ON FUJITA INVARIANTS OF SUBVARIATIES OF A
UNIRULED VARIETYCHRISTOPHER D. HACON AND CHEN JIANG
Abstract. We show that if X is a smooth uniruled projective varietyand L a big and semiample Q-divisor on X, then there exists a proper
closed subset W ⊂ X such that every subvariety Y satisfying a(Y, L) >a(X, L) is contained in W .

1. Introduction
If X is a smooth projective variety and L is a big Q-divisor on X, thenthe Fujita invariant, or a-constant is defined as followsa(X, L) = inf{t > 0 | KX + tL is big}.
Note that a(X, L) ∈ R≥0 is well defined since KX + tL is big for all t > 0
sufficiently large, and that a(X, L) > 0 if and only if KX is not pseudoeffective. It is easy to see that the a-constant is a birational invariant in
the sense that if ν : X 0 → X is a birational morphism of smooth varieties
and L0 = ν ∗ L, then a(X, L) = a(X 0 , L0 ). Therefore we may also define thea-constant for a big Q-Cartier Q-divisor L on an arbitrary normal projective
variety X by lettinga(X, L) := a(X 0 , L0 )where ν : X 0 → X is a resolution of singularities and L0 = ν ∗ L. Note that
if X is smooth, then the a-constant is the usual pseudo-effective threshold,however if X is singular, these numbers may be different.
In [8], motivated by a conjecture of Batyrev and Manin that relates arithmetic properties of varieties with ample anticanonical class to geometric invariants, a-constants were intensively studied by Lehmann, Tanimoto andTschinkel. They show that ([8, Theorem 1.1]), if X is a smooth uniruled
projective variety and L an ample Q-divisor on X, then there exists a countable union of proper closed subsets W ⊂ X such that every subvariety Y
satisfying a(Y, L) > a(X, L) is contained in W . For the purpose of applications, it is expected that one may choose W to be a proper closed subset ofX. The purpose of this note is to prove that this is indeed the case:
Theorem 1.1. Let X be a smooth uniruled projective variety and L a bigand semiample Q-divisor on X. Then there exists a proper closed subset
W ⊂ X such that every subvariety Y satisfying a(Y, L) > a(X, L) is contained in W .
Date: April 13, 2016.
The first author was supported by NSF research grants no: DMS-1300750, DMS1265285 and by a grant from the Simons Foundation; Award Number: 256202.
The second author was supported by the Program for Leading Graduate Schools andWorld Premier International Research Center Initiative (WPI), MEXT, Japan.
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CHRISTOPHER D. HACON AND CHEN JIANG

