On the K-stability of blow-ups of projective bundles

Daniel Mallory

Abstract.

We investigate the K-stability of certain blow-ups of $\mathbb{P}^{1}$ -bundles over a Fano variety $V$ , where the $\mathbb{P}^{1}$ -bundle is the projective compactification of a line bundle $L$ proportional to $-K_{V}$ and the center of the blow-up is the image along a positive section of a divisor $B$ also proportional to $L$ . When $V$ and $B$ are smooth, we show that, for $B\sim_{\mathbb{Q}}2L$ , the K-semistability and K-polystability of the blow-up is equivalent to the K-semistability and K-polystability of the log Fano pair $(V,aB)$ for some coefficient $a$ explicitly computed. We also show that, for $B\sim_{\mathbb{Q}}lL$ , $l\neq 2$ , the blow-up is K-unstable.

1. Introduction

Originally introduced in [Tia97, Don02] to provide a criterion for the existence of Kähler-Einstein metrics on Fano manifolds, K-stability has recently been a major area of research over the last decade. In the field of complex geometry, the work of Chen-Donaldson-Sun [CDS15] and Tian [Tia15] proved the Yau-Tian-Donaldson conjecture, showing that a Kähler-Einstein metric exists on a given Fano manifold precisely if and only the manifold is K-polystable. Recently, the algebraic theory of K-stability has seen remarkable progress, culminating in the discovery that K-stability provides a setting in which to construct moduli stacks of K-semistable Fano varieties with corresponding good moduli spaces of K-polystable Fano varieties, called K-moduli (see [Xu21, LXZ22, Xu24]).

Since K-stability provides the framework for a theory of moduli spaces for Fano varieties, a central aspect of research in the field of K-stability is developing methods to determine when a Fano variety is indeed K-stable (or K-polystable, or K-semistable, etc.). Key results towards this end include: Tian’s criterion on $\alpha$ -invariants of Fano varieties, providing a sufficient condition to show K-polystability ([Tia87, OS12]); the Fujita-Li criterion ([Fuj16, Li17]), relating the Futaki invariants of Donaldson’s algebraic reformulation of K-stability to $\beta$ -invariants consisting of birational data of the variety (or, more specifically, that of divisors on birational models of the variety); the development of stability thresholds ([BJ20]), relating $\beta$ -invariants to the so-called $\delta$ -invariant of a variety, thus relating K-stability to basis-type divisors, log canonical thresholds, and log canonical places of complements, which in turn led to an “inverse of adjunction”-type theorem by Abban-Zhuang ([AZ22]); and the equivalence of K-poly/semistability to a $G$ -equivariant setting of the same (with the additional restriction of $G$ reductive for K-polystability) ([DS16, LX20, LWX21, Zhu21]).

Much of the current literature on determining the K-stability of explicit Fano varieties are in low dimensions (i.e. dimensions $2$ and $3$ ). In dimension 2, the K-stability of smooth del Pezzo surfaces is well-investigated (see [Tia90, Che08, PW18]); as is that of singular del Pezzo surfaces (see [MM93, OSS16] for example). In dimension 3, the question of K-stability has been systematically approached via the Mori-Mukai classification of Fano threefolds into 105 deformation families. For example, in [ACC⁺23] they determine for which of the 105 families is a general member K-polystable. The wall-crossing phenomenon for certain log Fano pairs of dimension at most $3$ have been studied (see e.g. [ADL24, ADL23, Zha24, Zha23]). In higher dimensions, there is much less known about the K-stability of explicit Fano varieties. Much of what is known is related to hypersurfaces of projective space (see e.g. [Fuj19a, AZ22, AZ23]).

In this paper, we construct classes of K-polystable Fano varieties from K-polystable log Fano pairs of dimension one less, thus leading to many new examples of K-polystable Fano varieties in higher dimensions. Specifically, we relate the K-stability of Fano varieties that are blow-ups of certain $\mathbb{P}^{1}$ -bundles over Fano varieties to the log K-stability of the base of the $\mathbb{P}^{1}$ -bundle structure. More explicitly, let $V$ be a Fano variety of dimension $n-1$ , $\mathcal{L}$ an ample line bundle on $V$ such that $r\mathcal{L}\sim_{\mathbb{Q}}-K_{V}$ for some $r>1$ , and $B$ an effective divisor on $V$ such that $B\sim_{\mathbb{Q}}2\mathcal{L}$ . Let $Y=\mathrm{Bl}_{B_{\infty}}\mathbb{P}_{V}(\mathcal{L}\oplus\mathcal{O}_{V})$ , where $B_{\infty}$ is the image of $B$ under some positive section of the $\mathbb{P}^{1}$ -bundle structure of $\mathbb{P}_{V}(\mathcal{L}\oplus\mathcal{O}_{V})\to V$ .

This construction of $Y$ , although it may seem rather artificial, appears with some frequency among collections of Fano varieties. For example, smooth del Pezzo surfaces of degree 6 are of this form, as are the smooth members of the families №3.9, №3.19, and №4.2 in the Mori-Mukai classification of Fano threefolds. See Section 5 for more examples of Fano varieties arising from this construction.

Theorem 1.1.

Let $V$ , $B$ be as above. Further, let $V$ and $B$ be smooth. Then the variety $Y$ as constructed above is K-semistable (resp. K-polystable) if and only if $(V,aB)$ is K-semistable (resp. K-polystable), where

a=a(n,r):=\frac{r^{n+1}-(r-1)^{n+1}-(n+1)(r-1)^{n}}{2(n+1)(r^{n}-(r-1)^{n})}

We note the additional hypothesis of $V$ and $B$ being smooth. We believe that this statement holds in more generality, and this will be the focus of a future work that will also explore the implications on the K-moduli of the relevant Fano varieties that such a more general statement would imply.

We note that similar results regarding the K-stability of such varieties were obtained in [CGF⁺23]; specifically [CGF⁺23, Theorem 1.10], which gives a numerical criterion for the K-polystability of smooth Casagrande-Druel varieties. They further conjecture that such varieties are K-polystable if and only if the base space and the related double cover are both K-polystable. In the smooth case, we confirm their conjecture as a consequence of Theorem 1.1.

Corollary 1.2.

[CGF⁺23, Conjecture 1.16] Let $V,B$ be as above. Further, let $V$ and $B$ be smooth. Let $W\to V$ be the double cover of $V$ ramified over $B$ . Suppose that both $W$ and $V$ are K-polystable. Then $Y$ is also K-polystable.

Our approach to Theorem 1.1 is as follows: we first reduce the statement to that of $\mathbb{T}$ -equivariant K-stability with the standard $\mathbb{T}$ -action induced by the $\mathbb{P}^{1}$ -bundle structure where $\mathbb{T}=\mathbb{G}_{m}$ . Then for the reverse direction of the statement on K-semistability, we use [AZ22, Theorem 3.3] to bound the $\beta$ -invariants of most $\mathbb{T}$ -equivariant divisors over $Y$ (specifically, those with “vertical” centers on $Y$ ) and directly compute the $\beta$ -invariants of the rest (those with “horizontal” centers). For the forward direction, we construct an explicit destabilizing divisor on $Y$ given a destabilizing divisor on $(V,aB)$ . For the K-polystability case of the statement, we directly show that, given the K-polystability of $(V,aB)$ (resp. $Y$ ), a divisor $E$ over $Y$ (resp. $(V,aB)$ ) with $\beta$ -invariant $0$ must induce a product test configuration.

Theorem 1.1 reduces questions of the K-stability of certain families of Fano varieties to similar questions set in one dimension smaller, in exchange for the added complexity of now dealing with the K-stability of log pairs. However, this added complexity is often already of research interest; as seen in several examples, the K-stability of pairs is worked out in the investigation of the wall-crossing phenomenon of moduli spaces and, from some perspectives, is a main focus of investigating such wall-crossing phenomena. We direct readers towards [ADL24, Zho23] for more on the wall-crossing phenomenon for K-stability.

We also note that, in the construction of $Y$ , whose geometry we explore in Section 3, the choice of $B$ being $\mathbb{Q}$ -linearly equivalent to $2\mathcal{L}$ is necessary for $Y$ to be K-semistable. Analysis of the generalization of the construction when $B\sim_{\mathbb{Q}}l\mathcal{L}$ for $l\neq 2$ leads to the following:

Theorem 1.3.

Let $Y$ be constructed as above with $V$ and $B$ smooth and $B\sim_{\mathbb{Q}}l\mathcal{L}$ for $0\leq l<r+1,l\neq 2$ . Then $Y$ is K-unstable. Furthermore, either the strict transform of the image of the positive section containing $B_{\infty}$ , denoted as $\overline{V}_{\infty}$ , or the strict transform of the zero section, $V_{0}$ , is a destabilizing divisor for $Y$ .

We comment that the restriction on $l$ of $0\leq l<r+1$ is precisely the range such that $Y$ is also Fano, with the interpretation that for $l=0$ , $Y$ is simply the $\mathbb{P}^{1}$ -bundle $\mathbb{P}_{V}(\mathcal{L}\oplus\mathcal{O}_{V})$ . The case $l=0$ was previously known, see [ZZ22, Theorem 1.3]. From Theorem 1.3 we provide a simple construction of a K-unstable Fano family in each dimension. Specifically, by Theorem 1.3, The blow-up of $\mathbb{P}^{n}$ along a codimension $2$ linear subvariety is K-unstable; see Example 6.4.

Finally, during the preparation for our manuscript, we learned from Linsheng Wang that a different proof of Theorem 1.1 can be obtained from combining our Lemma 3.5 and [Wan24, Theorem 1.1].

1.1. Acknowledgements

I would like to thank Yuchen Liu for suggesting the problem and for many helpful conversations regarding it.

The author was partially supported by NSF Grant DMS-2148266 and NSF CAREER Grant DMS-2237139.

2. Preliminaries

First we review several definitions and theorems regarding the K-stability of Fano varieties. We work over $\mathbb{C}$ for the entirety. A log Fano pair is a klt projective pair $(X,D)$ such that $-(K_{X}+D)$ is ample. We recall the definitions of $\beta$ -invariants and product-type divisors; these contain all of the necessary data to define the K-semistability and K-polystability of log Fano pairs.

Definition 2.1 ( $\beta$ -invariant, [Fuj19b], [Li17]).

For $(X,D)$ a log Fano pair and $E$ some divisor over $X$ (that is to say, $E$ is a prime divisor on some birational model $\pi:Y\to X$ of $X$ ), we define the $\beta$ -invariant $\beta_{X,D}(E)$ as follows: Let $A_{X,D}(E)$ denote the log discrepancy of $E$ with respect to the log pair $(X,D)$ , and let

S_{X,D}(L;E)=\frac{1}{\operatorname{vol}(L)}\int_{0}^{\infty}\operatorname{vol}(\pi^{*}(L)-tE)dt

for $L$ a big $\mathbb{Q}$ -Cartier divisor on $X$ where $\operatorname{vol}(L)=\lim_{k\to\infty}\frac{\dim H^{0}(X,L^{\otimes k})}{k^{n}/n!}$ denotes the volume of $L$ . When $L=-(K_{X}+D)$ , we will usually omit the line bundle from the notation and simply write $S_{X,D}(E)$ , which is often called the expected order of vanishing of $E$ with respect to the pair $(X,D)$ . Then we define $\beta_{X,D}(E):=A_{X,D}(E)-S_{X,D}(E)$ .

Since we often will pass between divisors and divisorial valuations, we wish to mention the definition of the $\beta$ -invariant of a valuation $v$ . By a valuation on $X$ , we mean a valuation $v:\mathbb{C}(X)^{*}\to\mathbb{R}$ that is trivial on $\mathbb{C}$ . Since $X$ is projective, given a valuation $v$ on $X$ there will exist a unique point $\eta\in X$ such that $v\geq 0$ on $\mathcal{O}_{X,\eta}$ and $v>0$ on the maximal ideal $m_{X,\eta}\subset\mathcal{O}_{X,\eta}$ called the center of $v$ on $X$ . We will denote the closure of the center $\eta$ of a valuation $v$ as $c_{X}(v)$ . For a divisor $E$ over $X$ , we will write $c_{X}(E)$ for the center of the valuation $\operatorname{ord}_{E}$ .

We will define $\beta_{X,D}(v):=A_{X,D}(v)-S_{X,D}(v)$ , where $A_{X,D}(v)$ is the log discrepancy of the valuation (see [JM12, Proposition 5.1] and [BdFFU15, Theorem 3.1]), while

S_{X,D}(L;v)=\frac{1}{\operatorname{vol}(L)}\int_{0}^{\infty}\operatorname{vol}(L-tv)dt

where $\operatorname{vol}(L-tv)=\lim_{k\to\infty}\frac{\dim\{s\in H^{0}(X,L^{\otimes k})|v(s)\geq kt\}}{k^{n}/n!}$ . For $E$ a divisor over $X$ , we have that $S_{X,D}(L;\operatorname{ord}_{D})=S_{X,D}(L;E)$ .

Definition 2.2 (test configurations).

For a triple $(X,D,L)$ of a log Fano pair $(X,D)$ with $X$ dimension $n$ and $L$ an ample line bundle on $X$ with $L\sim-m(K_{X}+D)$ for some $m\in\mathbb{N}$ , a test configuration $(\mathcal{X},\mathcal{D},\mathcal{L})$ is a triple consisting of:

•

a normal variety $\mathcal{X}$ ;
•

an effective $\mathbb{Q}$ -divisor $\mathcal{D}$ on $\mathcal{X}$ ;
•

a flat projective morphism $\pi:(\mathcal{X},\mathcal{D})\to\mathbb{A}^{1}$ such that $\mathcal{L}$ is a $\pi$ -ample line bundle;
•

a $\mathbb{G}_{m}$ -action on the triple $(\mathcal{X},\mathcal{D},\mathcal{L})$ such that $\pi$ is equivariant with respect to the standard multiplicative action of $\mathbb{G}_{m}$ on $\mathbb{A}^{1}$ ;
•

a $\mathbb{G}_{m}$ -equivariant isomorphism $\nu:(\mathcal{X}\setminus\mathcal{X}_{0},\mathcal{D}|_{\mathcal{X}\setminus\mathcal{X}_{0}},\mathcal{L}|_{\mathcal{X}\setminus\mathcal{X}_{0}})\to(X,D,L)\times(\mathbb{A}^{1}\setminus\{0\})$ .

We say a test configuration is a product test configuration if the above isomorphism $\nu$ extends to an isomorphism $(\mathcal{X},\mathcal{D},\mathcal{L})\cong(X,D,L)\times\mathbb{A}^{1}$ . The restriction of the valuation $\operatorname{ord}_{\mathcal{X}_{0}}$ to $\mathbb{C}(X)\subset\mathbb{C}(\mathcal{X})\cong\mathbb{C}(X)(t)$ yields a divisorial valuation on $X$ ; we will call the divisor $E$ such that $\operatorname{ord}_{E}=\operatorname{ord}_{\mathcal{X}_{0}}|_{\mathbb{C}(X)}$ the divisor associated to the test configuration $(\mathcal{X},\mathcal{D},\mathcal{L})$ .

Definition 2.3 (product-type divisors).

Let $(X,D)$ be a log Fano pair and let $E$ be a divisor over $X$ . Let

\text{Rees}(E):=\bigoplus_{m\in\mathbb{N}}\bigoplus_{p\in\mathbb{Z}}\{s\in H^{0}(-mK_{X})|\operatorname{ord}_{E}(s)\geq p\}

be the Rees algebra of $E$ . Suppose $\text{Rees}(E)$ is finitely generated (in such case, we call $E$ a dreamy divisor). We define the test configuration associated to $E$ to be

TC(E):=\text{Proj}_{X}\left(\text{Rees}(E)\right)

We say $E$ is a product-type divisor over $X$ if $TC(E)$ is a product test configuration.

Definition 2.4.

(K-stability, [Fuj19b], [Li17], [LX20, Theorem E], [LWX21, Theorem 1.4], [Zhu21, Theorem 1.1]) Let $(X,D)$ be a log Fano pair. Suppose we have an action of an algebraic torus $\mathbb{T}=\mathbb{G}_{m}^{r}$ on $(X,D)$ . Then, we say that:

•

$(X,D)$ is K-semistable, if, for all $\mathbb{T}$ -invariant divisors $E$ over $X$ , $\beta_{X,D}(E)\geq 0$ .
•

Then $(X,D)$ is K-polystable, if $X$ is K-semistable and, for all $\mathbb{T}$ -invariant divisors $E$ over $X$ , $\beta_{X,D}(E)=0$ implies that $E$ is a product-type divisor.
•

$(X,D)$ is K-unstable if it is not K-semistable.

