [LP23, Lemma 2.7 (a)] repeatedly to the basis $\mathscr{B}_{m}$ to conclude that $I_{e}(m)=0$ for $S_{y^{\infty}}$ for each $y\in U$ and thus

\alpha_{e}(S_{y^{\infty}})\geq\frac{m}{rp^{e}}>\alpha_{0}-\varepsilon/2

for all $y\in U$ as required. This completes the proof of 5.8. ∎

6. Examples

In this section, we compute some examples of the $\alpha_{F}$ -invariant for non-toric varieties and highlight some interesting features.

6.1. Quadric hypersurfaces

Fix any algebraically closed field $k$ of characteristic $p\neq 2$ and let $Q_{d}\subset\mathbb{P}^{d+1}$ be the $d$ -dimensional smooth quadric hypersurface over $k$ . Note that by the adjunction formula, $-K_{Q_{d}}=dH$ where $H$ denotes a hyperplane section.

Example 6.1.

Then, $\alpha_{F}(Q_{d})=\frac{1}{d}$ . Equivalently, if $S$ denotes the section ring

S:=S(Q_{d},\mathcal{O}_{X}(1))\cong k[x_{0},\dots,x_{d+1}]/(x_{0}^{2}+\dots x_{d+1}^{2}),

then $\alpha_{F}(S)=1$ . This follows from a description of the structure of the sheaves $F^{e}_{*}(\mathcal{O}_{Q_{d}}(m))$ proved in [Lan08] and [Ach12]. More precisely, for any $e\geq 1$ and $0\leq m\leq p^{e}-1$ , [Ach12, Theorem 2] tells us that $F^{e}_{*}(\mathcal{O}_{Q_{d}}(m))$ is a direct sum of $\mathcal{O}_{Q_{d}}(-t)$ and $\mathcal{S}(-t)$ for $t\geq 0$ , where $\mathcal{S}$ is an ACM bundle that sits in an exact sequence of the form

0\to\mathcal{O}_{\mathbb{P}^{d+1}}(-2)^{\oplus a}\to\mathcal{O}_{\mathbb{P}^{d+1}}(-1)^{\oplus b}\to i_{*}\mathcal{S}\to 0

for suitable positive integers $a$ and $b$ . Here $i:Q_{d}\hookrightarrow\mathbb{P}^{d+1}$ is the inclusion. See [Ach12, Section 1.3] for the details. Since $H^{1}(\mathbb{P}^{d+1},\mathcal{O}_{\mathbb{P}^{d+1}}(-2))=0$ , we deduce from the exact sequence above that $\mathcal{S}(-t)$ has no global sections for any $t\geq 0$ . Therefore, all global sections of $F^{e}_{*}(\mathcal{O}_{Q_{d}}(m))$ appear in the trivial summands. In other words, $I_{e}(S(m))=0$ for any $e\geq 1$ and any $m\leq p^{e}-1$ . Moreover, since $S_{1}\neq 0$ , we know that $m_{e}=p^{e}-1$ . Therefore, by 3.8, we have

\alpha_{F}(S)=\lim_{e\to\infty}\frac{p^{e}-1}{p^{e}}=1.

Remark 6.2.

This example shows that the $\alpha_{F}$ -invariant does not characterize regularity of section rings, since the $\alpha_{F}$ -invariant of a polynomial ring is also equal to $1$ .

Remark 6.3.

Another interesting feature of this example is that the $\alpha_{F}$ -invariant of smooth quadrics is independent of the characteristic $p$ (for $p\neq 2$ ). This is far from true in general (as seen in the next example). Furthermore, for any $d>2$ , the $F$ -signature of $Q_{d}$ is known to depend on $p$ in a rather complicated way (see [Tri23]).

6.2. Comparison to the complex $\alpha$ -invariant

Let $k=\mathbb{F}_{p}$ for some prime number $p\geq 5$ and $X_{p}\subset\mathbb{P}^{3}$ be the diagonal cubic surface defined by $x^{3}+y^{3}+z^{3}+w^{3}=0$ over $k$ .

Example 6.4.

For each $p\geq 5$ , we have $\alpha_{F}(X_{p})<\frac{1}{2}$ . However,

\lim_{p\to\infty}\alpha_{F}(X_{p})=\frac{1}{2}.