Note that this result is proven in [8, Theorem 1.2] assuming that a weakversion of the BAB conjecture holds in dimension n − 1 = dim X − 1. We
expect that Theorem 1.1 holds also if we just assume that L is nef and big(rather than big and semiample).
Our idea is to replace the WBAB conjecture assumed in [8, Theorem 1.2]
by constructing non-klt centers (see Proposition 2.8) and applying finitenessof a-constants (see Corollary 2.15). This is an application of a recent resultof Di Cerbo [3] based on a boundedness result proved by Birkar [2].
Acknowledgment. The essential ideas in this note originated during
a visit of the second author to the University of Utah in February 2015.
The second author is grateful for the hospitality. The authors would like tothank Brian Lehmann for useful comments.
2. Preliminaries
In this paper we work over the field of complex numbers C.
2.1. Facts on a-constants. In this subsection, for the convenience of thereader, we collect several facts about a-constants that were proven in [8].
Proposition 2.1 ([8, Proposition 4.1]). Let X be a smooth projective variety
and L a big and nef Q-divisor. Let U → W be a family of subvarieties of X
such that U → X is dominant. Then a general member Y of the family Usatisfies a(Y, L) ≤ a(X, L).
Theorem 2.2 ([8, Theorem 4.2]). Let X be a smooth projective variety andL a big and nef Q-divisor. Let π : U → W be a family of subvarieties of X.
There exists a proper closed subset V ⊂ X such that if a member Y of thefamily U satisfies a(Y, L) > a(X, L) then Y ⊂ V .
Proposition 2.3 ([8, Proposition 4.6]). Let X be a smooth uniruled projective variety and L a big and nef Q-divisor. Then either(1) X is covered by proper subvarieties Y satisfying a(Y, L) = a(X, L),or(2) X is birational to a Q-factorial terminal Fano variety X 0 of Picard
number 1.
Lemma 2.4 ([8, Lemma 4.7]). Let X be a smooth projective variety and L a
big and nef Q-divisor on X. Fix a constant C. Then the subset of Chow(X)parametrizing subvarieties of X that are not contained in B+ (L) and are ofL-degree at most C is bounded.
2.2. Non-klt centers. We follow the standard notation and conventions ofthe minimal model program, see eg. [5].
Definition 2.5. Let (X, ∆) be a pair so that X is a normal variety, ∆ isan effective Q-divisor, and KX + ∆ is Q-Cartier. We say that a subvariety V ⊂ X is a non-klt center of (X, ∆) if it is the image of a divisor ofdiscrepancy at most −1. We will denote by Nklt(X, ∆) the union of all
non-klt centers of (X, ∆). A non-klt place is a valuation corresponding to adivisor of discrepancy at most −1. A non-klt center is pure if KX + ∆ is log
canonical at the generic point of V . If moreover there is a unique non-klt

ON FUJITA INVARIANTS OF SUBVARIATIES OF A UNIRULED VARIETY

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place lying over the generic point of V , we will say that V is an exceptional
non-klt center.
The following is a weak form of Kawamata’s subadjunction theorem.
Theorem 2.6 (Subadjunction, see [4, Proposition 5.1]). Let V ⊂ X be a
non-klt center of a pair (X, ∆) which is lc at a general point of V . Letν : V ν → V be the normalization. Then there is an effective Q-divisor ∆V νon V ν such thatν ∗ (KX + ∆)|Vν ∼Q KV ν + ∆V ν .
We have the following connectedness lemma of Kollár and Shokurov forthe non-klt locus (cf. Shokurov [9], Kollár [6, 17.4]).
Theorem 2.7 (Connectedness Lemma). Let f : X → Z be a proper morphism of normal varieties with connected fibers and D a Q-divisor suchthat −(KX + D) is Q-Cartier, f -nef, and f -big. Write D = D+ − D−where D+ and D− are effective with no common components. If D− is f exceptional (i.e. all of its components have image of codimension at least2), then Nklt(X, D) ∩ f −1 (z) is connected for any z ∈ Z.
We can use the following proposition to construct non-klt centers.
Proposition 2.8 (cf. [7, Lemma 3.2]). Let X be a Q-factorial terminal
Fano variety of dimension n. Assume (−KX )n > (wn)n for some positive
rational number w. Then for every point p ∈ X there is an effective Qdivisor ∆p ∼Q − w1 KX such that the unique minimal non-klt center Vp ⊂Nklt(X, ∆p ) containing p is exceptional.
Proof. Fix a point p. Fix a positive rational number w0 such that (−KX )n >(w0 n)n > (wn)n . By [5, 6.7.1 Theorem], there is an effective Q-divisor∆0p ∼Q − w10 KX such that (X, ∆0p ) is not lc at p. Take 0 < t ≤ 1 theunique rational number such that (X, t∆0p ) is log canonical but not klt atp. By [1, Proposition 3.2, Lemma 3.4], we can find an effective Q-divisor
Mp ∼Q − w10 KX and some rational number a > 0 such that for any rational
number 0 <   1, the pair (X, (1 − )t∆0p + aMp ) has a unique minimal
non-klt center Vp passing through p which is exceptional. Note that(1 − )t∆0p + aMp ∼Q −(1 − )t + aKXw0< w1 for 0 <   1. Since −KX is ample, by adding a Qand (1−)t+aw0divisor Q-linearly equivalent to a multiple of −KX to ∆0p , we conclude that
there exists an effective Q-divisor ∆p ∼Q − w1 KX and (X, ∆p ) has a uniqueminimal non-klt center Vp passing through p which is exceptional.
Lemma 2.9. Keep the notation in Proposition 2.8. If w > 2, then dim Vp >0 for a general point p.
Proof. Assume to the contrary that there exist p1 ∈ X such that Vp1 = {p1 }and p2 ∈ X\Supp(∆p1 ) such that Vp2 = {p2 }. Then p1 and p2 are containedin Nklt(X, ∆p1 +∆p2 ) and p2 is isolated by construction. On the other hand,
2
−(KX + ∆p1 + ∆p2 ) ∼Q 1 −(−KX )w

