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Extensions of two Chow stability criteria to positive characteristics

We extend two results on Chow (semi-)stability to positive characteristics. One is on the stability of non-singular projective hypersurfaces of degree greater than 2, and the other is the criterion by Y. Lee in terms of log canonical thresholds. Some properties of log-canonicity in positive characteristics are discussed with a couple of examples, in connection with the proof of the latter one. It is also proven in appendix that the sum of Chow (semi-)stable cycles are again Chow (semi-)stable.


Introduction
We work over an algebraically closed field k of an arbitrary characteristic.
Let X ⊂ P n k be an effective cycle of dimension r and degree d in a projective space of dimension n. Analysis of the Chow (semi-)stability of X is one of the basic problems in Geometric Invariant Theory (GIT). Contrary to the asymptotic Chow (semi-)stability, the precise classification of Chow (semi-)stable cycles is quite a subtle problem, and is known only for few cases, even for projective hypersurfaces. To name a few, J. Shah studied the case of plane sextics ( [Sh]) and recently R. Laza did the case of cubic fourfolds ( [L]), both in relation with period maps.
On the other hand, there are two sufficient conditions for Chow (semi-)stability in terms of the singularity of X or that of Chow divisor Z(X) ⊂ G = Grass k (n − r, n + 1), which deal with general situations. Both have been proven in characteristic zero, and the purpose of this paper is to extend them to arbitrary characteristics. Namely we prove the following two theorems: Theorem 1.1 (= Theorem 3.1). If d ≥ 3, any non-singular projective hypersurface of degree d is Chow stable.
Theorem 1.2 (= Theorem 4.1). Let X be an effective cycle of dimension r and degree d in P n k . Let (G, Z(X)) be the log pair defined by the Chow divisor Z(X) of X. If lct(G, Z(X)) > n+1 d (resp. ≥ n+1 d ), then X is Chow stable (resp. Chow semi-stable).
In the statement of Theorem 1.2, lct(G, Z(X)) stands for the log canonical threshold of (G, Z(X)), which measures how good the singularity of Z(X) is (see §2.2 for detail).
Characteristic zero case of Theorem 1.1 is due to Mumford ([GIT,Chapter 4 §2]), and that of Theorem 1.2 is due to Y. Lee ([Le]).
The original proof of Theorem 1.1 works only when the characteristic of the base field does not divide d (see §3). To prove the general case, we depend on the corresponding result in characteristic zero.
We sketch the proof of Theorem 1.1 in positive characteristics. First we take a suitable lift of the equation of given hypersurface over the ring of Witt vectors. This defines a family of projective hypersurfaces over the ring. We are assuming that the closed fiber is non-singular, hence the geometric generic fiber is again non-singular. Since we know that Theorem 1.1 holds in characteristic zero, we obtain some inequalities for the Hilbert-Mumford numerical functions of the lift. By the choice of the lift, those numerical functions coincide with those of the original hypersurface. Thus we obtain the inequalities for the numerical functions of the original one, concluding the proof.
The point is that the singularity of the hypersurface over the generic point is better than that of the special fiber, so that we can use the corresponding stability criterion in characteristic zero. This method seems to be applicable to other stability problems (see the remark at the beginning of §4).
In §3.2 it will also be shown that the complement of the locus of nonsingular hypersurfaces is an irreducible divisor, even when p divides d.
In general some multiple of the defining equation of this divisor lifts to the usual discriminant in characteristic zero. Theorem 1.2 will be proven along the same line as the proof given in [Le], but we must modify several points. This is due to the fact that some properties of log canonicity which hold in characteristic zero fail in positive characteristics, because of the existence of wild ramifications and inseparable morphisms.
We can prove that the property of log canonicity which we need still holds for finite separable morphisms. It turns out that this is enough for our purpose, for we can use a perturbation technique so that we need not to deal with the inseparable morphisms (see §4).
In §4 we also discuss some other properties of log canonicity, with a couple of (counter-)examples.