Definition 2.5 (filtrations).

For a finite dimensional vector space $V$ , we will say a filtration (or $\mathbb{R}$ -filtration) $\mathcal{F}$ on $V$ is a collection of subspaces $\mathcal{F}^{\lambda}V\subseteq V$ for $\lambda\in\mathbb{R}$ such that $\mathcal{F}^{\lambda_{0}}V=V$ for some $\lambda_{0}\in\mathbb{R}$ , $\mathcal{F}^{\lambda_{1}}V=0$ for some $\lambda_{1}\in\mathbb{R}$ , $\lambda\geq\lambda^{\prime}$ implies $\mathcal{F}^{\lambda^{\prime}}V\subseteq\mathcal{F}^{\lambda}V$ , and that the collection is left-continuous, which is to say that $\mathcal{F}^{\lambda-\epsilon}V=\mathcal{F}^{\lambda}$ for all $\lambda$ and sufficiently small $0<\epsilon$ .

A valuation $v$ on $V$ has a naturally associated filtration $\mathcal{F}_{v}V$ where $\mathcal{F}_{v}^{\lambda}V=\{x\in V|v(x)\geq\lambda\}$ . In particular, given a linear system $V\subseteq H^{0}(X,L)$ and a divisor $D$ on $X$ , there is a naturally associated filtration on $V$ given by the filtration associated to $\operatorname{ord}_{D}$ .

Given a filtration $\mathcal{F}$ on a vector space $V$ , we say a basis $\{s_{0},\ldots,s_{m}\}\subset V$ of $V$ is compatible with $\mathcal{F}$ if, for all $\lambda\in\mathbb{R}$ , $\mathcal{F}^{\lambda}V$ is spanned by some subset of the $s_{i}$ .

Next we recall details of $\delta$ -invariants, as related to both $\beta$ -invariants and basis-type divisors, to the end of recalling [AZ22, Theorem 3.3], which will be a key lemma in the proof of our main result.

Definition 2.6 ( $\delta$ -invariants, [BJ20]).

For a log Fano pair $(X,D)$ , a subvariety $Z\subseteq X$ , and a big $\mathbb{Q}$ -Cartier divisor $L$ on $X$ , we define the $\delta$ -invariant of the triple $(X,D,L)$ along $Z$ as

\delta_{Z}(X,D,L)=\inf_{E,Z\subseteq c_{X}(E)}\frac{A_{X,D}(E)}{S_{X,D}(L;E)}

where the infimum is taken over divisors $E$ over $X$ whose centers on $X$ contain $Z$ . As above, when $L=-(K_{X}+D)$ , we will often omit $L$ from the notation. Likewise, when $Z=X$ , we will omit the subscript. From Definition 2.4, we see that $(X,D)$ is K-semistable precisely when $\delta(X,D)\geq 1$ .

There is an alternate approach to determining $\delta$ -invariants due to [BJ20]; towards that end, we recall basis-type divisors.

Definition 2.7.

(basis-type divisors, [FO18, Definition 0.1]) Let $L$ be a big $\mathbb{Q}$ -Cartier $\mathbb{Q}$ -divisor on $(X,D)$ , and $V$ some linear series $V\subset H^{0}(X,mL)$ . For a global section $s\in V$ , let $\{s=0\}$ denote the divisor $\operatorname{Div}(s)+mL$ . Then, a basis-sum divisor of $V$ is a divisor $\Delta$ of the form

\Delta=\sum_{i=0}^{N_{m}}\{s_{i}=0\}

where $N_{m}=h^{0}(X,mL)$ and $\{s_{0},\ldots s_{N_{m}}\}$ form a basis of $V$ . We define a divisor $\Delta^{\prime}$ to be an $m$ -basis type $\mathbb{Q}$ -divisor of the line bundle $L$ if $\Delta^{\prime}=\frac{1}{mN_{m}}\Delta$ for $\Delta$ a basis-sum divisor of $H^{0}(X,mL)$ .

Given a valuation $v$ on $X$ , we say that an $m$ -basis type $\mathbb{Q}$ -divisor $\Delta$ is compatible with $v$ if the basis $\{s_{0},\ldots s_{N_{m}}\}$ is compatible with the filtration on $V$ associated to $v$ . If $v=\operatorname{ord}_{D}$ for a divisor $D$ over $X$ , then we will say $\Delta$ is compatible with $D$ .

Let $\delta_{Z,m}(X,D,L)=\sup\{t\geq 0|(X,D+t\Delta)\text{ is log canonical near }Z\}$ where the supremum is over all $m$ -basis type $\mathbb{Q}$ -divisors $\Delta$ of $L$ .

By [BJ20, Theorem 4.4], we have that $\delta_{Z}(X,D,L)=\lim_{m\to\infty}\delta_{Z,m}(X,D,L)$ , thus allowing us to approach K-stability via basis-type divisors.

We also can approximate $S_{X,D}$ . Let

S_{m}(L;v)=\sup_{D}v(D)

where the supremum is taken over $m$ -basis type $\mathbb{Q}$ -divisors of $L$ . Then, by [BJ20, Corollary 3.6], we have that $S_{X,D}(L;E)=\lim_{m\to\infty}S_{m}(L;E)$ .

Definition 2.8.

(multi-graded linear systems, [AZ22, Definition 2.11]) Let $L_{1},\ldots L_{k}$ be big $\mathbb{Q}$ -Cartier $\mathbb{Q}$ -divisors on $(X,D)$ . A multi-graded linear system $V_{\vec{\bullet}}$ graded by $\mathbb{N}^{k}$ on $(X,D)$ consists of a set of subspaces

V_{\vec{u}}\subset H^{0}(X,u_{1}L_{1}+\ldots u_{k}L_{k})

for all $\vec{u}\in\mathbb{N}^{k}$ such that $V_{\vec{0}}=\mathbb{C}$ , and, for any $\vec{u}_{1},\vec{u}_{2}\in\mathbb{N}$ , $V_{\vec{u_{1}}}\cdot V_{\vec{u_{2}}}\subset V_{\vec{u_{1}}+\vec{u_{2}}}$ . Denote

N_{m}=\sum_{\vec{u}\in\{m\}\times\mathbb{N}^{r}}h^{0}(X,V_{\vec{u}})

We can now analogously define an $m$ -basis type $\mathbb{Q}$ -divisor of a $N^{k+1}$ -graded linear system $V_{\vec{\bullet}}$ to be a divisor $\Delta$ of the form

\Delta=\frac{1}{mN_{m}}\sum_{\vec{u}\in\{m\}\times\mathbb{N}^{r}}\Delta_{\vec{u}}

where $\Delta_{\vec{u}}$ is a basis-sum divisor of $V_{\vec{u}}$ . We also analogously define $S_{m}(V_{\vec{\bullet}};E)$ , $S_{X,D}(V_{\vec{\bullet}};E)$ , $\delta_{m}(X,D,V_{\vec{\bullet}})$ and $\delta(X,D,V_{\vec{\bullet}})$ , and note that, for a big line bundle $L$ , we have that

\delta(X,D,W_{\bullet})=\delta(X,D,L)

for the $\mathbb{N}$ -graded linear system $W_{\bullet}$ with $W_{m}=H^{0}(X,L^{\otimes m})$ .

Definition 2.9.

(refinements of linear systems, [AZ22, Example 2.1]) Let $(X,D)$ be a log Fano pair, $V_{\vec{\bullet}}$ an $\mathbb{N}^{k}$ -graded linear system on $X$ for some big $\mathbb{Q}$ -Cartier $\mathbb{Q}$ -divisors $L_{1},\ldots L_{k}$ , and $E\subset Y\xrightarrow{\pi}X$ a divisor over $X$ . We define the refinement of $V$ by $E$ to be the $\mathbb{N}^{k+1}$ -graded linear series $W_{\vec{\bullet}}$ associated to the big $\mathbb{Q}$ -Cartier $\mathbb{Q}$ -divisors $L_{1}|_{E},\ldots L_{k}|_{E},-E|_{E}$ , with

	$\displaystyle W_{\vec{u},j}$	$\displaystyle=\text{Im}\left(\mathcal{F}_{j}V_{\vec{u}}\to H^{0}(X,\pi^{*}(u_{1}L_{1}+\ldots+u_{k}L_{k})-jE)\right.$
		$\displaystyle\left.\xrightarrow{\rho}H^{0}(E,\pi^{*}(u_{1}L_{1}+\ldots+u_{k}L_{k})\|_{E}-jE\|_{E})\right)$

where the map $\rho$ is the restriction map on global sections.

The invariant data of a multi-graded linear system has a close relation with that of its refinement, as is seen from [AZ22, Theorem 3.3]; this theorem will in turn be a key lemma in the argument for the semistable version of Theorem 1.1.

Theorem 2.10.

[AZ22, Theorem 3.3] Let $(X,D)$ be a log pair, $V_{\vec{\bullet}}$ a multi-graded linear system on $X$ , $E\subset Y\xrightarrow{\pi}X$ a primitive divisor over $X$ , and $Z\subset X$ a subvariety. Let $Z_{0}$ denote an irreducible component of $Z\cap c_{X}(E)$ , let $D_{Y}$ denote the strict transform of $D$ on $Y$ , and let $D_{F}$ denote the different (that is, the divisor on $E$ such that $(K_{Y}+D_{Y}+E)|_{E}=K_{E}+D_{E}$ . Let $W_{\vec{\bullet}}$ denote the refinement of $V_{\vec{\bullet}}$ by $E$ . Then, if $Z\subset c_{X}(E)$ , we have the following inequality of $\delta$ -invariants

\delta_{Z}(X,D,V_{\vec{\bullet}})\geq\min\left\{\frac{A_{X,D}(E)}{S_{X,D}(V_{\vec{\bullet}})},\inf_{Z^{\prime}}\delta_{Z^{\prime}}(E,D_{E},W_{\vec{\bullet}})\right\}

and, if $Z\not\subset c_{X}(E)$ ,

\delta_{Z}(X,D,V_{\vec{\bullet}})\geq\inf_{Z^{\prime}}\delta_{Z^{\prime}}(E,D_{E},W_{\vec{\bullet}})

with both infima over subvarieties $Z^{\prime}\subset Y$ such that $\pi(Z^{\prime})=Z_{0}$ .

Due to the natural torus action on $\mathbb{P}^{1}$ -bundles, we also recall the equivalence between K-stability and the notion of $\mathbb{T}$ -equivariant K-stability.

Theorem 2.11.

[LX20, Theorem E], [LWX21, Theorem 1.4], [Zhu21, Theorem 1.1] Let $(X,D)$ be a log Fano pair with a $\mathbb{T}$ -action for $\mathbb{T}\cong\mathbb{G}_{m}^{r}$ a torus group. Then the K-semistability (resp. K-polystability) of $(X,D)$ is equivalent to the $\mathbb{T}$ -equivariant K-semistability (resp $\mathbb{T}$ -equivariant K-polystability) of $(X,D)$ .

3. Blow-ups of Projective Compactifications of Proportional Line Bundles

Let $V$ be a smooth, $(n-1)$ -dimensional Fano variety. We construct an $n$ -dimensional Fano variety from the data of $V$ , a choice of an ample line bundle $\mathcal{L}$ on $V$ such that $r\mathcal{L}\sim_{\mathbb{Q}}-K_{V}$ for some proportionality constant $r>1$ , and a divisor $B$ on $V$ such that $B$ is smooth and $B\sim_{\mathbb{Q}}l\mathcal{L}$ for some proportionality constant $0\leq l<r+1$ .

Firstly we consider $X$ , the projective compactification of the total space of the line bundle $\mathcal{L}$ . $X$ can be geometrically realized as the projectivization $\mathbb{P}(\mathcal{L}\oplus\mathcal{O}_{V})$ of the total space of the vector bundle $\mathcal{L}\oplus\mathcal{O}_{V}$ . The morphism from the vector bundle structure $\mathcal{L}\oplus\mathcal{O}_{V}\to V$ induces a morphism $\phi:X\to V$ , giving $X$ the structure of a $\mathbb{P}^{1}$ -bundle over $V$ . With this structure, we denote the image of the zero section as $V_{0}$ . We have the following formula for the anti-canonical divisor $-K_{X}$ on $X$ in relation to that of $V$ :

-K_{X}=2H+\phi^{*}\left(-K_{V}-\text{det}(\mathcal{L}\oplus\mathcal{O}_{V})\right)=2H+\frac{r-1}{r}\phi^{*}(-K_{V})

where $H$ denotes the relative hyperplane section from the $\mathbb{P}^{1}$ -bundle structure of $X$ . Thus, we see that $X$ is also Fano exactly when $r>1$ . We also compute the anti-canonical volume of $X$ , that is, the top intersection power of $-K_{X}$ , as a function of $r,n,$ and the anti-canonical volume of $V$ :

Lemma 3.1.

$X$ is a Fano variety. Moreover, the anti-canonical volume of $X$ is

\operatorname{vol}(X):=(-K_{X})^{n}=\frac{(r+1)^{n}-(r-1)^{n}}{r^{n-1}}\operatorname{vol}(V).

Proof.

The anti-canonical divisor of $X$ is

-K_{X}=2H+\phi^{*}(-K_{V})-\text{det}(\mathcal{O}_{V}\oplus\mathcal{L})

where $H$ is the relative hyperplane section. Since $\mathcal{L}\sim\frac{-1}{r}K_{V}$ , the above simplifies to

-K_{X}=2H+\frac{r-1}{r}\phi^{*}(-K_{V})

We also have that $H=V_{0}+\phi^{*}(\mathcal{L})$ and thus

	$\displaystyle H^{2}$	$\displaystyle=V_{0}\cdot H+\phi^{*}(\mathcal{L})\cdot H$
		$\displaystyle=\phi^{*}(\mathcal{L})\cdot H$

and, more generally,

H^{k}=\phi^{*}(\mathcal{L})^{k-1}\cdot H

From this, we calculate $(-K_{X})^{n}$

	$\displaystyle(-K_{X})^{n}$	$\displaystyle=\left[2H+\frac{r-1}{r}\phi^{*}(-K_{V})\right]^{n}$
		$\displaystyle=\sum_{k=0}^{n}\binom{n}{k}2^{n-k}(\frac{r-1}{r})^{k}H^{n-k}\cdot\phi^{*}(-K_{V})^{k}$
		$\displaystyle=\sum_{k=0}^{n-1}\binom{n}{k}2^{n-k}(\frac{r-1}{r})^{k}\phi^{}(\mathcal{L})^{n-k-1}\cdot H\cdot\phi^{}(-K_{V})^{k}$
		$\displaystyle=\sum_{k=0}^{n-1}\binom{n}{k}2^{n-k}(\frac{r-1}{r})^{k}(\frac{1}{r})^{n-k-1}\phi^{*}(-K_{V})^{n-k-1}\cdot H\cdot(-K_{V})^{k}$
		$\displaystyle=\sum_{k=0}^{n-1}\binom{n}{k}2^{n-k}\frac{(r-1)^{k}}{r^{n-1}}H\cdot\phi^{*}(-K_{V})^{n-1}$
		$\displaystyle=\frac{(2+r-1)^{n}-(r-1)^{n}}{r^{n-1}}\operatorname{vol}(V)$

In particular, we see that $-K_{X}$ has strictly positive top intersection power.