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CHRISTOPHER D. HACON AND CHEN JIANG

is ample. By the connectedness lemma, Nklt(X, ∆p1 + ∆p2 ) is connected,
which is a contradiction.
2.3. Finiteness of a-constants. We recall the main result of [3] in thissubsection.
Definition 2.10. Let X be a normal projective variety and H a big Qdivisor. We define the pseudo-effective threshold to beτ (X, H) := inf{t ≥ 0 | KX + tH is big}.
Note that if X is smooth, a-constant and pseudo-effective threshold justcoincide.
Definition 2.11 (cf. [3, Definition 3.1]). Fix a positive integer n and two
positive real numbers  and δ. We define Dn (, δ) to be the set of lc pairs(X, ∆) such that:(1) X is a normal projective variety of dimension n,(2) ∆ is a big Q-divisor with coefficients ≥ δ, and(3) (X, t∆) is -lc and KX + t∆ is pseudo-effective for some 0 ≤ t ≤ 1.
Definition 2.12 (cf. [3, Definition 3.2]). Fix a positive integer n and two
positive real numbers  and δ. We define the setTn (, δ) := {τ (X, ∆) | (X, ∆) ∈ Dn (, δ)}.
Theorem 2.13 ([3, Corollary 3.6]). Fix a positive integer n and three positive real numbers , δ and η. Then the set Tn (, δ) ∩ [η, 1] is a finite set.
To apply this theorem in our situation, we have the following corollary.
Definition 2.14. Fix a positive integer n. We define Pn to be the set ofpairs (Y, L) such that:(1) Y is a normal projective variety of dimension n,(2) L is a base point free big Cartier divisor.
Corollary 2.15. Fix a positive integer n and a positive real number η. Thenthe set{a(Y, L) | (Y, L) ∈ Pn } ∩ [η, ∞)is a finite set.
1.
Proof. We may assume that η ≤ 4(n+1)
Firstly, we show that the seth 1i{a(Y, L) | (Y, L) ∈ Pn } ∩ η,2
is a finite set. Take (Y, L) ∈ Pn and assume that a(Y, L) ∈ [η, 12 ]. Note thata(Y, 12 L) = 2a(Y, L) ∈ [2η, 1]. By taking a resolution, we may assume thatY is smooth. In this case a(Y, 21 L) = τ (Y, 21 L). Replacing L by a generalelement in |L|, we may assume that L is irreducible and smooth. Moreover,(Y, 12 L) is 21 -lc and KY + 12 L is pseudo-effective, that is, (Y, 21 L) ∈ Dn ( 21 , 12 ).
This implies that the setn  1 oa Y, L (Y, L) ∈ Pn ∩ [2η, 1]
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ON FUJITA INVARIANTS OF SUBVARIATIES OF A UNIRULED VARIETY