In Appendix A we prove the following This proposition may be well-known to experts, but the author could not find it in the literature. The proof is a simple application of the fact that the stability can be checked 1-PS wise, which is essentially the same as the numerical criterion. But the conclusion itself seems to be rather surprising: if we have two Chow stable cycles, the sum of them is always Chow stable no matter how badly they touch. Proposition 1.3 will be used to give a family of stable projective hypersurfaces whose stability can not be detected by Theorem 1.2 (see Example A.5).
Acknowledgement. The author would like to express his sincere gratitude to his supervisor Professor Yujiro Kawamata for his valuable advice, many suggestions to improve this paper and warm encouragement. He is indebted to Prof. Yongnam Lee for kindly answering the author's question and Prof. Shunsuke Takagi for informing the author of the paper [Mus] in relation with Conjecture 4.2. He would also like to thank the referee for useful comments and corrections for the draft of this paper. Thanks are also due to Prof. Kei-ichi Watanabe for his useful comments and interest in my work, and to Dr. Shohei Ma for his nice comments. This paper is an extended version of the author's master thesis submitted to the University of Tokyo in 2010.

Preliminary
2.1. Notations from scheme theory. We need some notations from [Ha].
Let R be an N-graded ring. For a homogeneous ideal I of R we denote by V (I) the corresponding closed subscheme Proj(R/I) of ProjR.
For a homogeneous element f ∈ R, we denote ProjR \ V (f ) by D + (f ). This open subscheme is known to be affine, with coordinate ring

Notions of singularities.
In this subsection, we summarize the notions of singularities of pairs which we need later.
Definition 2.1 (discrepancy, log canonical). Let X be a normal variety over k and ∆ be an effective R-Weil divisor on X such that K X + ∆ is R-Cartier. Let π : Y → X be a birational morphism from another normal variety Y over k and E ⊂ Y be a prime divisor. Then in a neighborhood of the generic point of E, the following canonical bundle formula holds: The real number a in the above equation is called the discrepancy of E with respect to (X, ∆), and denoted by a(E; X, ∆). It is independent of the choice of Y and π, depending only on the valuation of k(X) which corresponds to E.
We say that the log pair (X, ∆) is log canonical (lc, for short) if a(E; X, ∆) ≥ −1 holds for all the divisors E as above.
A finer version is: Definition 2.2. Let x ∈ X be a point. We say that the log pair (X, ∆) is log canonical at x if the restriction of (X, ∆) to an open neighborhood of x is log canonical. Definition 2.3 (log canonical threshold). Let (X, ∆) be a log canonical pair and D be an effective R-Cartier divisor on X. The log canonical threshold of D with respect to (X, ∆) is defined as follows: lct(X, ∆; D) = sup{t ∈ R|(X, ∆ + tD) is log canonical}. For a point x ∈ X, we set lct x (X, ∆; D) = sup{t ∈ R|(X, ∆ + tD) is log canonical at x}.
It is easy to see by definition that "sup" in the above definition is actually "max".

2.3.
Chow stability and the numerical criterion. Let X ⊂ P n k be an effective r-dimensional cycle of degree d. We associate to X its Chow divisor Z(X), which is a hypersurface of degree d of the Grassmannian G = Grass k (n − r, n + 1), as follows (one may consult either [Ko2] or [GKZ] for detail). If X itself is a variety, set Z(X) = {L ∈ G|L∩X = φ}. For a general cycle X, define Z(X) additively. The defining equation of Z(X) is called the Chow form of X (Chow form is determined by X only up to scalar multiplication). The homogeneous coordinate ring of G with respect to the Plücker embedding is denoted by B = d≥0 B d . This is the subring of the polynomial ring of (n + 1)(n − r) indeterminants U (j) i 's, where (i, j) runs through the range i = 0, . . . , n and j = 1, . . . , n − r, generated by all the (n − r) × (n − r) minors of the matrix (U (j) i ). The Chow form of a cylce X is an element of B d (up to scalar multiplication), so that the Chow divisor Z(X) of X can be regarded as an element of the projective space P * B d . The canonical action of SL(n + 1, k) on P n k naturally induces a linear action on B d , hence we can discuss the GIT (semi-)stability of an element of P * B d (here we are using the terminology "stable" in the sense of "properly stable" in [GIT], which requires the finiteness of the stabilizer subgroup. We heavily rely on the numerical criterion, so we follow this terminology 1 ). Chow (semi-)stability of X is defined to be the (semi-)stability of Z(X) in the above sense.