Now, we consider the intersection of $-K_{X}$ with effective classes of curves on $X$ . In particular, since $-K_{X}$ is invariant under the $\mathbb{T}$ -action of the $\mathbb{P}^{1}$ -bundle, we consider intersection products of $-K_{X}$ with effective classes of curves fixed by the $\mathbb{T}$ -action. Since $\mathbb{T}$ is $1$ dimensional, a $\mathbb{T}$ -fixed curve is either the closure of a single orbit or a curve that is fixed point-wise. The closure of a single orbit would be a fibre under the $\mathbb{P}^{1}$ -bundle structure; a curve that is fixed point-wise would be contained in the fixed locus of the $\mathbb{T}$ -action, which is $V_{0}\sqcup V_{\infty}$ . Thus, we concern ourselves with the following classes of curves: curves $c_{0}\subset V_{0}$ , curves $c_{\infty}\subset V_{\infty}$ , and fibres $f$ of the $\mathbb{P}^{1}$ -bundle structure. We have the following intersection products:

	$\displaystyle-K_{X}\cdot c_{0}$	$\displaystyle=(2V_{\infty}+\frac{r-1}{r}\phi^{}(-K_{V}))\cdot c_{0}=\frac{r-1}{r}\phi^{}(-K_{V})\cdot c_{0}>0$
	$\displaystyle-K_{X}\cdot c_{\infty}$	$\displaystyle=(2V_{0}+\frac{r+1}{r}\phi^{}(-K_{V}))\cdot c_{\infty}=\frac{r+1}{r}\phi^{}(-K_{V})\cdot c_{\infty}>0$
	$\displaystyle-K_{X}\cdot f$	$\displaystyle=(2H+\frac{r-1}{r}\phi^{*}(-K_{V}))\cdot f=2H\cdot f>0$

Thus, we see $-K_{X}$ is strictly nef, and thus the computation of the top intersection power of $-K_{X}$ shows that it is big. Thus, by the Basepoint-free Theorem (see [KM98, Theorem 3.3]), $-K_{X}$ is ample. ∎

With $X$ Fano, it is natural to question whether $X$ is K-semistable. In this case, a quick computation of a specific $\beta$ -invariant shows that such $X$ is always K-unstable. This was previously shown in [ZZ22]; however, we include the computation of the $\beta$ -invariant here as it is indicative of the general procedure we will apply in Section 6 to show various blow-ups of $X$ are also K-unstable.

Proposition 3.2.

$X$ is $K$ -unstable, with $\beta(V_{0})<0$ .

Proof.

\displaystyle\beta_{X}(V_{0})=A_{X}(V_{0})-S_{X}(V_{0})=1-S_{X}(V_{0})

Thus it suffices to show that $S_{X}(V_{0})>1$ . We note that $V_{0}$ has pseudo-effective threshold $\tau=2$ , and that for $0\leq t\leq 2$ , $-K_{X}-tV_{0}$ is nef, thus:

	$\displaystyle S_{X}(V_{0})$	$\displaystyle=\frac{1}{\operatorname{vol}(X)}\int_{0}^{2}(-K_{X}-tV_{0})^{n}dt$
		$\displaystyle=\frac{1}{\operatorname{vol}(X)}\int_{0}^{2}\frac{\operatorname{vol}(V)}{r^{n-1}}\left((r+1)^{n}-(r+t-1)^{n}\right)dt$
		$\displaystyle=\frac{\operatorname{vol}(V)}{r^{n-1}\operatorname{vol}(X)}\left(2(r+1)^{n}-\frac{(r+1)^{n+1}-(r-1)^{n+1}}{n+1}\right)$
		$\displaystyle=\frac{1}{(r+1)^{n}-(r-1)^{n}}\left(2(r+1)^{n}-\frac{(r+1)^{n+1}-(r-1)^{n+1}}{n+1}\right)$
		$\displaystyle=\frac{1}{(n+1)((r+1)^{n}-(r-1)^{n})}\left(f(r-1)-f(r+1)\right)+1$

Where $f(x)=x^{n+1}-(n+1)x^{n}$ which is strictly decreasing on the range $0<x<n$ , thus showing the first term is positive and thus $S_{X}(V_{0})>1$ as desired. ∎

Due to the $\mathbb{P}^{1}$ -bundle structure of $\phi:X\to V$ , we have a natural $\mathbb{T}$ -action on $X$ where $\mathbb{T}\cong\mathbb{G}_{m}$ acts fibre-wise, and on each fiber of $\phi$ , $\mathbb{T}$ acts with the standard $\mathbb{G}_{m}$ -action on $\mathbb{P}^{1}$ . Given this action, $X$ has two divisors that are point-wise fixed by $\mathbb{T}$ : $V_{0}$ and the image of a positive section $s_{\infty}:V\to X$ of the $\mathbb{P}^{1}$ -bundle structure, which we will denote as $V_{\infty}$ (hereafter also referred to as the infinity section).

We continue with the construction. We label the the image of the divisor $B$ under $s_{\infty}$ as $B_{\infty}$ . This image is a codimension $2$ subvariety of $X$ isomorphic to $B$ since $\mathrm{s}_{\infty}$ is an embedding. We denote the blow-up of $X$ along $B_{\infty}$ as $\pi:Y=\mathrm{Bl}_{B_{\infty}}X\to X$ , and the exceptional divisor as $E$ . For ease of notation, we also denote the strict transform of the pullback of $B$ along $\phi$ as $F:=\phi^{-1}_{*}(B)$ and the strict transform of $V_{\infty}$ as $\overline{V}_{\infty}:=\phi^{-1}_{*}(V_{\infty})$ . By abuse of notation, we will denote the pullbacks $\pi^{*}(H)$ and $\pi^{*}(V_{0})$ as $H$ and $V_{0}$ respectively. We note that $Y\to V$ has the structure of a conic bundle, and the fibers are reducible conics precisely over $B\subset V$ . We note that, since $B_{\infty}$ is fixed under the $\mathbb{T}$ -action on $X$ , said action lifts to an action on $Y$ , with the locus of fixed points of the $\mathbb{T}$ -action is precisely $\overline{V}_{\infty}\sqcup V_{0}\sqcup(E\cap F)$ .

Lemma 3.3.

$Y$ is a Fano variety for $0\leq l<r+1$ (where we interpret $l=0$ to mean $Y=X$ ). Moreover, the anti-canonical volume of $Y$ is

\operatorname{vol}(Y)=\begin{cases}\left(\frac{r^{n}-(r+1-l)^{n}}{l-1}+r^{n}-(r-1)^{n}\right)\frac{\operatorname{vol}(V)}{r^{n-1}},&\text{if }l\neq 1\\ \left(nr^{n-1}+r^{n}-(r-1)^{n}\right)\frac{\operatorname{vol}(V)}{r^{n-1}},&\text{if }l=1\end{cases}

In particular, when $l=2$ , the anti-canonical volume of $Y$ is

\operatorname{vol}(Y)=\frac{2(r^{n}-(r-1)^{n})}{r^{n-1}}\operatorname{vol}(V).

Proof.

	$\displaystyle(-K_{Y})^{n}$	$\displaystyle=(2\pi^{}H+\frac{r-1}{r}\pi^{}\phi^{*}(-K_{V})-E)^{n}$
		$\displaystyle=(\pi^{}H+\frac{r-1}{r}\pi^{}\phi^{*}(-K_{V})+\overline{V}_{\infty})^{n}$
		$\displaystyle=(\pi^{}V_{0}+\pi^{}\phi^{*}(-K_{V})+\overline{V}_{\infty})^{n}$
		$\displaystyle=(\pi^{}V_{0}+\pi^{}\phi^{}(-K_{V}))^{n}+(\pi^{}\phi^{}(-K_{V})+\overline{V}_{\infty})^{n}-(\pi^{}\phi^{*}(-K_{V}))^{n}$
		$\displaystyle=(V_{0}+\pi^{}\phi^{}(-K_{V}))^{n}+(\pi^{}\phi^{}(-K_{V})+\overline{V}_{\infty})^{n}$

with $V_{0},\overline{V}_{\infty}$ as denote above. Now, computing monomial products, we have

	$\displaystyle V_{0}^{k}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$	$\displaystyle=V_{0}^{k-1}\cdot V_{0}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$
		$\displaystyle=(H-\frac{1}{r}\pi^{}\phi^{}(-K_{V}))^{k-1}\cdot V_{0}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$
		$\displaystyle=(-\frac{1}{r}\pi^{}\phi^{}(-K_{V}))^{k-1}\cdot V_{0}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$
		$\displaystyle=(\frac{-1}{r})^{k-1}\operatorname{vol}(V)$

and, similarly,

	$\displaystyle\overline{V}_{\infty}^{k}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$	$\displaystyle=\overline{V}_{\infty}^{k-1}\cdot\overline{V}_{\infty}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$
		$\displaystyle=(V_{0}+\frac{1}{r}\pi^{}\phi^{}(-K_{V})-E)^{k-1}\cdot\overline{V}_{\infty}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$
		$\displaystyle=(\frac{1}{r}\pi^{}\phi^{}(-K_{V})-E)^{k-1}\cdot\overline{V}_{\infty}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$
		$\displaystyle=(\frac{1-l}{r})^{k-1}\operatorname{vol}(V)$

Substituting yields

	$\displaystyle(-K_{Y})^{n}$	$\displaystyle=\sum_{i=1}^{n}\binom{n}{i}\overline{V}_{\infty}^{i}\pi^{}\phi^{}(-K_{V})^{n-i}+\sum_{j=1}^{n}\binom{n}{j}V_{0}^{j}\pi^{}\phi^{}(-K_{V})^{n-j}$
		$\displaystyle=\sum_{i=1}^{n}\binom{n}{i}(\frac{1-l}{r})^{i-1}\operatorname{vol}(V)+\sum_{j=1}^{n}\binom{n}{j}(\frac{-1}{r})^{j-1}\operatorname{vol}(V)$

=\begin{cases}\left(\frac{r^{n}-(r+1-l)^{n}}{l-1}+r^{n}-(r-1)^{n}\right)\frac{\operatorname{vol}(V)}{r^{n-1}},&\text{if }l\neq 1\\ \left(nr^{n-1}+r^{n}-(r-1)^{n}\right)\frac{\operatorname{vol}(V)}{r^{n-1}},&\text{if }l=1\end{cases}

Note that, under our assumptions on $r$ and $V$ , the top intersection power of $-K_{Y}$ is positive when $0\leq l<r+1$ .

Now, we consider the intersection of $-K_{Y}$ with effective classes of curves on $Y$ . In particular, since $-K_{Y}$ is invariant under the $\mathbb{T}$ -action, we consider intersection products of $-K_{Y}$ with effective classes of curves fixed by the $\mathbb{T}$ -action. A curve fixed by the $\mathbb{T}$ -action will either be the closure of a single orbit or a curve that is fixed point-wise. A curve that is the closure of a single orbit would either be the strict transform of a fibre from the $\mathbb{P}^{1}$ -bundle structure $X\to V$ or a fibre of the $\mathbb{P}^{1}$ -bundle $E\to B_{\infty}$ . A curve that is fixed point-wise would be contained in the fixed locus of the $\mathbb{T}$ -action, which is $\overline{V}_{\infty}\sqcup V_{0}\sqcup(E\cap F)$ . Thus, we consider intersection products of $-K_{Y}$ with curves $c_{0}\in V_{0}$ , curves $c_{\infty}\in\overline{V}_{\infty}$ , fibres $e\in E$ , fibres $f$ of the conic bundle structure over $V\setminus B$ , and fibres that lie in $F$ in the class $f-e$ . We have the following intersection products:

	$\displaystyle-K_{Y}\cdot c_{0}$	$\displaystyle=(\pi^{}H+\frac{r-1}{r}\pi^{}\phi^{}(-K_{V})+\overline{V}_{\infty})\cdot c_{0}=\frac{r-1}{r}\pi^{}\phi^{*}(-K_{V})\cdot c_{0}>0$
	$\displaystyle-K_{Y}\cdot c_{\infty}$	$\displaystyle=(2\pi^{}V_{0}+\frac{r+1}{r}\pi^{}\phi^{}(-K_{V})-E)\cdot c_{\infty}=\frac{r-1}{r}\pi^{}\phi^{*}(-K_{V})\cdot c_{\infty}>0$
	$\displaystyle-K_{Y}\cdot f$	$\displaystyle=(2H+\frac{r-1}{r}\pi^{}\phi^{}(-K_{V})-E)\cdot f=2H\cdot f=2>0$
	$\displaystyle-K_{Y}\cdot e$	$\displaystyle=(2H+\frac{r-1}{r}\pi^{}\phi^{}(-K_{V})-E)\cdot e=-E\cdot e=1>0$
	$\displaystyle-K_{Y}\cdot(f-e)$	$\displaystyle=(2H+\frac{r-1}{r}\pi^{}\phi^{}(-K_{V})-E)\cdot(f-e)=2H-E\cdot(f-e)=1>0$

Thus, we see $-K_{Y}$ is strictly nef, and thus the computation of the top intersection power of $-K_{Y}$ shows that it is big, and thus ample. ∎

For the next few sections, until Section 6, we will work with this construction under the additional assumption that $l=2$ . We note that there exists an alternate construction of $Y$ when $l=2$ related to Fano double covers, see [CGF⁺23, Section 2]. One property of such $Y$ that we use frequently is the existence of an involution $\iota:Y\to Y$ such that $\iota(\overline{V}_{\infty})=V_{0}$ , $\iota(E)=F$ , and $\iota$ respects the conic bundle structure of $Y$ over $V$ . The existence of such an involution is readily apparent from the point of view of [CGF⁺23]. From this involution we see that $Y$ permits a second contraction morphism $Y\to X$ distinct from $\pi$ : the blow-down of $F$ , i.e. the composition $\pi\circ\iota$ of the involution and the blow-up morphism.

With the construction of $Y$ from the triple $(V,B,\mathcal{L})$ , we now compute certain key $\beta$ -invariants. In particular, given the induced $\mathbb{T}$ -action on $Y$ , we can consider the test configurations induced by the coweights of the $\mathbb{T}$ -action. Since the torus action on $Y$ is that of a $1$ -dimensional algebraic torus, the co-character lattice $N$ is isomorphic to $\mathbb{Z}$ , with $1$ the identity co-character.

Definition 3.4 (Futaki character).

We define the Futaki character on $Y$ to be $Fut|_{N}:\mathbb{Z}\to\mathbb{R}$ where

Fut|_{N}(i)=\beta_{Y}(\mathrm{wt}_{i})

with $\mathrm{wt}_{i}$ the valuation induced by the coweight $i\in\mathbb{Z}$ ,

Lemma 3.5.

On $Y$ , we have $\operatorname{Fut}|_{N}=0$ . In particular, $\beta(\overline{V}_{\infty})=\beta(V_{0})=0$ .

Proof.

The test configuration induced by $\xi$ has $\operatorname{ord}_{t}$ as its associated valuation under the isomorphism $\mathbb{C}(Y)\cong\mathbb{C}(V)(t)$ induced by the $\mathbb{T}$ -action. This valuation is divisorial with associated divisor $V_{0}$ , so $\operatorname{Fut}(\xi)=\beta_{Y}(V_{0})$ . Similarly, the test configuration induced by $-\xi$ has associated divisor $\overline{V}_{\infty}$ . Any other coweight is a multiple of either of these, and $\operatorname{Fut}(b\xi)=b\operatorname{Fut}(\xi)$ , it suffices to show $\beta_{Y}(V_{0})=\beta_{Y}(\overline{V}_{\infty})=0$ . We compute both $\beta$ -invariants simultaneously. As both $V_{0}$ and $\overline{V}_{\infty}$ are prime divisors on $Y$ , each has log discrepancy of $1$ . Thus it remains to show that $S_{Y}(V_{0})=S_{Y}(\overline{V}_{\infty})=1$ . Let $P_{\infty}(t),N_{\infty}(t)$ (resp. $P_{0}(t),N_{0}(t)$ ) be the positive and negative parts of the Zariski decomposition of $-K_{Y}-t\overline{V}_{\infty}$ (resp. $-K_{Y}-tV_{0}$ ). Then

P_{\infty}(t)=\begin{cases}-K_{Y}-t\overline{V}_{\infty}=(1-t)\overline{V}_{\infty}+\pi^{*}\phi^{*}(-K_{V})+V_{0},&\text{if }0\leq t\leq 1\\ (2-t)H+\frac{r-1}{r}\pi^{*}\phi^{*}(-K_{V})=\frac{r+1-t}{r}\pi^{*}\phi^{*}(-K_{V})+(2-t)V_{0},&\text{if }1\leq t\leq 2\end{cases}

N_{\infty}(t)=\begin{cases}0,&\text{if }0\leq t\leq 1\\ (1-t)E,&\text{if }1\leq t\leq 2\end{cases}

P_{0}(t)=\begin{cases}-K_{Y}-tV_{0}=\overline{V}_{\infty}+\pi^{*}\phi^{*}(-K_{V})+(1-t)V_{0},&\text{if }0\leq t\leq 1\\ (2-t)\overline{V}_{\infty}+\frac{r+1-t}{r}\pi^{*}\phi^{*}(-K_{V}),&\text{if }1\leq t\leq 2\end{cases}

N_{0}(t)=\begin{cases}0,&\text{if }0\leq t\leq 1\\ (t-1)F=(t-1)(\overline{V}_{\infty}+\frac{1}{r}\pi^{*}\phi^{*}(-K_{V})-V_{0}),&\text{if }1\leq t\leq 2\end{cases}

We observe that $\iota(P_{\infty})=P_{0}$ and $\iota(N_{\infty})=N_{0}$ . From this and the computations that $V_{0}\cdot\overline{V}_{\infty}=0$ and $V_{0}^{k}\cdot\pi^{*}\phi^{*}(-K_{V})^{n-k}=\overline{V}_{\infty}^{k}\cdot\pi^{*}\phi^{*}(-K_{V})^{n-k}$ , we conclude that $S_{Y}(V_{0})=S_{Y}(\overline{V}_{\infty})$ , and thus we need only compute one of them (this can also be more directly seen from $\iota(V_{0})=\overline{V}_{\infty}$ ).