5is finite by Theorem 2.13, and so is {a(Y, L) | (Y, L) ∈ Pn } ∩ [η, 21 ].
Then we show that the seth1
{a(Y, L) | (Y, L) ∈ Pn } ∩ , ∞2
is a finite set. Take (Y, L) ∈ Pn and assume that a(Y, L) ≥ 12 . By taking
a resolution, we may assume that Y is smooth. By [8, Proposition 2.10],a(Y, L) ≤ n + 1. Now we consider (Y, 2(n + 1)L) ∈ Pn . Note that a(Y, 2(n +11a(Y, L), hence a(Y, 2(n + 1)L) ∈ [ 4(n+1), 12 ]. By the first step,1)L) = 2(n+1)
a(Y, 2(n + 1)L) belongs to a finite set. Hence a(Y, L) belongs to a finiteset.
3. Proof of Theorem 1.1
We prove the following proposition suggested by B. Lehmann.
Proposition 3.1. Fix a positive real number t. Let X be a smooth projective
variety and L a big and semiample Q-divisor. Then there is a boundedfamily U of subvarieties of X such that any subvariety Y not contained in
B+ (L), with a(Y, L) > t is dominated by some members Z of U, such thata(Z, L) = a(Y, L).
Proof. Note that for a subvariety Y not contained in B+ (L), L|Y is nefand big, and so a(Y, L) is well defined. Therefore we will only consider
subvarieties not contained in B+ (L).
Replacing L by some multiple, we may assume that L is a base point free
Cartier divisor.
We construct U inductively by increasing induction on the dimension ofY.
For a subvariety Y with a(Y, L) > t and dim Y = 1, it is easy to see that
Y is a rational curve with22< .
degY (L) = Y · L =a(Y, L)t
By Lemma 2.4, such Y form a bounded family U1 .
Suppose that we have constructed a bounded family Ui of subvarieties
such that every subvariety Y with a(Y, L) > t and dim Y ≤ i is dominatedby some members Z of U such that a(Z, L) = a(Y, L). We construct Ui+1
as follows. Suppose that Y is an (i + 1)-dimensional subvariety satisfyinga(Y, L) > t. By taking a resolution, we may assume that Y is smooth.
Proposition 2.3 shows that either(1) Y is covered by proper subvarieties Z with a(Z, L) = a(Y, L), or(2) Y is birational to a Q-factorial terminal Fano variety Y 0 of Picard
number 1.
In Case (1), by induction, Z is dominated by some members Z 0 of Ui suchthat a(Z 0 , L) = a(Z, L), and so is Y .
In Case (2), by taking a resolution, we may assume φ : Y 99K Y 0 is amorphism. By the proof of [8, Proposition 4.6], KY 0 + a(Y, L)φ∗ (L|Y ) ≡ 0.
We define constant c0 < 1 and w > 2 as follows: since L is base point
free, we know that the set{a(Z, L) | Z is a subvariety of X} ∩ (t, ∞]6CHRISTOPHER D. HACON AND CHEN JIANGis finite by Corollary 2.15. Hence we may take a rational number c0 < 1such that the set{a(Z, L) | Z is a subvariety of X} ∩ [c0 a(Z 0 , L), a(Z 0 , L))1
is empty for any subvariety Z 0 with a(Z 0 , L) > t. Take w = 1−c. We may0
assume w > 2 by decreasing c0 .
If (−KY 0 )i+1 ≤ (w(i + 1))i+1 , then