Next we recall the Hilbert-Mumford numerical criterion (numerical criterion, for short) for stability and obtain an explicit description of the numerical function µ following [GIT,Proposition 2.3]. We start with some preparations.
For a subset I ⊂ [n] with #I = n − r, let ∆ I be the (n − r) × (n − r) minor of the matrix (U (j) i ) obtained by picking out the n − r rows according to I. Recall that B d is a k-vector space generated by the set Now fix X. Take any g ∈ SL(n + 1, k) and let F be the Chow form of g * X (= the defining equation of g * Z(X)). Set (2) An element r of R corresponds to a non-trivial one-parameter subgroup (1-PS for short) λ : Using these notations, the numerical function of X with respect to g ∈ SL(n + 1, k) and r ∈ R is defined as follows: Definition 2.4 (numerical function). Let I(F ) be the set of such d- Remark 2.5. µ(Z(X), g, r) depends only on g, r and the set I(F ).
Next we rephrase Proposition 2.6 in such a way as to prove Theorem 4.1. This reinterpretation is just a generalization of [Le,Lemma 2.1]. Before that, we need some preparations. Take an arbitrary g ∈ SL(n + 1, k) and let F be the Chow form of g * X.
Let f be the local equation (1)) satisfying the following two conditions: Therefore f is a polynomial in x I 's. Now assign nontrivial integral weights r = (r 0 , . . . , r n ) ∈ R to X 0 , . . . , X n , so that the induced weight w(x I ) on x I satisfies which is non-negative by the assumption r 0 ≤ r 1 ≤ · · · ≤ r n . Now Proposition 2.6 is equivalent to ) holds for all g ∈ SL(n + 1, k) and r ∈ R (see (2) for the definition of R).
Above f is the local equation on D + (∆ [n−r−1] ) of the Chow form of g * X as before. In the left hand side of (5), w(f ) denotes the weighted multiplicity of f (= the lowest weight of the monomials occurring in f ) with respect to the weight (w(x I )) I .
Proof. We only discuss the stable case. Semi-stable case can be proven similarly.
The inequality (5) is equivalent to Combining the calculation of w(x I ) (see (4)) with the definition of w(f ), we see that the left hand side of (6) equals to Recalling the conditions (3) posed on I's we see I i∈I A little calculation shows that the left hand side of (6) boils down to d(n − r) n i=0 r i − (n + 1)µ(X, g, r) = −(n + 1)µ(X, g, r), since we assumed that n i=0 r i = 0. Therefore (5) is equivalent to the condition µ(X, g, r) < 0.
2.4. Chow stability in characteristic p from characteristic zero. Let k be a field of characteristic p > 0 and X be a cycle in P n k . In this subsection we want to propose a method to deduce the Chow (semi-)stability of X from the corresponding results in characteristic zero.
From now on, we denote by W = W (k) the ring of Witt vectors. This is a discrete valuation ring (DVR for short) of characteristic zero, whose residue field is isomorphic to k (see [S,Chapter 2 §5 Theorem 5]). Actually these are all the properties of W which we need in this paper. We denote by K the field of fractions of W and by m W the unique maximal ideal of W .