	$\displaystyle S_{Y}(\overline{V}_{\infty})$	$\displaystyle=\frac{1}{\operatorname{vol}(Y)}\int_{0}^{\infty}\operatorname{vol}(-K_{Y}-t\overline{V}_{\infty})dt$
		$\displaystyle=\frac{1}{\operatorname{vol}(Y)}\int_{0}^{2}\left(P_{\infty}(t)\right)^{n}dt$
		$\displaystyle=\frac{1}{\operatorname{vol}(Y)}\int_{0}^{1}((1-t)\overline{V}_{\infty}+\pi^{}\phi^{}(-K_{V})+V_{0})^{n}dt$
		$\displaystyle+\frac{1}{\operatorname{vol}(Y)}\int_{1}^{2}(\frac{r+1-t}{r}\pi^{}\phi^{}(-K_{V})+(2-t)V_{0})^{n}dt$
		$\displaystyle=\frac{1}{\operatorname{vol}(Y)}\int_{0}^{1}\frac{2r^{n}-(r-1)^{n}-(r+t-1)^{n}}{r^{n-1}}\operatorname{vol}(V)dt$
		$\displaystyle+\frac{1}{\operatorname{vol}(Y)}\int_{1}^{2}\frac{(r+t-1)^{n}-(r-1)^{n}}{r^{n-1}}\operatorname{vol}(V)dt$
		$\displaystyle=\frac{1}{\operatorname{vol}(Y)}\left(\frac{(r-1)^{n+1}-r^{n+1}}{n+1}+2r^{n}-(r-1)^{n}\right)\frac{\operatorname{vol}(V)}{r^{n-1}}$
		$\displaystyle+\frac{1}{\operatorname{vol}(Y)}\left(\frac{r^{n+1}-(r-1)^{n+1}}{n+1}-(r-1)^{n}\right)\frac{\operatorname{vol}(V)}{r^{n-1}}$
		$\displaystyle=1$

We note that there is a second, simpler proof. Since $\beta_{Y}(bv)=b\beta_{Y}(v)$ . By the linearity of the Futaki character we have that $\beta_{Y}(V_{0})=\beta_{Y}(\mathrm{wt}_{1})=-\beta_{Y}(-\mathrm{wt}_{1})=-\beta_{Y}(\mathrm{wt}_{-1})=-\beta_{Y}(\overline{V}_{\infty})$ , and that $\beta_{Y}(V_{0})=\beta_{Y}(\overline{V}_{\infty})$ . ∎

4. Proof of Main Theorem

4.1. K-Semistability

In the light of Theorem 2.11 and the natural $\mathbb{T}$ -action on $Y$ , we investigate the $\mathbb{T}$ -equivariant K-semistability of $Y$ in relation to the K-semistability of $(V,aB)$ .

4.1.1. Reverse Implication

Definition 4.1.

(vertical and horizontal divisors [Che08, Definition 1.8]) Let $D$ be a $\mathbb{T}$ -invariant divisor over $Y$ . $D$ is called a vertical divisor if the maximal $\mathbb{T}$ -orbit in $D$ has the same dimension as $\mathbb{T}$ , and is otherwise called a horizontal divisor.

Lemma 4.2.

For $D$ a vertical divisor over $Y$ , there exists a vertical divisor $D^{\prime}$ over $Y$ such that $\beta_{Y}(D)=\beta_{Y}(D^{\prime})$ and $c_{Y}(D^{\prime})\cap\overline{V}_{\infty}$ is nonempty.

Proof.

Let $D$ be a vertical divisor over $Y$ . Then $c_{Y}(D)$ is a $\mathbb{T}$ -invariant subvariety of $Y$ . Either the closed points of $c_{Y}(D)$ are all fixed by the $\mathbb{T}$ -action or $c_{Y}(D)$ consists of closures of orbits of points not fixed by $\mathbb{T}$ . In the later case, these closures of orbits are either fibres of $E$ thought of as a $\mathbb{P}^{1}$ -bundle over $B_{\infty}$ , pullbacks of fibres of $X\to V$ away from $B\in V$ , or the strict transform of fibres of $X\to V$ over $B$ . If $c_{Y}(D)$ contains any fibres from the first two cases, these fibres, and thus $c_{Y}(D)$ , have nonempty intersection with $\overline{V}_{\infty}$ , so we set $D^{\prime}:=D$ . If this is not the case, then $c_{Y}(D)\subset F$ . From here, we consider the divisorial valuation $v^{\prime}:=\operatorname{ord}_{D}\circ\iota$ and let $D^{\prime}$ be a divisor associated to the valuation $v^{\prime}$ . Then $c_{Y}(D^{\prime})=\iota(c_{Y}(D))\subset E$ and $c_{Y}(D^{\prime})$ consists of fibres of $E\to B_{\infty}$ , so $c_{Y}(D^{\prime})\cap\overline{V}_{\infty}$ is nonempty. Since $\operatorname{ord}_{D^{\prime}}=\operatorname{ord}_{D}\circ\iota$ , $\beta_{Y}(D)=\beta_{Y}(D^{\prime})$ .

If all closed points $y\in c_{Y}(D)$ are fixed by the $\mathbb{T}$ -action, then we have that $c_{Y}(D)$ is either contained in $\overline{V}_{\infty}$ , $V_{0}$ , or $E\cap F$ . If $c_{Y}(D)\subset\overline{V}_{\infty}$ , then take $D^{\prime}:=D$ . If $c_{Y}(D)\subset V_{0}$ , we consider the divisorial valuation $v^{\prime}:=\operatorname{ord}_{D}\circ\iota$ and let $D^{\prime}$ be a divisor associated to the valuation $v^{\prime}$ . Then $c_{Y}(D^{\prime})=\iota(c_{Y}(D))$ and $c_{Y}(D^{\prime})\cap\overline{V}_{\infty}=\iota(c_{Y}(D)\cap V_{0})\neq\emptyset$ . Again, since $\operatorname{ord}_{D^{\prime}}=\operatorname{ord}_{D}\circ\iota$ , $\beta_{Y}(D)=\beta_{Y}(D^{\prime})$ .

Suppose $c_{Y}(D)\subset E\cap\iota(E)$ , and denote $\operatorname{ord}_{D}$ as $v$ . Let $\mu:=\operatorname{ord}_{D}|_{\mathbb{C}(V)}$ where the inclusion $\mathbb{C}(V)\subset\mathbb{C}(Y)$ is induced by the composition $\pi\circ\phi$ .

Let $\mu$ be a valuation on $V$ and $c_{V}(\mu)\subset U\subset V$ be a Zariski-open neighborhood of $c_{V}(\mu)$ . Choose a trivialization of $\mathcal{L}$ over $U$ . This trivialization gives a birational map $X\dashrightarrow V\times\mathbb{A}^{1}$ , and composing by $\pi$ gives us a birational map $Y\dashrightarrow V\times\mathbb{A}^{1}$ . This birational map induces an isomorphism on function fields, so we have $\mathbb{C}(Y)\cong\mathbb{C}(V\times\mathbb{A}^{1})\cong\mathbb{C}(V)(t)$ .

We define the valuation $v_{\mu,b}$ on $Y$ for $\mu$ a valuation on $V$ and $b\in\mathbb{Z}$ as follows:

v_{\mu,b}(f)=\min_{i\in\mathbb{Z}}\{\mu(f_{i})+bi\}

where $f=\sum_{i}f_{i}t^{i}$ is the weight decomposition under the $\mathbb{G}_{m}$ action via the above isomorphism of function fields. Moreover, every $\mathbb{T}$ -invariant valuation on $Y$ is of this form for some (not necessarily unique) choice of $\mu,$ $U,$ and $b$ . Indeed, suppose $v$ is a $\mathbb{T}$ -invariant valuation and let $\mu=v\circ\overline{s_{0}}$ where $\overline{s_{0}}:V\to Y$ is the lifting of the zero section $s_{0}:V\to X$ . Choose $U$ such that $\mathcal{L}$ has a trivialization over $U$ and $c_{V}(\mu)\subset U$ . Let $b=v(t)$ under the isomorphism $\mathbb{C}(Y)\cong\mathbb{C}(V)(t)$ induced by the trivialization. Then, for $f=\sum f_{i}t^{i}\in\mathbb{C}(Y)$ ,

\displaystyle v(f)=\min_{i\in\mathbb{Z}}\{v(f_{i}t^{i})\}=\min_{i\in\mathbb{Z}}\{v(f_{i})+bi\}=v_{\mu,b}(f)

We then have by [Li22, Proposition 3.12],

\beta(v_{b})=\beta(v)+b\operatorname{Fut}(\xi)=\beta(v)=\beta(\mu^{*}),

with $\mu^{*}=v_{\mu,0}$ . The second equality follows from Lemma 3.5. Now, we note that $c_{Y}(\mu^{*})=\phi^{-1}_{*}(c_{V}(\mu))$ and thus consists of $1$ -dimensional orbits of the $\mathbb{T}$ -action. If $c_{Y}(\mu^{*})\cap\overline{V}_{\infty}$ is nonempty, we take $D^{\prime}$ to be a divisor over $Y$ such that $D^{\prime}$ is the associated divisor to the divisorial valuation $\mu^{*}$ . If $c_{Y}(\mu^{*})\cap\overline{V}_{\infty}$ is empty, then $c_{Y}(\mu^{*})\subset F$ , so we consider the valuation $\mu^{*^{\prime}}:=\mu^{*}\circ\iota$ , and take $D^{\prime}$ to be a divisor associated to $\mu^{*^{\prime}}$ . From there we see that $\beta_{Y}(D^{\prime})=\beta_{Y}(\mu^{*})=\beta_{Y}(D)$ . ∎

Lemma 4.3.

The only horizontal divisors over $Y$ are $\overline{V}_{\infty}$ and $V_{0}$ .

Proof.

Any horizontal divisor must solely consist of points fixed by the torus action since the maximal orbit contained in the divisor is strictly less than 1 and thus must be zero dimensional. Thus, the only horizontal $\mathbb{T}$ -invariant divisors on $Y$ are $\overline{V}_{\infty}$ and $V_{0}$ , as all fixed points are either contained with these two divisors or in the intersection $E\cap F$ .

A nontrivial horizontal valuation $v$ when restricted to $\mathbb{C}(Y)^{\mathbb{T}}$ will be the trivial valuation, and thus will be equal to $\mathrm{wt}_{b}$ for some $b\in\mathbb{Z},b\neq 0$ . If $b>0$ , then the associated divisor will be $V_{0}$ , and if $b<0$ then the associated divisor will be $\overline{V}_{\infty}$ . ∎

In order to compare $\delta$ -invariants using the techniques developed in [AZ22], we want to consider the refinement of the linear system $T_{\bullet}$ where $T_{m}=H^{0}(Y,-mK_{Y})$ by the divisor $\overline{V}_{\infty}$ , which we will denote as $W_{\bullet,\bullet}$ . This is defined (see Definition 2.9) as

W_{m,j}:=\text{Im}\left(H^{0}(Y,-mK_{Y}-j\overline{V}_{\infty})\xrightarrow{\rho}H^{0}(\overline{V}_{\infty},(-mK_{Y}-j\overline{V}_{\infty})|_{\overline{V}_{\infty}})\right)

where the map $\rho$ is induced by the inclusion of $\overline{V}_{\infty}$ into $Y$ .

Lemma 4.4.

The refinement of $T_{\bullet}$ by $\overline{V}_{\infty}$ is

W_{m,j}=\begin{cases}H^{0}(\overline{V}_{\infty},(-mK_{Y}-j\overline{V}_{\infty})|_{\overline{V}_{\infty}}),&\text{if }j\leq m\\ (j-m)B+H^{0}(\overline{V}_{\infty},P(m,j)|_{\overline{V}_{\infty}}),&\text{if }m<j\leq 2m\\ 0,&\text{if }j>2m\end{cases}

The movable part of $W_{m,j}$ , denoted as $M_{m,j}$ , is

M_{m,j}=\begin{cases}H^{0}(V,-mK_{V}-(m-j)\mathcal{L}),&\text{if }j\leq m\\ H^{0}(V,-mK_{V}+(m-j)\mathcal{L}),&\text{if }m<j\leq 2m\\ 0,&\text{if }j>2m\end{cases}

and the fixed part of $W_{m,j}$ , denote as $F_{m,j}$ , is

F_{m,j}=\begin{cases}0,&\text{if }j\leq m\\ (j-m)B,&\text{if }n<j\leq 2m\\ 0,&else\end{cases}

Proof.

For $j>2m$ , $-mK_{Y}-j\overline{V}_{\infty}$ is no longer psuedo-effective, and thus the image of $\rho$ is $0$ .

Denoting the positive (resp. negative) part of the Zariski decomposition of the divisor $-mK_{Y}-j\overline{V}_{\infty}$ with $P(m,j)$ (resp. $N(m,j)$ ), we have

P(m,j)=\begin{cases}-mK_{Y}-j\overline{V}_{\infty},&\text{if }j\leq m\\ -mK_{Y}-j\pi^{*}(V_{\infty}),&\text{if }m<j\leq 2m\end{cases}

and

N(m,j)=\begin{cases}0,&\text{if }j\leq m\\ (j-m)E,&\text{if }m<j\leq 2m\end{cases}

Again, we can see that the above is the Zariski decomposition as follows: for $j\leq m$ , the divisor is nef, and thus has no negative part. For $m<j\leq 2m$ , $P(m,j)$ is the pullback of a nef divisor along the contraction $Y\to X$ and $N(m,j)$ is supported on the contracted locus.

To compute $W_{\bullet,\bullet}$ we use the following long exact sequence from the restriction $\rho$ :

	$\displaystyle\dots\rightarrow H^{0}(Y,-mK_{Y}-(j+1)\overline{V}_{\infty})\rightarrow H^{0}(Y,-mK_{Y}-j\overline{V}_{\infty})\xrightarrow{\rho}$
	$\displaystyle H^{0}(\overline{V}_{\infty},(-mK_{Y}-j\overline{V}_{\infty})\|_{\overline{V}_{\infty}})\rightarrow H^{1}(Y,-mK_{Y}-(j+1)\overline{V}_{\infty})\rightarrow\dots$

Thus, we see that the map on global sections induced by restriction is surjective when $H^{1}(Y,-mK_{Y}-(j+1)\overline{V}_{\infty})$ vanishes, which occurs when $j\leq m$ by the Kawamata-Viehweg Vanishing Theorem. More specifically, when $j\leq m$ , $t:=\frac{j+1}{m+1}\leq 1$ and $-K_{Y}-t\overline{V}_{\infty}$ is both big and nef, and thus so is $(m+1)(-K_{Y}-t\overline{V}_{\infty})=-(m+1)K_{Y}-(j+1)\overline{V}_{\infty}$ , so by Kawamata-Viehweg Vanishing Theorem, $H^{i}(Y,-mK_{Y}-(j+1)\overline{V}_{\infty})=0$ for $i>0$ .