To see this, we recall the following result proved by Shideler (see [Shi, Example 4.2.2 and Section 5.1]), building on the techniques of Han and Monsky: Let $\mathscr{s}_{p}$ denote the $F$ -signature of $X_{p}$ , equivalently, of the ring $\mathbb{F}_{p}[x,y,z,w]/(x^{3}+y^{3}+z^{3}+w^{3})$ . Then for any $p\geq 5$ , we have $\mathscr{s}_{p}<\frac{1}{8}$ . Moreover,

\lim_{p\to\infty}\mathscr{s}_{p}=\frac{1}{8}.

Using this, our claims about the $\alpha_{F}$ -invariant of $X_{p}$ follow from 4.7 and 4.8, once we observe that

\frac{\mathrm{vol}(-K_{X_{p}})}{2^{2}\,3!}=\frac{1}{8}.

Remark 6.5.

The complex $\alpha$ -invariant of the cubic surface defined by $x^{3}+y^{3}+z^{3}+w^{3}$ is equal to $2/3$ (see [Che08, Theorem 1.7]). Note that by [HY03], we know that for a fixed divisor $D$ on a variety $X$

\lim_{p\to\infty}\text{fpt}(X_{p},D_{p})=\text{lct}(X,D)

where $X_{p}$ and $D_{p}$ denote the reduction to characteristic $p$ of $X$ and $D$ respectively. 6.4 points to limitations of approximating the log canonical threshold by $F$ -pure threshold for an unbounded family of divisors on $X$ .

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A Frobenius version of Tian’s Alpha-Invariant

Abstract.

1. Introduction

Definition 1.1.

Theorem 1.2.

Theorem 1.3.

Theorem 1.4.

Acknowledgements

2. Preliminaries

Notation 2.1.

Notation 2.2 (Divisors and Pairs).

2.1. Section Rings and Modules

Definition 2.3.

Lemma 2.4.

Proof.

2.2. Affine and Projective Cones

2.3. Cones over ℚ\mathbb{Q}-divisors

2.4. FF-signature

Definition 2.5 (Splitting Ideals).

Definition 2.6.

2.5. FF-signature of cones over projective varieties.

Lemma 2.7.

Proof.

Definition 2.8.

Remark 2.9.

Remark 2.10.

Lemma 2.11.

Proof.

2.6. Duality and the Trace Map

Lemma 2.12.

Proof.

Lemma 2.13.

Proof.

Proposition 2.14.

Proof.

2.7. FF-regularity:

Definition 2.15 (Sharp FF-splitting).

Definition 2.16 (FF-regularity).

Remark 2.17.

Remark 2.18.

Theorem 2.19.

Remark 2.20.

Theorem 2.21 ([Smi00], Corollary 4.3).

Proposition 2.22.

Proof.

Proposition 2.23.

Proof.

3. The αF\alpha_{F}-invariant of section rings

3.1. Definitions

Definition 3.1.

Definition 3.2.

Lemma 3.3.

Proof.

Proposition 3.4.

Remark 3.5.

Lemma 3.6.

Proof.

Proof of 3.4.

3.2. Finite-degree approximations

Definition 3.7.

Theorem 3.8.

Lemma 3.9.

Proof.

Proof of 3.8.

3.3. Positivity and comparison to the FF-signature.

Theorem 3.10.

Lemma 3.11.

Proof.

Proof of 3.10.

Remark 3.12.

3.4. Behaviour under certain ring extensions.

Proposition 3.13.

Proof.

Remark 3.14.

Proposition 3.15.

Proof.

Corollary 3.16.

Proof.

Remark 3.17.

Remark 3.18.

2.3. Cones over $\mathbb{Q}$ -divisors

2.4. $F$ -signature

2.5. $F$ -signature of cones over projective varieties.

2.7. $F$ -regularity:

Definition 2.15 (Sharp $F$ -splitting).

Definition 2.16 ( $F$ -regularity).

3. The $\alpha_{F}$ -invariant of section rings

3.3. Positivity and comparison to the $F$ -signature.

4. The $\alpha_{F}$ -invariant of globally $F$ -regular Fano varieties.

Definition 4.3 ( $F$ -signature).

The $\alpha_{F}$ -invariant of toric Fano varieties

5.2. Semicontinuity for a family of $\mathbb{Q}$ -Fano varieties

6.2. Comparison to the complex $\alpha$ -invariant