Take g ∈ SL(n + 1, k) and let F be the Chow form of g * X as in the previous subsection. Let F W be a lift of F over W such that a monomial which does not appear in F never appears in F W , which is equivalent to the assumption I(F ) = I(F W ) (see Definition 2.4 for the definition of Theorem 2.8. Assume that for any g ∈ SL(n + 1, k) we can take F W such that I(F ) = I(F W ) holds and V (F W ) is stable (resp. semi-stable) with respect to the induced action of SL(n + 1, K). Then X is Chow stable (resp. Chow semi-stable).
Remark 2.9. By the result of C. S. Seshadri ([Se, Proposition 6], see also [GIT, Appendix to Chapter 1, §G]), the converse of Theorem 2.8 also holds: if X is Chow stable (resp. Chow semi-stable), any lift F W of F is also stable (resp. semi-stable) with respect to the induced action of SL(n + 1, K).

Chow stability of non-singular hypersurfaces
In this section X denotes a hypersurface of degree d in P n k . In §3.1, we prove the stability of non-singular hypersurfaces of degree at least three. This is an easy application of Theorem 2.8. In §3.2 we study the complement of the locus of non-singular hypersurfaces via geometric arguments. It turns out that the complement is an irreducible divisor and that some multiple of its defining equation lifts to the usual discriminant in characteristic zero.
3.1. A proof via lifting to characteristic zero. First of all we recall that the characteristic zero case of Theorem 3.1 was settled in [GIT,Chapter 4 §2]. Thanks to a theorem by Matsumura and Monsky, the proof given there also works for characteristic p cases if p does not divide d. We briefly recall the proof and see why it does not work for the cases when p do divide d.
Let F (X 0 , X 1 , . . . , X n ) be a homogeneous polynomial of degree d. We have the Euler's lemma: Therefore we see that provided that p does not divide d. The emptiness of the latter is equivalent to the vanishing of the discriminant of F when d ≥ 2. This shows the semi-stability of non-singular hypersurfaces of degree at least 2. Furthermore, when d ≥ 3, it is known (see [MM,Theorem 1]) that only finitely many projective linear transformations preserve the given non-singular hypersurface. This means that any non-singular hypersurface is stable, provided d ≥ 3 and p |d.
The above argument does not work in general, for the equality (7) may break down when p divides d. Actually when p divides d and d ≥ 3, the right hand side of the equality (7) can not be empty. This will be proven in the next subsection (see Proposition 3.4).
Even when p divides d, a closer look at the numerical criterion shows that non-singular hypersurfaces are always (semi-)stable if d > n + 1 (resp. d ≥ n + 1) (see [N,Lemma 4.2]. This may also be deduced from Theorem 4.1, since the pair (P n k , X) is log canonical when X is a non-singular hypersurface). Now we prove that the stability is always the case: Theorem 3.1. If d ≥ 3, any non-singular projective hypersurface of degree d is Chow stable.
Proof. The theorem is already established when chark = 0, so we assume chark > 0. We use Theorem 2.8. Let X ⊂ P n k be an nonsingular projective hypersurface of degree at least three. Take any g ∈ SL(n + 1, k) and let F k be the equation of g * X. Note that in this case F k itself is the Chow form of g * X. Take a lift F W of F k over the ring of Witt vectors W satisfying I(F k ) = I(F W ) (see Definition 2.4 for the definition of I). Then it is easy to see the Proof. Since W is a DVR, W [X 0 , . . . , X n ] is a UFD. So it is enough to show that F W is an irreducible element of W [X 0 , . . . , X n ]. Suppose for a contradiction that F W = G · H holds for some G, H ∈ W [X 0 , . . . , X n ] such that neither G nor H is a unit. Note that both G and H are homogeneous and non-zero, since F W is. Therefore G, H are homogeneous polynomials of degrees at least one, since neither of them is a unit. This means that either G = 0 or deg G = deg G (here G denotes the reduction modulo m W of G) must hold (similar for H). On the other hand, G · H = F W = F k = 0 holds. Therefore deg G = deg G ≥ 1 (resp. deg H ≥ 1), contradicting the irreducibility of F k .