Now we consider when $m<j\leq 2m$ . In this case, the restriction of the negative part of the Zariski decomposition gives us the fixed part of the refinement, which is $(j-m)E_{\overline{V}_{\infty}}=(j-m)B$ . For the movable part, we see that $P(m,j)$ is again big and nef, so $H^{1}(Y,P(m,j))=0$ . Thus, by the analogous long exact sequence from the restriction of $P(m,j)$ to $\overline{V}_{\infty}$ , we see that the restriction map is surjective, implying that the free part of the refinement is the space of global sections of the restriction of $P(m,j)$ . ∎

Note that, since $r\mathcal{L}\sim_{\mathbb{Q}}-K_{Y}$ , we have

	$\displaystyle-mK_{Y}\|_{\overline{V}_{\infty}}-j\overline{V}_{\infty}\|_{\overline{V}_{\infty}}$	$\displaystyle=-m(K_{Y}+\overline{V}_{\infty})\|_{\overline{V}_{\infty}}-(j-m)\overline{V}_{\infty}\|_{\overline{V}_{\infty}}$
		$\displaystyle=-mK_{\overline{V}_{\infty}}-(j-m)(V_{\infty}-E)\|_{\overline{V}_{\infty}}$
		$\displaystyle=-mK_{\overline{V}_{\infty}}-(j-m)(\mathcal{L}-B)=mrL-(j-m)(-\mathcal{L})$
		$\displaystyle=(mr-m+j)\mathcal{L}$

When $j\leq m$ , this is then the free part of $W_{m,j}$ . When $m<j\leq 2m$ , we see that the fixed part $F_{m,j}$ of $W_{m,j}$ is $(j-m)B=2(j-m)\mathcal{L}$ , and thus the free part is $(mr-m+j)\mathcal{L}-2(j-m)\mathcal{L}=(mr+m-j)\mathcal{L}$ .

Let $N_{m,j}$ denote the dimension of $H^{0}(V,W_{m,j})$ , and $N_{m}=\sum_{j}N_{m,j}$ . Let

a(n,r):=\frac{r^{n+1}-(r-1)^{n+1}-(n+1)(r-1)^{n}}{2(n+1)(r^{n}-(r-1)^{n})}

and let $a_{m}B$ be the fixed part of an $m$ -basis type $\mathbb{Q}$ -divisor of $W_{\bullet,\bullet}$ . Then from the above description of $W_{\bullet,\bullet}$ , we have that

a_{m}=\frac{1}{mN_{m}}\sum_{j=0}^{2m}N_{m,j}a_{m,j}

Lemma 4.5.

\lim_{m\to\infty}a_{m}(n,r)=a(n,r)

Proof.

We note that, asymptotically,

	$\displaystyle N_{m}$	$\displaystyle=\sum_{j=0}^{m}\frac{(mr-m+j)^{n-1}\text{vol}(\mathcal{L})}{(n-1)!}$
		$\displaystyle+\sum_{j=m+1}^{2m}\frac{(mr+m-j)^{n-1}\text{vol}(\mathcal{L})}{(n-1)!}+O(m^{n-2})$
		$\displaystyle=\frac{m^{n}\text{vol}(\mathcal{L})}{(n-1)!}\left(\sum_{j=0}^{m}\frac{1}{m}(r-1+\frac{j}{m})^{n-1}\right.$
		$\displaystyle+\left.\sum_{j=m+1}^{2m}\frac{1}{m}(r+1-\frac{j}{m})^{n-1}+\frac{(n-1)!}{m^{n}\text{vol}(\mathcal{L})}O(m^{n-2})\right)$

Thus, we have

	$\displaystyle\sum_{j=0}^{2m}N_{m,j}a_{m,j}$	$\displaystyle=\frac{\text{vol}(\mathcal{L})}{(n-1)!}\left(\sum_{m+1}^{2m}(mr+m-j)^{n-1}(j-m)+O(m^{n-2})\right)$
		$\displaystyle=\frac{m^{n+1}\text{vol}(\mathcal{L})}{(n-1)!}\left(\sum_{m+1}^{2m}\frac{1}{m}(r+1-\frac{j}{m})^{n-1}(\frac{j}{m}-1)+\frac{1}{m^{n}+1}O(m^{n-2})\right)$

Thus we compute the limit as follows:

	$\displaystyle\lim_{m\to\infty}a_{m}$	$\displaystyle=\lim_{m\to\infty}\frac{1}{mN_{m}}\sum_{j=0}^{2m}N_{m,j}a_{m,j}$
		$\displaystyle=\lim_{m\to\infty}\frac{\sum_{j=0}^{m}\frac{1}{m}(r-1+\frac{j}{m})^{n-1}+\sum_{j=m+1}^{2m}\frac{1}{m}(r+1-\frac{j}{m})^{n-1}}{\sum_{m+1}^{2m}\frac{1}{m}(r+1-\frac{j}{m})^{n-1}(\frac{j}{m}-1)}$
		$\displaystyle=\frac{\int_{1}^{2}(r+1-t)^{n-1}(t-1)dt}{2\int_{1}^{2}(r+1-t)^{n-1}dt}$
		$\displaystyle=\frac{1}{2}\left[\frac{-(r-1)^{n}-\frac{(r-1)^{n+1}-r^{n+1}}{n+1}}{r^{n}-(r-1)^{n}}\right]$
		$\displaystyle=a$

∎

In the terminology of [AZ22], $a(n,r)$ is the coefficient of $B$ in the asymptotic fixed part $F(W_{\bullet,\bullet})$ of $W_{\bullet,\bullet}$ .

Lemma 4.6.

Let $D$ be a divisor over $Y$ and $Z\subset c_{Y}(D)\subset Y$ be a subvariety of $Y$ contained in the center of $D$ on $Y$ . If $\delta_{Z}(Y)\geq 1$ , then $\beta_{Y}(D)\geq 0$ .

Proof.

Suppose $\delta_{Z}(Y)\geq 1$ . Then, by definition of the local $\delta$ -invariant, $\frac{A_{Y}(D^{\prime})}{S_{Y}(D^{\prime})}\geq 1$ for all divisors $D^{\prime}$ over $Y$ whose center on $Y$ contains $Z$ . This includes $D$ . Thus, by rearranging terms, $\beta_{Y}(D)=A_{Y}(D)-S_{Y}(D)\geq 0$ . ∎

Lemma 4.7.

Let $Z$ be a subvariety of $Y$ such that $Z\subset\overline{V}_{\infty}$ . Suppose $(V,aB)$ is K-semistable. Then $\delta_{Z}(Y)\geq 1$ .

Proof.

By [AZ22, Theorem 3.3], we have that

\delta_{Z}(Y)\geq\text{min}\left\{\frac{A_{Y}(\overline{V}_{\infty})}{S_{Y}(\overline{V}_{\infty})},\delta_{Z}(\overline{V}_{\infty},W_{\bullet,\bullet})\right\}

Where $W_{\bullet,\bullet}$ is the refinement of the graded linear series associated to $-K_{Y}$ by $\overline{V}_{\infty}$ . The first value in the minimum is $1$ by Lemma 3.5. Thus, it remains to show that $\delta_{Z}(\overline{V}_{\infty},W_{\bullet,\bullet})\geq 1$ .

By the assumption that the pair $(V,aB)$ is K-semistable, we have that $\delta(V,aB)\geq 1$ , which in particular means that, for a choice of $\epsilon>0$ , $\delta_{M}(V,aB)>1-\epsilon$ for all $M$ sufficiently large. So, for any $M$ -basis type $\mathbb{Q}$ divisor $\Delta_{M}$ of $-K_{V}-aB$ , we have that $(V,(1-\epsilon)\Delta_{M}+aB)$ is lc.

Since $-K_{V}-aB=(r-2a)\mathcal{L}$ , for any such $\Delta_{M}$ , $(r-2a)\Delta_{M}$ is an $M(r-2a)$ -basis type $\mathbb{Q}$ -divisor of $\mathcal{L}$ , and thus we have $\delta_{M(r-2a)}(V,aB;\mathcal{L})>(r-2a)(1-\epsilon)$ .

Now, let $D_{m}$ be an $m$ -basis type divisor of $W_{\bullet,\bullet}$ , then

D_{m}=\frac{1}{mN_{m}}\sum_{j=0}^{2m}D_{m,j}

where each $D_{m,j}$ is a basis sum divisor of $W_{m,j}$ . By our computation of $W_{m,j}$ , we see that $D_{m,j}=\Delta_{mr-|m-j|}+a_{m,j}N_{m,j}B$ , where $\Delta_{mr-|m-j|}$ is a basis sum divisor of $H^{0}((mr-|m-j|)\mathcal{L})$ and $a_{m,j}=j-m$ for $2m\geq j>m$ and $0$ otherwise. In light of the previous discussion, we can choose $m$ such that $mr-|m-j|$ is sufficiently large and hence the pair

\left(V,(r-2a)(1-\epsilon)\frac{1}{(mr-|m-j|)N_{m,j}}\Delta_{mr-|m-j|}+aB\right)

is log canonical for all $0\leq j\leq 2m$ . Letting $w_{m,j}=\frac{(mr-|m-j|)N_{m,j}}{m(r-2a_{m})N_{m}}$ . Then we see that $\sum_{j=0}^{2m}w_{m,j}=1$ , so by the convexity of log canonicity, we have that the pair

		$\displaystyle\left(V,(r-2a)(1-\epsilon)\sum_{j=0}^{2m}w_{m,j}\left(\frac{1}{(mr-\|m-j\|)N_{m,j}}\Delta_{mr-\|m-j\|}+aB\right)\right)$
		$\displaystyle=\left(V,(1-\epsilon)\frac{1}{mN_{m}}\left(\sum_{j=0}^{2m}\Delta_{mr-\|m-j\|}+a_{m,j}N_{m,j}B\right)\right)$
		$\displaystyle=\left(V,(1-\epsilon)D_{m}\right)$

is log canonical. Thus, we see that $\delta_{Z}(\overline{V}_{\infty},W_{\bullet,\bullet})\geq 1$ as desired. ∎

Proposition 4.8.

If the pair $(V,aB)$ is K-semistable, then $Y$ is K-semistable.

Proof.

We will show that the K-semistability of the pair $(V,aB)$ implies the $\mathbb{T}$ -equivariant K-semistability of $Y$ , which then implies the K-semistability of $Y$ by Theorem 2.11. Suppose $D$ is a $\mathbb{T}$ -invariant divisor over $Y$ . We proceed to show $\beta_{Y}(D)\geq 0$ . If $D$ is a horizontal divisor, then $\beta_{Y}(D)=0$ by Lemma 4.3 and Lemma 3.5. Thus, we suppose $D$ is vertical. Then, by Lemma 4.2, we may assume $c_{Y}(D)\cap\overline{V}_{\infty}$ is nonempty and thus contains some subvariety $Z$ . By Theorem 4.7, $\delta_{Y}(Z)\geq 1$ , and thus $\beta_{Y}(D)\geq 0$ by Lemma 4.6. ∎

4.1.2. Forward Implication

First we proceed with a $\mathbb{G}_{m}$ -equivariant strengthening of [AZ22, Lemma 3.1] and [AZ22, Proposition 3.2].

Lemma 4.9.

Let $H$ be a finite dimensional vector space over $\mathbb{C}$ with an action of $\mathbb{G}_{m}$ and two $\mathbb{G}_{m}$ -equivariant filtrations $\mathcal{F},\mathcal{G}$ . Then there exists a $\mathbb{G}_{m}$ -invariant basis of $H$ compatible with both $\mathcal{F}$ and $\mathcal{G}$ .

Proof.

The proof mostly follows that of the analogous statement in [AZ22]. Consider the induced filtrations $\mathcal{G}_{i}$ on $Gr_{i}^{\mathcal{F}}H$ for each $i$ . The equivariance of $\mathcal{F}$ implies that the $\mathbb{G}_{m}$ action on $H$ restricts to an action on each $Gr_{i}^{\mathcal{F}}H$ ; the equivariance of $\mathcal{G}$ implies the equivariance of each $\mathcal{G}_{i}$ and thus the aforementioned action of $\mathbb{G}_{m}$ restricts to an action on $Gr_{j}^{\mathcal{G}_{i}}Gr_{i}^{\mathcal{F}}H$ for all $i,j$ . For each $i$ and $j$ , take a basis $B_{i,j}$ of $\mathbb{G}_{m}$ -eigenvectors of $Gr_{j}^{\mathcal{G}_{i}}Gr_{i}^{\mathcal{F}}H$ ; for each $i$ lift these bases to a basis $B_{i}$ of $Gr_{i}^{\mathcal{F}}H$ . Each of these bases will consist of $\mathbb{G}_{m}$ -eigenvectors for the action on $Gr_{i}^{\mathcal{F}}H$ ; these then lift to a $\mathcal{F}$ -compatible $\mathbb{G}_{m}$ -invariant basis $B$ of $H$ . Since $Gr_{j}^{\mathcal{G}_{i}}Gr_{i}^{\mathcal{F}}H=Gr_{i}^{\mathcal{F}_{j}}Gr_{j}^{\mathcal{G}}H$ for all $i,j$ , we see that for each $j$ , $\bigcup_{i}B_{i,j}$ forms a basis of $Gr_{j}^{\mathcal{G}}H$ and that $B$ is the lift of these bases and is thus $\mathcal{G}$ -compatible. ∎

Lemma 4.10.

For a linear system $H_{\bullet}$ , a filtration $\mathcal{F}$ on $H_{\bullet}$ , and a valuation $v$ on $H_{\bullet}$ , we have

S(H_{\bullet};v)=S_{\mathbb{G}_{m}}(H_{\bullet};\mathcal{F};v)

Proof.

We have that $S_{m}(H_{\bullet};v)\geq S_{\mathbb{G}_{m},m}(H_{\bullet};v)$ by definition. Let $D_{m}$ be a m-basis type $\mathbb{Q}$ -divisor of $H_{\bullet}$ that is $\mathbb{G}_{m}$ -invariant and compatible with $\mathcal{F}$ and $\mathcal{F}_{v}$ ; such a $D_{m}$ exists by Lemma 4.9. Then $S_{m}(H_{\bullet};v)=v(D_{m})\leq S_{\mathbb{G}_{m},m}(H_{\bullet};v)$ . Taking the limit of each side as $m$ approaches infinity yields the lemma. ∎

Lemma 4.11.

Fix $m$ such that $h^{0}(Y,-mK_{Y})\neq 0$ . There is a one-to-one correspondence between $\mathbb{T}$ -invariant $m$ -basis type $\mathbb{Q}$ -divisors of $-K_{Y}$ and $m$ -basis type $\mathbb{Q}$ -divisors of $W_{\bullet,\bullet}$ .

Proof.

Since there is a clear one-to-one correspondence between $m$ -basis type $\mathbb{Q}$ divisors and bases of global sections modulo scaling each basis element, this lemma follows from an analogous correspondence between bases of $R_{m}:=H^{0}(-mK_{Y})$ invariant under the $\mathbb{T}$ -action and sets of bases for $H^{0}(W_{m,j})$ for $0\leq j\leq 2m$ . We can decompose $R_{m}=\oplus R_{m,j}$ into the direct sum of eigenspaces under the $\mathbb{T}$ -action, where $\mathbb{T}$ acts on $s\in R_{m,j}$ by $\sigma(\lambda)s=\lambda^{j}s$ ; a $\mathbb{T}$ -invariant basis on $R_{m}$ is exactly a union of bases on each $R_{m,j}$ . In particular, each $R_{m,j}$ is isomorphic to the $j$ -th graded piece of the weight filtration on $R_{m}$ , which is exactly the filtration induced by $\mathrm{wt}_{-1}=\operatorname{ord}_{\overline{V}_{\infty}}$ . Thus, by construction, $R_{m,j}\cong H^{0}(W_{m,j})$ , and the correspondence between bases follows. ∎

With lemmata in hand, we begin our argument for the forward implication of the main theorem by constructing divisors over $Y$ whose $\beta$ -invariants are dictated by divisors over $V$ .

Convention 4.12.