(L|Y )i+1 ≤ (φ∗ (L|Y ))i+1 ≤(w(i + 1))i+1(w(i + 1))i+1<.
a(Y, L)i+1a(X, L)i+10 .
Then by Lemma 2.4, such Y form a bounded family Ui+1
Now we assume that (−KY 0 )i+1 > (w(i+1))i+1 . By Proposition 2.8, for ageneral point p ∈ Y 0 , there exists an effective Q-divisor ∆0p ∼Q − w1 KY 0 suchthat Vp0 ⊂ Nklt(Y 0 , ∆0p ) is the minimal exceptional non-klt center containing
p. Note that by Lemma 2.9 and w > 2, dim Vp0 > 0. Let ν : Ṽpν → Vp0 be thenormalization. For any Q-Cartier divisor G on Vp0 , we denote G|Ṽ ν = ν ∗ G.
pBy Theorem 2.6, there is an effective Q-divisor ∆Ṽ ν such thatp(KY 0 + ∆0p )|Ṽ ν ∼Q KṼ ν + ∆Ṽ ν .
ppp
Note that since KY 0 + a(Y, L)φ∗ L ≡ 0, we have
1KṼ ν + ∆Ṽ ν + 1 −a(Y, L)φ∗ L|Ṽ ν ∼Q 0.
pppw
Let Vp be the strict transform of Vp0 on Y . Let Ṽp be a common resolutionof Ṽpν and Vp , f : Ṽp → Vp , g : Ṽp → Ṽpν . Then
1KṼp + 1 −a(Y, L)f ∗ (L|Vp )w
1a(Y, L)φ∗ L|Ṽ ν − g∗−1 ∆Ṽ ν + E= g ∗ KṼ ν + ∆Ṽ ν + 1 −ppppw−1∼Q − g∗ ∆Ṽ ν + E,pwhere E is a g-exceptional Q-divisor. Note that the Q-divisor −g∗−1 ∆Ṽ ν + E
pis not big. Hence KṼp + (1 − w1 )a(Y, L)f ∗ (L|Vp ) is not big and therefore1a(Y, L) = c0 a(Y, L).
a(Vp , L) ≥ 1 −w
By the definition of c0 , this implies that a(Vp , L) ≥ a(Y, L). Since p is ageneral point, Y is dominated by such Vp . By induction, Vp is dominatedby some members Z of Ui such that a(Z, L) = a(Vp , L) ≥ a(Y, L). Hence
Y is dominated by some members Z of Ui such that a(Z, L) ≥ a(Y, L).
By Proposition 2.1, by taking general members, Y is dominated by somemembers Z of Ui such that a(Z, L) = a(Y, L).
0 , and the proof is completed.
Hence we may take Ui+1 = Ui ∪ Ui+1
Proof of Theorem 1.1. Take t = a(X, L) in Proposition 3.1, there is a boundedfamily U of subvarieties of X such that any subvariety Y not contained in
B+ (L), with a(Y, L) > a(X, L) is dominated by some members Z of U, suchthat a(Z, L) = a(Y, L) > a(X, L). By Theorem 2.2, there exists a properON FUJITA INVARIANTS OF SUBVARIATIES OF A UNIRULED VARIETY

7

closed subset W ⊂ X such that any member Z of the family U satisfying a(Z, L) > a(X, L) is contained in W . Hence any subvariety Y witha(Y, L) > a(X, L) is contained in W .
References[1] F. Ambro, The locus of log canonical singularities, arXiv:math/9806067.
[2] C. Birkar, Anti-pluricanonical systems on Fano varieties, arXiv:1603.05765.
[3] G. Di Cerbo, On Fujita’s spectrum conjecture, arXiv:1603.09315.
[4] X. Jiang, On the pluricanonical maps of varieties of intermediate Kodaira dimension,
Math. Ann. 356 (2013), 979–1004.
[5] J. Kollár, Singularities of pairs, Algebraic Geometry, Santa Cruz 1995, Proc. of Symp.
in Pure Math. Vol. 62., Amer. Math. Soc., 1997.
[6] J. Kollár, et al, Flips and abundance for algebraic threefolds, A summer seminar at
the University of Utah, Salt Lake City, 1991, Astérisque, 211 (1992)[7] C-J. Lai, Bounding the volumes of singular Fano threefolds, Nagoya Math. J., toappear.
[8] B. Lehmann, S. Tanimoto, Y. Tschinkel, Balanced line bundles on Fano varieties, J.
Reine Angew. Math., to appear.
[9] V. V. Shokurov, 3-fold log flips, Izv. A. N. SSSR, Ser. Math. 56 (1992), 105–201 &57 (1993), 141–175; English transl. Russian Acad. Sci. Izv. Math. 40 (1993), 93–202& 43 (1994), 527–558.
CDH:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112,
USA
E-mail address: hacon@math.utah.edu
CJ:
Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 2778583, Japan.
E-mail address: chen.jiang@ipmu.jp