Since V (F W ) dominates the generic point of Spec W , the above claim means that V (F W ) is flat over Spec W (see [Ha,Chapter III,Proposition 9.7]). Also it is projective over Spec W .
The closed fiber of V (F W ) → Spec W is g * X, which is non-singular. Therefore the geometric generic fiber is also non-singular (see [EGA IV, (12.2.4)(iii)]). Since the characteristic of the generic fiber is zero and deg(F W ) ≥ 3, we already know that it is stable. By Theorem 2.8, we see that X is stable too.
3.2. The defining equation. Let Hyp d (n) be the projective space of degree d hypersurfaces in P n k , and U ns ⊂ Hyp d (n) be the locus of non-singular hypersurfaces. In this subsection we study the defining equation for the complement of the locus of non-singular hypersurfaces, Hyp d (n) \ U ns , via geometric arguments. This is a version of the arguments given in [Mu,Chapter 5 §2].
The defining equation is well-known when p does not divide d, the discriminant. Therefore we are interested in the cases when p divide d.
When chark = 2, D mod 2 = (T mod 2) 2 and the defining equation for Hyp 4 (1) \ U is given by T mod 2 = a 0 a 2 3 + a 1 a 2 a 3 + a 2 1 a 4 . Proof of Theorem 3.2. Let W = W (k) be the ring of Witt vectors. Set where Hyp d (n) = |O P n W (d)| is the projective space of families of degree d projective hypersurfaces over Spec W . Let p : I → P n W , q : I → Hyp d (n) be the natural projections.
First we show the following Claim. p is a smooth morphism with connected fibers.
Proof. Let x : Spec Ω → P n W be a geometric point, where Ω is an algebraically closed field. By the definition of I above, it is easy to see that I x ⊂ Hyp d (n) Ω := Hyp d (n) × Spec W Spec Ω is a linear subspace.
Next we calculate the fiber I (1:0:···:0) , where (1 : 0 : · · · : 0) ∈ P n Ω . Note that if we write F (X) = |α|=d C α X α by using multi-indices, (C α ; |α| = d) gives a system of coordinates for the projective space Hyp d (n) Ω . Then we can show the following equality (note that the equality is independent of the characteristic of Ω): In order to show that the dimension of the linear subspace I x is independent of x, we show that it is isomorphic to I (1:0:···:0) . Consider the action of SL Ω (n + 1) on P n W × Spec W Hyp d (n) Ω which is defined by g · (x, X) = (gx, g * X) for g ∈ SL(n + 1, Ω). It can be easily checked that this action preserves I × Spec W Spec Ω and that we obtain an isomorphism between I (1:0:···:0) and I x via this action.
By the claim we see that both I and I k , the restriction of I over the closed point Spec k ⊂ Spec W , are integral schemes. Now consider the integral closed subscheme q(I) ⊂ Hyp d (n). Note that the defining equation for q(I) is the usual discriminant, and that q(I) k , restriction of q(I) over Spec k ⊂ Spec W , coincides with q(I k ) as sets.
Similar arguments as above show the following Proposition 3.4. Assume d is divided by p and either d ≥ 3 or d = p = 2 and n is even. Let X = V (F ) ⊂ P n k be an arbitrary hypersurface of degree d. Then Proof. Set Z = (x, X); x ∈ V ∂F ∂X 0 , ∂F ∂X 1 , . . . , ∂F ∂X n ⊂ P n k × Hyp d (n) and let p : Z → P n k and q : Z → Hyp d (n) be the natural projections. As in the proof of Theorem 3.2, we can show the following Claim. p is a smooth morphism with connected fibers.