Let $D_{V}$ be a divisor over $V$ . Then $D_{V}$ defines a divisor $D_{Y}$ over $Y$ as follows:

Let $W\to V$ be a model on which $D_{V}$ is a divisor. We consider the birational model $X_{W}:=W\times_{V}X\xrightarrow{\eta}X$ of $X$ , and the divisor $D_{X}:={\phi_{W}}^{*}(D_{V})$ . $D_{X}$ defines a divisorial valuation on $X$ , which we pullback along the blow-up map $\pi$ to a divisorial valuation $\operatorname{ord}_{D_{X}}$ on $Y$ . We then pull back $\operatorname{ord}_{D_{X}}$ along $\iota$ to a divisorial valuation $\operatorname{ord}_{D_{Y}}$ achieved by some divisor $D_{Y}$ over $Y$ .

Lemma 4.13.

Under the inclusion $\mathbb{C}(V)\subset\mathbb{C}(Y)$ of function fields induced by the composition $\pi\circ\phi$ , for a divisor $D_{V}$ over $V$ and $D_{Y}$ as in Convention 4.12, we have the identification $\operatorname{ord}_{D_{Y}}|_{\mathbb{C}(V)}=\operatorname{ord}_{D_{V}}$ .

Proof.

It is clear that $\operatorname{ord}_{D_{X}}|_{\mathbb{C}(V)}=\operatorname{ord}_{D_{V}}$ via the inclusion of function fields induced by $\phi$ . Now, $\operatorname{ord}_{D_{Y}}=\operatorname{ord}_{D_{X}}\circ\iota$ by construction, so $\operatorname{ord}_{D_{Y}}=\operatorname{ord}_{D_{X}}$ on meromorphic functions that are fixed by composition with $\iota$ . Since $\iota$ fixes fibres of the conic bundle structure of $\pi\circ\phi:Y\to V$ , it in particular fixes $\mathbb{C}(V)\subset\mathbb{C}(Y)$ , and the lemma follows. ∎

Lemma 4.14.

Let $D_{V}$ be a divisor over $V$ and define $D_{Y}$ as in Convention 4.12. Then $A_{Y}(D_{Y})=A_{V}(D_{V})$ .

Proof.

First we note that $A_{Y}(D_{Y})=A_{Y}(\operatorname{ord}_{D_{X}})$ since $\operatorname{ord}_{D_{X}}\circ\iota=\operatorname{ord}_{D_{Y}}$ . Then, we compute $A_{Y}(D_{X})$ as follows:

	$\displaystyle A_{Y}(D_{X})$	$\displaystyle=\operatorname{ord}_{D_{X}}(K_{-/Y})+1$
		$\displaystyle=\operatorname{ord}_{D_{X}}(K_{-/X})-\operatorname{ord}_{D_{X}}(E)+1$
		$\displaystyle=A_{X}(D_{X})-\operatorname{ord}_{D_{X}}(E)$
		$\displaystyle=A_{X}(D_{X})$
		$\displaystyle=A_{V}(D_{V})$

where the notation $K_{-/Z}$ for $Z=X,Y$ denotes the relative canonical divisor of an appropriate choice of resolution of $Z$ , and the last equality follows from the canonical bundle formula for projective bundles. In particular, $K_{X_{W}}-\eta^{*}(K_{X})=\phi^{*}(K_{W}-f^{*}K_{V})$ . ∎

Proposition 4.15.

For $D_{V}$ a divisor over $V$ and $D_{Y}$ as in Convention 4.12, $\beta_{Y}(D_{Y})=\beta_{V,aB}(D_{V})$ .

Proof.

Using Lemma 4.9, let $\Delta_{m}^{Y}$ be a $\mathbb{T}$ -invariant $m$ -basis type $\mathbb{Q}$ -divisor of $-K_{Y}$ compatible with $D_{Y}$ and $\overline{V}_{\infty}$ . Then $\Delta_{m}^{Y}$ decomposes $\mathbb{T}$ -equivariantly as below, with $\operatorname{Supp}(\Gamma)$ consisting of $\mathbb{T}$ -invariant divisors distinct from $\overline{V}_{\infty}$ (see [AZ22, Section 3.1]).

\Delta_{m}^{Y}=\Gamma+S_{m}(-K_{Y},\overline{V}_{\infty})\cdot\overline{V}_{\infty}

In particular, $\Gamma|_{\overline{V}_{\infty}}$ is the $m$ -basis type $\mathbb{Q}$ -divisor of $W_{\bullet,\bullet}$ corresponding to $\Delta_{m}^{Y}$ under Lemma 4.11, i.e. $\Gamma|_{\overline{V}_{\infty}}=\frac{1}{mN_{m}}\sum_{j}(\Delta_{m,j}+a_{m,j}N_{m,j}B)$ , where $\Delta_{m,j}$ is a basis-sum divisor of the movable part of the refinement $M_{m,j}$ . Since $\Delta_{m}^{Y}$ is compatible with $D_{Y}$ , each $\Delta_{m,j}$ is compatible with the restriction of $\operatorname{ord}_{D_{Y}}$ to $\mathbb{C}(V)$ , which in particular means that each $\Delta_{m,j}$ is compatible with $D_{V}$ . Since the irreducible components of $\operatorname{Supp}(\Gamma)$ are invariant under the $\mathbb{T}$ action, we see that

\Gamma=\frac{1}{mN_{M}}\left(\sum_{j}\Gamma_{j}+a_{m,j}N_{m,j}E\right)

where $\Gamma_{j}|_{\overline{V}_{\infty}}=\Delta_{m,j}$ and each $\Gamma_{j}$ is strictly supported on the $\mathbb{T}$ -invariant divisors on $Y$ . We compute $\operatorname{ord}_{D_{Y}}(\Gamma_{j})$ by decomposing $\Gamma_{j}$ into its parts supported on specific $\mathbb{T}$ -invariant divisors. Let $D$ denote all divisors on $V$ (other than $B$ ) such that $\Gamma$ has support on ${\pi}_{*}^{-1}(\phi^{*}D)$ . We note that $\operatorname{ord}_{D_{Y}}(\overline{V}_{\infty}),\operatorname{ord}_{D_{Y}}(V_{0})$ and $\operatorname{ord}_{D_{Y}}(F)$ are all $0$ : for a choice any one of these three divisors we can find an open neighborhood $U$ of $c_{Y}(D_{Y})$ that has empty intersection with the chosen divisor. Thus, on $U$ , the chosen divisor is defined locally by a nonvanishing holomorphic function $f$ , in particular $f$ has vanishing order $0$ at $c_{Y}(D_{Y})$ . Similarly, we compute $\operatorname{ord}_{D_{Y}}(E)=\operatorname{ord}_{{\pi}_{*}^{-1}(D_{X})}(F)=\operatorname{ord}_{D_{X}}(\phi^{*}(B))=\operatorname{ord}_{V}(B)$ and $\operatorname{ord}_{D_{Y}}({\pi}_{*}^{-1}(\phi^{*}D))=\operatorname{ord}_{D_{V}}(D)$ . Thus, we have the following computation:

	$\displaystyle\operatorname{ord}_{D_{Y}}(\Gamma_{j})$	$\displaystyle=\mathrm{coeff}_{V_{0}}(\Gamma_{j})\cdot\operatorname{ord}_{D_{Y}}(V_{0})+\mathrm{coeff}_{\overline{V}_{\infty}}(\Gamma_{j})\cdot\operatorname{ord}_{D_{Y}}(\overline{V}_{\infty})$
		$\displaystyle+\mathrm{coeff}_{E}(\Gamma_{j})\cdot\operatorname{ord}_{D_{Y}}(E)+\mathrm{coeff}_{F}(\Gamma_{j})\cdot\operatorname{ord}_{D_{Y}}(F)$
		$\displaystyle+\mathrm{coeff}_{{\pi}_{}^{-1}(\phi^{}D)}(\Gamma_{j})\cdot\operatorname{ord}_{D_{Y}}({\pi}_{}^{-1}(\phi^{}D))$
		$\displaystyle=\mathrm{coeff}_{E}(\Gamma_{j})\cdot\operatorname{ord}_{D_{Y}}(E)+\mathrm{coeff}_{{\pi}_{}^{-1}(\phi^{}D)}(\Gamma_{j})\cdot\operatorname{ord}_{D_{Y}}({\pi}_{}^{-1}(\phi^{}D))$
		$\displaystyle=\mathrm{coeff}_{B}(\Delta_{m,j})\cdot\operatorname{ord}_{D_{V}}(B)+\mathrm{coeff}_{D}(\Delta_{m,j})\operatorname{ord}_{D_{V}}(D)$
		$\displaystyle=\operatorname{ord}_{D_{V}}(\Delta_{m,j})$

As an aside, we note that $\mathrm{coeff}_{\overline{V}_{\infty}}(\Gamma_{j})=0$ from the decomposition of $\Delta_{m}^{Y}$ .

With this, we have the following computation of $S_{m}(D_{Y})$ :

	$\displaystyle S_{m}(D_{Y})=\operatorname{ord}_{D_{Y}}\Delta_{m}^{Y}$	$\displaystyle=\operatorname{ord}_{D_{Y}}\left(\Gamma+S_{m}(-K_{Y},\overline{V}_{\infty})\cdot\overline{V}_{\infty}\right)$
		$\displaystyle=\frac{1}{mN_{m}}\operatorname{ord}_{D_{Y}}\left(\sum_{j}\Gamma_{j}+a_{m,j}N_{m,j}E\right)$
		$\displaystyle+S_{m}(-K_{Y},\overline{V}_{\infty})\cdot\operatorname{ord}_{D_{Y}}(\overline{V}_{\infty})$
		$\displaystyle=\frac{1}{mN_{m}}\operatorname{ord}_{D_{Y}}\left(\sum_{j}\Gamma_{j}+a_{m,j}N_{m,j}E\right)$
		$\displaystyle=\frac{1}{mN_{m}}\left(\sum_{j}\operatorname{ord}_{D_{Y}}(\Gamma_{j})+a_{m,j}N_{m,j}\operatorname{ord}_{D_{Y}}(E)\right)$
		$\displaystyle=\frac{1}{mN_{m}}\left(\sum_{j}\operatorname{ord}_{D_{V}}(\Delta_{m,j})+a_{m,j}N_{m,j}\operatorname{ord}_{D_{V}}(B)\right)$
		$\displaystyle=\frac{1}{mN_{m}}\left(\sum_{j}mN_{m,j}S_{mr-\|m-j\|}(D_{V})+a_{m,j}N_{m,j}\operatorname{ord}_{D_{V}}B\right)$

Taking the limit as $m$ tends to $\infty$ , we are left with:

\displaystyle S(D_{Y})

\displaystyle=S(D_{V})+a\operatorname{ord}_{D_{V}}B

Also by Lemma 4.14 we have $A_{V}(D_{V})=A_{Y}(D_{Y})$ , so we have

	$\displaystyle\beta_{V,aB}(D_{V})$	$\displaystyle=A_{V,aB}(D_{V})-S(D_{V})$
		$\displaystyle=A_{V}(D_{V})-a\operatorname{ord}_{D_{V}}(B)-S(D_{V})$
		$\displaystyle=A_{Y}(D_{Y})-S(D_{Y})$
		$\displaystyle=\beta_{Y}(D_{Y})$

∎

Proposition 4.16.

If $(V,aB)$ is K-unstable, then so is $Y$ .

Proof.

Assume that $(V,aB)$ is K-unstable. Let $D_{V}$ be a destabilizing divisor of the pair $(V,aB)$ , that is a divisor on a birational model $W$ of $V$ such that $\beta_{V,aB}(D_{V})<0$ . Then by Proposition 4.15 $\beta_{Y}(D_{Y})<0$ . ∎

4.2. K-Polystability

By Proposition 4.16 and Proposition 4.8, if either of $(V,aB)$ or $Y$ is not K-semistable, then neither are K-semistable. Since K-polystability implies K-semistability, we assume, for this section, that both of $Y$ and the pair $(V,aB)$ are K-semistable.

4.2.1. Forward Implication

Proposition 4.17.

Suppose $Y$ is K-polystable. Then the log Fano pair $(V,aB)$ is K-polystable.

Proof.

Suppose $Y$ is K-polystable. Let $D_{V}$ be a divisor over the pair $(V,aB)$ such that $\beta_{V,aB}(D)=0$ . To show $(V,aB)$ is K-polystable it suffices to show that $D_{V}$ is a product-type divisor. We know that $\beta_{Y}(D_{Y})=\beta_{V,aB}(D_{V})=0$ with $D_{Y}$ as in Convention 4.12, so $D_{Y}$ is a product-type divisor. Denote this test configuration as $\mathcal{Y}\cong Y\times\mathbb{A}^{1}$ , and denote the $\mathbb{G}_{m}$ -action of the test configuration as $\sigma_{D_{Y}}:\mathbb{G}_{m}\times\mathcal{Y}\to\mathcal{Y}$ . Further, since $D_{Y}$ is invariant with respect to the $\mathbb{T}$ -action on $Y$ , $\mathcal{Y}$ is a $\mathbb{T}$ -equivariant test configuration.

Consider the closure of the orbit $\overline{\text{orb}}_{\sigma_{D}}(\overline{V}_{\infty}\times\{1\})\subset\mathcal{Y}$ under the action of $\sigma_{D_{Y}}$ on $\mathcal{Y}$ . Let $\mathcal{V}$ denote this closure. Since $\mathcal{Y}$ is a $\mathbb{T}$ -invariant test configuration, the $\mathbb{G}_{m}$ -action on $Y$ induced by $\sigma_{D_{Y}}$ commutes with the $\mathbb{T}$ -action $\sigma$ . We will denote this restriction as $\sigma_{D_{Y}}^{0}$ . For $t\in\mathbb{T},s\in\mathbb{G}_{m},$ and $y\in\overline{V}_{\infty}\subset Y$ , we have

	$\displaystyle\sigma(t)\sigma_{D_{Y}}^{0}(s)(y)$	$\displaystyle=\sigma_{D_{Y}}^{0}(s)\sigma(t)(y)$
		$\displaystyle=\sigma_{D_{Y}}^{0}(s)(y)$

So $\sigma_{D_{Y}}^{0}(s)(y)$ is in the fixed locus of $\sigma$ , which is $\overline{V}_{\infty}\sqcup V_{0}\sqcup(E\cap F)$ . However, since $\sigma_{D_{Y}}^{0}$ has connected orbits, we see that $\sigma_{D_{Y}}^{0}(s)(y)\in\overline{V}_{\infty}$ . Thus, $\sigma_{D_{Y}}^{0}$ maps points in $\overline{V}_{\infty}$ to points in $\overline{V}_{\infty}$ , which implies that $\sigma_{D_{Y}}$ maps points in $\overline{V}_{\infty}\times\mathbb{A}^{1}\subset\mathcal{Y}$ to points in $\overline{V}_{\infty}\times\mathbb{A}^{1}$ . From this we see that $\mathcal{V}\cong V\times\mathbb{A}^{1}$ and that $\mathcal{V}$ with the restriction of the $\mathbb{G}_{m}$ -action $\sigma_{D_{Y}}$ to $\mathcal{V}$ forms a product test configuration of $V$ .

The valuation associated to $\mathcal{V}$ is, by definition, is $\operatorname{ord}_{\mathcal{V}_{0}}|_{\mathbb{C}(V)}$ . We see that the $\mathbb{T}$ -action $\sigma$ extends to $\mathcal{Y}$ , and induces a weight decomposition on $\mathbb{C}(\mathcal{Y})$ (similar to that on $\mathbb{C}(Y)$ ) such that $\mathbb{C}(\mathcal{Y})^{\mathbb{T}}\cong\mathbb{C}(\mathcal{V})$ . From this, we see the $\operatorname{ord}_{\mathcal{Y}_{0}}|_{\mathbb{C}(\mathcal{V})}=\operatorname{ord}_{\mathcal{V}_{0}}$ . Since $\operatorname{ord}_{D_{Y}}=b\operatorname{ord}_{\mathcal{Y}_{0}}|_{\mathbb{C}(\mathcal{Y})}$ for some $b\in\mathbb{Z}_{>0}$ , we have that $b\operatorname{ord}_{\mathcal{V}_{0}}|_{\mathbb{C}(V)}=b\operatorname{ord}_{\mathcal{Y}_{0}}|_{\mathbb{C}(V)}=\operatorname{ord}_{D_{Y}}|_{\mathbb{C}(V)}=\operatorname{ord}_{D_{V}}$ . Thus, we see that the test configuration associated to $D_{V}$ is isomorphic to $\mathcal{V}$ and is thus a product test configuration. ∎

4.2.2. Reverse Implication

Proposition 4.18.