Next we calculate the dimension of Z (1:0:···:0) . With the notations in the proof of Theorem 3.2, we can write Now all we have to show is that q : Z → q(Z) is generically finite, because then we see that dim q(Z) = dim Z = dim Hyp d (n), hence q(Z) = Hyp d (n). In order to show it, we check the finiteness of the fiber of q at for all i = 0, 1, . . . , n, where X −1 = X n and X n+1 = X 0 . Suppose a = (a 0 : · · · : a n ) ∈ V ∂F ∂X 0 , ∂F ∂X 1 , . . . , ∂F ∂Xn . From (8), one see that a 0 · · · a n = 0. Hence we assume a 0 = 1. Using (8) recursively, we obtain the following equation: The exponent of a 1 above is non-zero under our assumptions on (d, p, n). Therefore (9) poses a non-trivial condition on a 1 . a 2 , a 3 , . . . , a n are uniquely determined from a 1 , because of (8). Thus the finiteness is shown.
Remark 3.5. When d = p = 2 and n is odd, we have the following counter-example: It is easy to see that V ∂F ∂X 0 , ∂F ∂X 1 , . . . , ∂F ∂Xn = ∅.

Y. Lee's criterion in characteristic p
In this section we prove the following Theorem 4.1. Let X be an effective cycle of dimension r and degree d in P n k . Let (G, Z(X)) be the log pair defined by the Chow divisor Z(X) of X. If lct(G, Z(X)) > n+1 d (resp. ≥ n+1 d ), then X is Chow stable (resp. Chow semi-stable).
See §2.3 for notations. Our proof goes along the same line as the original one by Y. Lee ([Le]), but we need to modify several points.
Before the proof, we point out that we might prove Theorem 4.1 via Theorem 2.8 as in the previous section, provided that the following conjecture would be true (below W is the ring of Witt vectors and K, k are the field of fractions and the residue field of W, respectively): Conjecture 4.2. Let X W → Spec W be a smooth proper morphism where X W is an integral scheme. Let D W be an effective R-divisor on X W , such that no irreducible component is contained in a fiber of the projection to Spec W . By X K and D K we denote the restrictions of X W and D W over the generic point of Spec W . Similarly X k , D k denote the restrictions of X W and D W over the closed point of Spec W . Then if (X k , D k ) is log canonical, so is (X K , D K ). In particular lct(X k , D k ) ≤ lct(X K , D K ).
Note that the above conjecture can be proven when (X k , D k ) has a good log resolution (here "good" means that it is isomorphic outside of the support of D k ), following [Mus]. In [Mus], the lower semi-continuity of log canonical thresholds in a family of projective log pairs with nonsingular ambient varieties is proven when the base scheme of the family is defined in characteristic zero. We need the last assumption because the existence of good log resolution is not yet established in positive characteristics in full generality. Today basic results on motivic integrations are established over arbitrary perfect fields (see [Y]), so the arguments in [Mus] can be applied to our case without change under the existence of good log resolutions. 4.1. Log canonicity in positive characteristics. In this subsection, we discuss how the log canonicity of log pairs are preserved under finite morphisms. Some properties of log canonicity which hold in characteristic zero fail in characteristic p > 0, but we can circumvent those difficulties and obtain Proposition 4.9, which is the key for the proof Theorem 4.1.
When the characteristic of the base field is zero, it is well known that the log canonicity is preserved under finite dominant morphisms (see [KoM,Proposition 5.20(4)]). Namely: Theorem 4.3. Let g : X ′ → X be a finite dominant morphism of normal varieties over a field of characteristic zero. Let ∆ (resp. ∆ ′ ) be a Q-divisor on X (resp. X ′ ) such that K X + ∆ is Q-Cartier and We should note that the canonical divisors K X and K X ′ in Theorem 4.3 above are chosen in such a way that K X ′ = g * K X + R holds, where R is the ramification divisor of g.
When the characteristic of the base field is positive, we need to modify Theorem 4.3. First we consider the case when g is separable. In this case we may have wild ramifications, so we only have a weaker version of the ramification formula: Lemma 4.4. Let g : X → Y be a finite separable morphism between normal varieties over k. Let E ⊂ X be a prime divisor on X and r be the ramification index of g along E. Then there exists a non-negative integer b ≥ r − 1 such that K X = g * K Y + bE holds around the generic point of E.