Suppose the log pair $(V,aB)$ is K-polystable. Then $Y$ is K-polystable.

Proof.

Suppose that the pair $(V,aB)$ is K-polystable; we seek to show the same for the construction $Y$ . In fact, due to 2.11, it suffices to show that $Y$ is $\mathbb{T}$ -equivariantly K-polystable.

Suppose $D$ is a $\mathbb{T}$ -invariant divisor over $Y$ such that $\beta_{Y}(D)=0$ ; we seek to show that the test configuration induced by $D$ is a product test configuration. If $D$ is horizontal, then $D=V_{0}$ or $\overline{V}_{\infty}$ ; both of these induce product test configurations. Indeed, $\operatorname{ord}_{V_{0}}=\mathrm{wt}_{\xi}$ and $\operatorname{ord}_{\overline{V}_{\infty}}=\mathrm{wt}_{-\xi}$ , so the test configurations induced by $D$ is the product test configuration given by either the $\mathbb{T}$ -action on $Y$ or its inverse.

Now, assume $D$ is vertical. Since the underlying space of two test configurations are isomorphic if one of the associated valuations is achieved by twisting the other by a suitable cocharacter, using Lemma 4.2 we may assume $c_{Y}(D)\cap\overline{V}_{\infty}\neq\emptyset$ .

Let $\mu:=\operatorname{ord}_{D}|_{\mathbb{C}(V)}$ , in which case we see via [Li22] that $\operatorname{ord}_{D}=v_{\mu,n\xi}$ for some $n\in\mathbb{N}$ , and denote the divisor associated to $\mu$ as $D_{\mu}$ . Then $\beta_{V,aB}(D_{\mu})=\beta_{Y}(D_{\mu,Y})$ , with $D_{\mu,Y}$ as in Convention 4.12 with respect to $D_{\mu}$ . Since $\operatorname{ord}_{D_{\mu,Y}}=v_{\mu,0}$ , we see that $D_{\mu,Y}$ and $D$ differ by a twist and thus $\beta_{V,aB}(D_{\mu})=\beta_{Y}(D_{\mu,Y})=\beta_{Y}(D)=0$ . Thus, by our assumption that the log pair $(V,aB)$ is K-polystable, the test configuration associated to $D_{\mu}$ is a product test configuration.

In what follows, we use a linearization on $\mathcal{L}$ of the $\mathbb{G}_{m}$ action $\sigma_{\mu}$ on $V$ induced by the product test configuration associated to $D_{\mu}$ . Such a linearization exists by [KKLV89, Proposition 2.4] and its antecedent remark.

Denote the test configuration associated to $\mu$ as $\mathcal{V}_{\mu}\cong V\times\mathbb{A}^{1}$ . Lifting $\mathcal{L}$ along the first projection to $\text{pr}_{2}^{*}(\mathcal{L})=:\mathbb{L}$ , we can then consider the variety $\mathcal{X}:=\mathbb{P}_{\mathcal{V}_{\mu}}(\mathbb{L}\oplus\mathcal{O}_{\mathcal{V}_{\mu}})$ (which in fact is the test configuration of $X$ associated to some $\mathbb{T}$ -invariant lift of the valuation $\mu$ ). The aforementioned $\mathbb{G}_{m}$ -linearization of $\sigma_{\mu}$ on $\mathcal{L}$ lifts to a linearization on $\mathbb{L}\oplus\mathcal{O}_{V\times A^{1}}$ , thus we have an induced $\mathbb{G}_{m}$ action on $\mathcal{X}$ that fixes a positive section of the $\mathbb{P}^{1}$ -bundle $\mathcal{X}\to V\times\mathbb{A}^{1}$ . Thus, we may define the variety $\mathcal{Y}$ to be the blow-up of $\mathcal{X}$ along the image of $B\times\mathbb{A}^{1}$ along the $\mathbb{T}$ -invariant positive section. Thus, the $\mathbb{G}_{m}$ action lifts to $\mathcal{Y}$ , which is in fact a product test configuration for $Y$ .

By construction we have $\mathbb{C}(\mathcal{Y})\cong\mathbb{C}(Y)(s)\cong\mathbb{C}(V)(t,s)$ , and the valuation associated to the test configuration $\mathcal{Y}$ is the restriction $w:=\operatorname{ord}_{s=0}|_{\mathbb{C}(Y)}$ . The further restriction of $w$ to $\mathbb{C}(V)$ is in fact $\mu$ , so $w=v_{\mu,k\xi}$ for some $k\in\mathbb{N}$ , and thus $w$ is equivalent to some twist of $\operatorname{ord}_{D}$ , so their associated test configurations are isomorphic as varieties by [Li22, Example 3.7], and thus the test configuration associated to $D$ is a product test configuration. ∎

Proof of Theorem 1.1.

By combining Proposition 4.16 and Proposition 4.8, we have that $(V,aB)$ is K-semistable if and only if $Y$ is K-semistable. Similarly, Proposition 4.17 and Proposition 4.18 together show that $(V,aB)$ is K-polystable if and only if $Y$ is K-polystable. ∎

With Theorem 1.1 established, we now prove Corollary 1.2 through an application of interpolation of K-stability.

Proof of Corollary 1.2.

By [LZ22, Zhu21], the K-polystability of $W$ is equivalent to that of the pair $(V,\frac{1}{2}B)$ . A quick calculation shows that $0<a(n,r)<\frac{1}{2}$ for all $n,r$ , so by interpolation of K-stability (see e.g. [ADL24, Proposition 2.13]), we have that $(V,aB)$ is K-polystable, so by Theorem 1.1, $Y$ is K-polystable. ∎

5. Examples

Example 5.1.

The Fano family № $3.9$ (previously known, see [ACC⁺23]): Letting $V=\mathbb{P}^{2}$ , $r=\frac{3}{2}$ , and $B$ be a smooth quartic curve, then $\mathcal{L}$ is $\mathcal{O}(2)$ , $X$ is $\mathbb{P}(\mathcal{O}(2)\oplus\mathcal{O})$ , and we have then that by Theorem 1.1 $Y$ is a member of the Fano family № $3.9$ with K-poly/semistability equivalent to that of the pair $(V,\frac{9}{52}B)$ . By [ADL24], we see that such pairs are K-poly/semistable exactly when the quartic plane curve $B$ is poly/semistable in the GIT sense, both of which are implied in this case by the smoothness of $B$ . This provides another proof that all smooth members of family № $3.9$ are K-polystable, as previously shown in [ACC⁺23].

Example 5.2.

The Fano family № $3.19$ (previously known, see [ACC⁺23]): Letting $V=\mathbb{P}^{2}$ , $r=3$ , and $B$ be a smooth conic, we have then that $\mathcal{L}=\mathcal{O}_{V}(1)$ , $X$ is $\mathbb{P}^{3}$ blown up at a point, and $Y$ is a member of the Fano family № $3.19$ with K-poly/semistability equivalent to that of the pair $(V,\frac{33}{152}B)$ by Theorem 1.1. By [LS14], we see that such a pair K-polystable, thus providing another proof that the unique smooth Fano variety in family № $3.19$ is K-polystable, originally shown in [IS17].

Example 5.3.

The Fano family № $4.2$ (previously known, see [ACC⁺23]): Let $V=\mathbb{P}^{1}\times\mathbb{P}^{1}$ , $r=2$ , and $B$ be a smooth curve of bidegree $(2,2)$ . Then, $\mathcal{L}=\mathcal{O}_{V}(1,1)$ , $X=\mathbb{P}(\mathcal{O}(1,1)\oplus\mathcal{O})$ , and $Y$ is a smooth member of the Fano family № $4.2$ and, by the above theorem, the K-polystability of $Y$ is equivalent to that of the pair $(V,\frac{11}{56}B)$ . The K-polystability of this pair follows from the interpolation of K-stability (see [ADL24, Proposition 2.13] and [ADL23, Theorem 2.10]) since $V$ is K-polystable and $(V,B)$ is a plt log Calabi-Yau pair. Thus, by Theorem 1.1, such $Y$ is K-polystable, giving a new proof of this result previously shown in [ACC⁺23].

Example 5.4.

New examples from blow-ups related to quartic surfaces in $\mathbb{P}^{3}$ and higher dimensional analogs: Let $V=\mathbb{P}^{3}$ , $r=2$ , and $B$ a smooth quartic surface in $\mathbb{P}^{3}$ . Then, $\mathcal{L}=\mathcal{O}_{V}(2)$ , and $Y$ is the blow-up of the cone over the second Veronese embedding of $\mathbb{P}^{3}$ in $\mathbb{P}^{9}$ along the cone point and a quartic surface in the base. In this case, $(V,aB)=(\mathbb{P}^{3},\frac{13}{75}B)$ . By [ADL23], such pairs are K-poly/semistable exactly when they are poly/semistable in the GIT sense. As $B$ is smooth, it is GIT polystable, thus all smooth fourfolds constructed in this manner are K-polystable by Theorem 1.1.

We generalize this to higher dimensions: let $V=\mathbb{P}^{n-1}$ for $n$ even, $r=2$ , and $B$ a smooth degree $n$ hypersurface in $V$ . Then $\mathcal{L}=\mathcal{O}_{V}(\frac{n}{2})$ , and $Y$ is the blow-up of the cone over the embedding $\phi_{|\mathcal{L}|}:V\hookrightarrow\mathbb{P}^{N}$ along the cone point and the inclusion of $B$ in the base of the cone. By [ADL24, Theorem 1.4], there exists some $c_{1}$ such that $(V,cB)$ is K-semi/polystable if and only if $B$ is semi/polystable in the GIT sense for all $c<c_{1}$ . If $a=a(n,2)<c_{1}$ , then $B$ smooth implies $B$ is polystable in the GIT sense and thus $(V,aB)$ is K-polystable, implying by Theorem 1.1 that $Y$ is K-polystable. If $c_{1}\leq a$ , then again since $B$ is smooth, $(V,cB)$ is K-polystable for some $c<a$ . Then, since $(V,B)$ is a plt log Calabi-Yau pair, $(V,(1-\epsilon)B)$ is K-polystable for some sufficiently small $\epsilon$ (see [ADL23, Theorem 2.10]), and thus by interpolation of K-stability (see [ADL24, Proposition 2.13]), we again have that $(V,aB)$ is K-polystable, implying the K-polystability of $Y$ by Theorem 1.1. This improves [CGF⁺23, Theorem 1.9].

Example 5.5.

New K-unstable example: Let $V=\mathrm{Bl}_{p}\mathbb{P}^{3}$ , $r=2$ , and $B$ a smooth member of $|-K_{V}|$ . Then, $\mathcal{L}=\pi^{*}(2H)-E$ where $\pi:V\to\mathbb{P}^{3}$ is the blow-up map and $E$ is the exceptional divisor of the blow-up, $X=\mathbb{P}_{V}(\mathcal{L}\oplus\mathcal{O}_{V})$ , and $Y$ is $\mathrm{Bl}_{B_{\infty}}X$ . As in the previous example, $a=\frac{13}{75}$ . A straightforward computation shows that the pair $(V,aB)$ is K-unstable with $\beta_{V,aB}(\pi^{*}(H))<0$ , where $\pi^{*}(H)$ is the pullback of the hyperplane section of $\mathbb{P}^{3}$ along the blow-up. Thus, by Theorem 1.1, $Y$ is K-unstable.

6. Other Blow-ups of Projective Compactifications of Proportional Line Bundles

In this section, we see that, when replacing the assumption that $l=2$ in the construction of $Y$ with $l\neq 2$ (where $B\sim_{\mathbb{Q}}l\mathcal{L}$ ), the resulting Fano varieties $Y$ are always K-unstable.

Theorem 6.1 (Theorem 1.3).

Let $Y$ be constructed as above with $V$ and $B$ smooth and $B\sim_{\mathbb{Q}}l\mathcal{L}$ for $0<l<r+1,l\neq 2$ . Then $Y$ is K-unstable. Furthermore, either the strict transform of the image of the positive section containing $B_{\infty}$ , denoted as $\overline{V}_{\infty}$ , or the strict transform of the zero section, $V_{0}$ , is a destabilizing divisor for $Y$ .

Lemma 6.2.

\displaystyle\beta_{Y}(V_{0})+\beta_{Y}(\overline{V}_{\infty})=0

Proof.

As $V_{0},\overline{V}_{\infty}$ are both divisors on $Y$ , $A_{Y}(V_{0})=A_{Y}(\overline{V}_{\infty})=1$ , so $\beta_{Y}(V_{0})+\beta_{Y}(\overline{V}_{\infty})=2-(S_{Y}(V_{0})+S_{Y}(\overline{V}_{\infty}))$ . It remains to show the sum $S_{Y}(V_{0})+S_{Y}(\overline{V}_{\infty})$ is equal to $2$ .

The Zariski decompositions of $-K_{Y}-t\overline{V}_{\infty},-K_{Y}-tV_{0}$ on $Y$ are as follows:

P_{\infty}(t)=\begin{cases}-K_{Y}-t\overline{V}_{\infty}=(1-t)\overline{V}_{\infty}+\pi^{*}\phi^{*}(-K_{V})+V_{0},&\text{if }0\leq t\leq 1\\ (2-t)H+\frac{r-1}{r}\pi^{*}\phi^{*}(-K_{V})=\frac{r+1-t}{r}\pi^{*}\phi^{*}(-K_{V})+(2-t)V_{0},&\text{if }1\leq t\leq 2\end{cases}

N_{\infty}(t)=\begin{cases}0,&\text{if }0\leq t\leq 1\\ (1-t)E,&\text{if }1\leq t\leq 2\end{cases}

P_{0}(t)=\begin{cases}-K_{Y}-tV_{0}=\overline{V}_{\infty}+\pi^{*}\phi^{*}(-K_{V})+(1-t)V_{0},&\text{if }0\leq t\leq 1\\ (2-t)\overline{V}_{\infty}+\frac{r-(t-1)(l-1)}{r}\pi^{*}\phi^{*}(-K_{V}),&\text{if }1\leq t\leq 2\end{cases}

N_{0}(t)=\begin{cases}0,&\text{if }0\leq t\leq 1\\ (t-1)F=(t-1)(\overline{V}_{\infty}+\frac{l-1}{r}\pi^{*}\phi^{*}(-K_{V})-V_{0}),&\text{if }1\leq t\leq 2\end{cases}

This can be seen as follows: for the range $0\leq t\leq 1$ , $-K_{Y}-t\overline{V}_{\infty}$ and $-K_{Y}-tV_{0}$ are both nef, thus the negative parts of their Zariski decompositions are trivial. For $-K_{Y}-t\overline{V}_{\infty}$ with $1\leq t\leq 2$ , we observe that $P_{\infty}$ is the pullback along the contraction $\pi$ of a nef class on $X$ and $N_{\infty}$ is supported on the exceptional locus of the same contraction. Thus by [Oka16, Proposition 2.13], we see that this is the Zariski decomposition of $-K_{Y}-t\overline{V}_{\infty}$ . For $l\neq 1$ , a similar argument shows that $P_{0}+N_{0}$ is the Zariski decomposition of $-K_{Y}-tV_{0}$ , using the contraction $\pi^{\prime}:Y\to X^{\prime}:=\mathbb{P}_{V}(\mathcal{L}^{\prime}\oplus\mathcal{O}_{V})$ (for some line bundle $\mathcal{L}^{\prime}$ such that $l^{\prime}\mathcal{L}^{\prime}\sim_{\mathbb{Q}}-K_{V}$ where $(l^{\prime}-1)(l-1)=1$ ) that contracts $F$ . For $l=1$ , we have the same contraction map $\pi^{\prime}:Y\to X^{\prime}:=\mathbb{P}_{V}(\mathcal{O}_{V}\oplus\mathcal{O}_{V})$ but now for $\mathcal{L}^{\prime}=\mathcal{O}_{V}$ .

We have the following intersection products on $Y$ , for $k>0$ :

	$\displaystyle V_{0}^{k}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$	$\displaystyle=V_{0}^{k-1}\cdot V_{0}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$
		$\displaystyle=(H-\frac{1}{r}\pi^{}\phi^{}(-K_{V}))^{k-1}\cdot V_{0}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$
		$\displaystyle=(-\frac{1}{r}\pi^{}\phi^{}(-K_{V}))^{k-1}\cdot V_{0}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$
		$\displaystyle=(\frac{-1}{r})^{k-1}\operatorname{vol}(V)$

	$\displaystyle\overline{V}_{\infty}^{k}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$	$\displaystyle=\overline{V}_{\infty}^{k-1}\cdot\overline{V}_{\infty}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$
		$\displaystyle=(V_{0}+\frac{1}{r}\pi^{}\phi^{}(-K_{V})-E)^{k-1}\cdot\overline{V}_{\infty}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$
		$\displaystyle=(\frac{1}{r}\pi^{}\phi^{}(-K_{V})-E)^{k-1}\cdot\overline{V}_{\infty}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$
		$\displaystyle=(\frac{1-l}{r})^{k-1}\operatorname{vol}(V)$

Let us assume $l\neq 1$ , then we have for $\operatorname{vol}(Y)S_{Y}(\overline{V}_{\infty})$ :

	$\displaystyle\operatorname{vol}(Y)S_{Y}(\overline{V}_{\infty})$	$\displaystyle=\int_{0}^{1}\left((1-t)\overline{V}_{\infty}+\pi^{}\phi^{}(-K_{V})+V_{0}\right)^{n}dt$
		$\displaystyle+\int_{1}^{2}\left(\frac{r+1-t}{r}\pi^{}\phi^{}(-K_{V})+(2-t)V_{0}\right)^{n}dt$
		$\displaystyle=\frac{\operatorname{vol}(V)}{r^{n-1}}(\int_{0}^{1}\frac{1}{1-l}\left((1-t)(1-l)+r)^{n}-r^{n}\right)dt$
		$\displaystyle+\int_{0}^{1}r^{n}-(r-1)^{n}dt-\int_{1}^{2}(r-1)^{n}-(r+1-t)^{n}dt)$
		$\displaystyle=\frac{\operatorname{vol}(V)}{r^{n-1}}\int_{0}^{1}\frac{((1-t)(1-l)+r)^{n}-r^{n}}{1-l}+r^{n}-2(r-1)^{n}+(r+t-1)^{n}dt$

Similarly, for $\operatorname{vol}(Y)S_{Y}(\overline{V}_{\infty})$ :

	$\displaystyle\operatorname{vol}(Y)S_{Y}(V_{0})$	$\displaystyle=\int_{0}^{1}\left(\overline{V}_{\infty}+\pi^{}\phi^{}(-K_{V})+(1-t)V_{0}\right)^{n}dt$
		$\displaystyle+\int_{1}^{2}\left((2-t)\overline{V}_{\infty}+\frac{r-(t-1)(l-1)}{r}\pi^{}\phi^{}(-K_{V})\right)^{n}dt$
		$\displaystyle=\frac{\operatorname{vol}(V)}{r^{n-1}}(\int_{0}^{1}\frac{1}{1-l}\left((r+1-l)^{n}-r^{n}\right)dt+\int_{0}^{1}r^{n}-(r+t-1)^{n}dt$
		$\displaystyle+\int_{1}^{2}\frac{1}{1-l}\left((r+1-l)^{n}-(r-(t-1)(l-1))^{n}\right)dt)$
		$\displaystyle=\frac{\operatorname{vol}(V)}{r^{n-1}}\int_{0}^{1}\frac{(r+1-l)^{n}-r^{n}}{1-l}+r^{n}-(r+t-1)^{n}$
		$\displaystyle+\frac{(r+1-l)^{n}-(r+(t-1)(l-1))^{n}}{1-l}dt$

Summing these two terms we get precisely $2\operatorname{vol}(Y)$ , and so $\beta(V_{0})+\beta(\overline{V}_{\infty})=0$ .

Now, for $l=1$ , we have for $\operatorname{vol}(Y)S_{Y}(\overline{V}_{\infty})$ :

	$\displaystyle\operatorname{vol}(Y)S_{Y}(\overline{V}_{\infty})$	$\displaystyle=\int_{0}^{1}\left((1-t)\overline{V}_{\infty}+\pi^{}\phi^{}(-K_{V})+V_{0}\right)^{n}dt$
		$\displaystyle+\int_{1}^{2}\left(\frac{r+1-t}{r}\pi^{}\phi^{}(-K_{V})+(2-t)V_{0}\right)^{n}dt$
		$\displaystyle=\frac{\operatorname{vol}(V)}{r^{n-1}}(\int_{0}^{1}n(1-t)r^{n-1}+r^{n}-(r-1)^{n}dt$
		$\displaystyle-\int_{1}^{2}(r-1)^{n}-(r+1-t)^{n}dt)$
		$\displaystyle=\frac{\operatorname{vol}(V)}{r^{n-1}}\int_{0}^{1}n(1-t)r^{n-1}+r^{n}-2(r-1)^{n}+(r+t-1)^{n}dt$

and similarly for $\operatorname{vol}(Y)S_{Y}(V_{0})$ :

	$\displaystyle\operatorname{vol}(Y)S_{Y}(V_{0})$	$\displaystyle=\int_{0}^{1}\left(\overline{V}_{\infty}+\pi^{}\phi^{}(-K_{V})+(1-t)V_{0}\right)^{n}dt$
		$\displaystyle+\int_{1}^{2}\left((2-t)\overline{V}_{\infty}+\frac{r-(t-1)(l-1)}{r}\pi^{}\phi^{}(-K_{V})\right)^{n}dt$
		$\displaystyle=\frac{\operatorname{vol}(V)}{r^{n-1}}(\int_{0}^{1}nr^{n-1}+r^{n}-(r+t-1)^{n}dt+\int_{1}^{2}n(2-t)r^{n-1}dt)$

So for the $l=1$ case, summing these two terms we again get $2\operatorname{vol}(Y)$ , and so $\beta(V_{0})+\beta(\overline{V}_{\infty})=0$ . ∎

Thus, to finish the prove of Theorem 1.3, we simply must show that $\beta_{Y}(\overline{V}_{\infty})\neq 0$ for $l\neq 2$ .

Proof.

In fact, we show that $\frac{r^{n-1}\operatorname{vol}(Y)}{\operatorname{vol}(V)}\beta_{Y}(\overline{V}_{\infty})\neq 0$ .

	$\displaystyle\operatorname{vol}(Y)S_{Y}(\overline{V}_{\infty})$	$\displaystyle=\int_{0}^{1}\left((1-t)\overline{V}_{\infty}+\pi^{}\phi^{}(-K_{V})+V_{0}\right)^{n}dt$
		$\displaystyle+\int_{1}^{2}\left(\frac{r+1-t}{r}\pi^{}\phi^{}(-K_{V})+(2-t)V_{0}\right)^{n}dt$
		$\displaystyle=\frac{\operatorname{vol}(V)}{r^{n-1}}(\int_{0}^{1}\frac{1}{1-l}\left((1-t)(1-l)+r)^{n}-r^{n}\right)dt$
		$\displaystyle+\int_{0}^{1}r^{n}-(r-1)^{n}dt+\int_{1}^{2}(r-1)^{n}-(r+1-t)^{n}dt)$
		$\displaystyle=\int_{0}^{1}\frac{((1-t)(1-l)+r)^{n}-r^{n}}{1-l}+r^{n}-2(r-1)^{n}+(r+t-1)^{n}dt$

	$\displaystyle\frac{r^{n-1}\operatorname{vol}(Y)}{\operatorname{vol}(V)}\beta_{Y}(\overline{V}_{\infty})$	$\displaystyle=\frac{r^{n-1}\operatorname{vol}(Y)}{\operatorname{vol}(V)}-\frac{r^{n-1}\operatorname{vol}(Y)}{\operatorname{vol}(V)}S_{Y}(\overline{V}_{\infty})$
		$\displaystyle=\left(\frac{r^{n}-(r+1-l)^{n}}{l-1}+r^{n}-(r-1)^{n}\right)$
		$\displaystyle-\int_{0}^{1}\frac{((1-t)(1-l)+r)^{n}-r^{n}}{1-l}+r^{n}-2(r-1)^{n}+(r+t-1)^{n}dt$
		$\displaystyle=\int_{0}^{1}\left(\frac{r^{n}-(r+1-l)^{n}}{l-1}+r^{n}-(r-1)^{n}\right)$
		$\displaystyle-\frac{((1-t)(1-l)+r)^{n}-r^{n}}{1-l}-r^{n}+2(r-1)^{n}-(r+t-1)^{n}dt$
		$\displaystyle=\int_{0}^{1}\frac{(r-(t-1)(1-l))^{n}-(r+1-l)^{n}}{l-1}+(r-1)^{n}-(r+t-1)^{n}dt$

We will show the integrand is strictly positive (resp. negative) when $r>2$ (resp. $r<2$ ) for $0<t<1$ . Let $f(x)=(r-x)^{n}$ , $w=1-t$ and $y=l-1$ . Then the integrand is equal to

(1-w)\left(\frac{f(1)-f(w)}{1-w}-\frac{f(wy)-f(y)}{wy-y}\right)

Then, since $f$ is strictly convex on the interval $(0,r)$ , we have, for $l>2$ ,

\frac{f(1)-f(w)}{1-w}<\frac{f(1)-f(wx)}{1-wx}<\frac{f(wy)-f(y)}{wy-y}

with the inequalities reversing for $l<2$ . Thus the integrand is strictly positive (resp. negative), so $\beta_{Y}(\overline{V}_{\infty})\neq 0$ as desired. ∎

6.1. Examples of Other Blow-ups

Example 6.3.

The Fano family № $3.14$ : Letting $V$ be $\mathbb{P}^{2}$ , $r=3$ , and $B$ a smooth planar cubic curve, so that $l=3$ , we have that $X=\mathrm{Bl}_{p}\mathbb{P}^{3}$ is the blow-up of $\mathbb{P}^{3}$ at a point and $Y$ is a smooth member of the family of Fano threefolds № $3.14$ , and all smooth members of said family are obtained in this way. Thus, Theorem 1.3 recovers the K-unstability of members of this family, originally due to Fujita [Fuj16, Theorem 1.4].

Example 6.4.

K-unstable in each dimension: Let $V$ be $\mathbb{P}^{n-1}$ , $r=n$ , and $B$ a degree $d$ hypersurface in $V$ , for $d<n$ . Then $X=\mathrm{Bl}_{p}\mathbb{P}^{n}$ the blow up of $\mathbb{P}^{n}$ at a point and $Y$ is the blow-up of projective $n$ -space along a codimension $2$ subvariety contained in the pullback of a hyperplane in $X$ that doesn’t contain $p$ . Such $Y$ is K-unstable by Theorem 1.3. Thus, we have for every $n$ several examples of a K-unstable Fano variety of dimension $n$ .

References

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	$\displaystyle(-K_{Y})^{n}$	$\displaystyle=(2\pi^{}H+\frac{r-1}{r}\pi^{}\phi^{*}(-K_{V})-E)^{n}$
		$\displaystyle=(\pi^{}H+\frac{r-1}{r}\pi^{}\phi^{*}(-K_{V})+\overline{V}_{\infty})^{n}$
		$\displaystyle=(\pi^{}V_{0}+\pi^{}\phi^{*}(-K_{V})+\overline{V}_{\infty})^{n}$
		$\displaystyle=(\pi^{}V_{0}+\pi^{}\phi^{}(-K_{V}))^{n}+(\pi^{}\phi^{}(-K_{V})+\overline{V}_{\infty})^{n}-(\pi^{}\phi^{*}(-K_{V}))^{n}$
		$\displaystyle=(V_{0}+\pi^{}\phi^{}(-K_{V}))^{n}+(\pi^{}\phi^{}(-K_{V})+\overline{V}_{\infty})^{n}$

	$\displaystyle V_{0}^{k}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$	$\displaystyle=V_{0}^{k-1}\cdot V_{0}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$
		$\displaystyle=(H-\frac{1}{r}\pi^{}\phi^{}(-K_{V}))^{k-1}\cdot V_{0}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$
		$\displaystyle=(-\frac{1}{r}\pi^{}\phi^{}(-K_{V}))^{k-1}\cdot V_{0}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$
		$\displaystyle=(\frac{-1}{r})^{k-1}\operatorname{vol}(V)$

	$\displaystyle\overline{V}_{\infty}^{k}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$	$\displaystyle=\overline{V}_{\infty}^{k-1}\cdot\overline{V}_{\infty}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$
		$\displaystyle=(V_{0}+\frac{1}{r}\pi^{}\phi^{}(-K_{V})-E)^{k-1}\cdot\overline{V}_{\infty}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$
		$\displaystyle=(\frac{1}{r}\pi^{}\phi^{}(-K_{V})-E)^{k-1}\cdot\overline{V}_{\infty}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$
		$\displaystyle=(\frac{1-l}{r})^{k-1}\operatorname{vol}(V)$

	$\displaystyle-mK_{Y}\|_{\overline{V}_{\infty}}-j\overline{V}_{\infty}\|_{\overline{V}_{\infty}}$	$\displaystyle=-m(K_{Y}+\overline{V}_{\infty})\|_{\overline{V}_{\infty}}-(j-m)\overline{V}_{\infty}\|_{\overline{V}_{\infty}}$
		$\displaystyle=-mK_{\overline{V}_{\infty}}-(j-m)(V_{\infty}-E)\|_{\overline{V}_{\infty}}$
		$\displaystyle=-mK_{\overline{V}_{\infty}}-(j-m)(\mathcal{L}-B)=mrL-(j-m)(-\mathcal{L})$
		$\displaystyle=(mr-m+j)\mathcal{L}$

	$\displaystyle V_{0}^{k}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$	$\displaystyle=V_{0}^{k-1}\cdot V_{0}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$
		$\displaystyle=(H-\frac{1}{r}\pi^{}\phi^{}(-K_{V}))^{k-1}\cdot V_{0}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$
		$\displaystyle=(-\frac{1}{r}\pi^{}\phi^{}(-K_{V}))^{k-1}\cdot V_{0}\cdot\pi^{}\phi^{}(-K_{V})^{n-k}$
		$\displaystyle=(\frac{-1}{r})^{k-1}\operatorname{vol}(V)$

On the K-stability of blow-ups of projective bundles

Abstract.

1. Introduction

Theorem 1.1.

Corollary 1.2.

Theorem 1.3.

1.1. Acknowledgements

2. Preliminaries

Definition 2.1 (β\beta-invariant, [Fuj19b], [Li17]).

Definition 2.2 (test configurations).

Definition 2.3 (product-type divisors).

Definition 2.4.

Definition 2.5 (filtrations).

Definition 2.6 (δ\delta-invariants, [BJ20]).

Definition 2.7.

Definition 2.8.

Definition 2.9.

Theorem 2.10.

Theorem 2.11.

3. Blow-ups of Projective Compactifications of Proportional Line Bundles

Lemma 3.1.

Proof.

Proposition 3.2.

Proof.

Lemma 3.3.

Proof.

Definition 3.4 (Futaki character).

Lemma 3.5.

Proof.

4. Proof of Main Theorem

4.1. K-Semistability

4.1.1. Reverse Implication

Definition 4.1.

Lemma 4.2.

Proof.

Lemma 4.3.

Proof.

Lemma 4.4.

Proof.

Lemma 4.5.

Proof.

Lemma 4.6.

Proof.

Lemma 4.7.

Proof.

Proposition 4.8.

Proof.

4.1.2. Forward Implication

Lemma 4.9.

Proof.

Lemma 4.10.

Proof.

Lemma 4.11.

Proof.

Convention 4.12.

Lemma 4.13.

Proof.

Lemma 4.14.

Proof.

Proposition 4.15.

Proof.

Proposition 4.16.

Proof.

4.2. K-Polystability

4.2.1. Forward Implication

Proposition 4.17.

Proof.

4.2.2. Reverse Implication

Proposition 4.18.

Proof.

Proof of Theorem 1.1.

Proof of Corollary 1.2.

5. Examples

Example 5.1.

Example 5.2.

Example 5.3.

Example 5.4.

Example 5.5.

6. Other Blow-ups of Projective Compactifications of Proportional Line Bundles

Theorem 6.1 (Theorem 1.3).

Definition 2.1 ( $\beta$ -invariant, [Fuj19b], [Li17]).

Definition 2.6 ( $\delta$ -invariants, [BJ20]).