Proof. Set V = Y \ Sing Y and U = g −1 (V ) − Sing X. Note that the closed subsets we have through away have codimension greater than 1. Over U we have the following exact sequence: Since g is separable, Ω U/V generically vanishes (see [M,Theorem 59]). Hence f is generically isomorphic.
be the highest exterior product of the morphism f in (10) above. This is also generically isomorphic. Therefore ker F is a torsion subsheaf of the torsion free sheaf g * O V (K V ), so is trivial. Hence we see that F is injective.
Take a generic closed point e of E ∩ U which is contained in no other irreducible component of Supp Ω U/V except for E. Set e ′ = g(e), E ′ = g(E). Choose systems of local coordinates x 1 , . . . , x n at e and y 1 , . . . , y n at e ′ , satisfying the following conditions: (a) E = div(x 1 ) near e (resp. E ′ = div(y 1 )). (b) g * y i = x i holds for all i = 2, . . . , n. (c) there exists an invertible function u at e such that g * y 1 = u · x r 1 . In (c), r denotes the ramification index of g along E. Now hence there exists some non-negative integer b such that holds in a neighborhood of e.
If r ≡ 0 (mod p), b = r − 1. Now assume that E is wildly ramifying. Then (11) = ∂u ∂x 1 x r 1 dx 1 ∧ dx 2 · · · ∧ dx n = 0, since otherwise F is not generically isomorphic. In this case we see that where val E denotes the valuation corresponding to E.
Remark 4.5. Ramification formula for inseparable morphisms are discussed in [RS]. In this case the ramification divisor is defined only up to linear equivalence. If we adopt this version of ramification formula, the 'only if' part of Theorem 4.3 does not hold in general. For our purpose we need not to deal with inseparable cases.
With the weaker version of ramification formula above, we can prove that the 'only if' part of the Theorem 4.3 still holds for separable morphisms: Proposition 4.6. Let k be an algebraically closed field of characteristic p > 0. Let g : X ′ → X be a finite separable morphism of normal varieties over k. Let ∆ (resp. ∆ ′ ) be a Q-divisor on X (resp. X ′ ) such that K X + ∆ is Q-Cartier and g * (K X + ∆) = K X ′ + ∆ ′ . Then if (X, ∆) is log canonical, so is (X ′ , ∆ ′ ).
Proof. The proof goes along the same line as the proof of ( [KoM,Prop 5.20(4)]), once we replace the ramification formula by the weaker version given above.
Remark 4.7. In general, the 'if' part of Theorem 4.3 holds only when there exists no wildly ramifying divisor. In such a case, the proof goes as in characteristic zero. If some of the ramification divisors are wildly ramifying it may not hold. An example is: Note that (X, ∆) is not lc, but (X ′ , ∆ ′ ) is.
Step 1. First we establish the inequality for those w's such that w(x i ) > 0 holds for all i = 1, . . . , n, and p divides none of the w(x i )'s. In this case the inequality can be established along the same line as the original proof, since we have Proposition 4.6. For the sake of completeness, we reestablish the argument.
Consider g : A n k → A n k given by g( . By the assumptions on w(x i )'s, g is dominant and separable. Take a real number c ∈ R ≥0 and assume (A n k , c · div(f )) is lc at 0. Now calculate the pull-back of K A n k + c · div(f ) by g: By Proposition 4.6, we see (A n k , ∆ ′ ) is lc at 0. Let E be the exceptional divisor of the blow-up of A n k at the origin. We know that a(E; A n k , ∆ ′ ) ≥ −1 holds. With a calculation we see that a(E; A n k , ∆ ′ ) equals to −1 + i w(x i ) − cw(f ), obtaining the inequality.
Step 2. Now consider the continuous function ϕ : , as in the case when w(x i )'s are integers. If we replace w by some positive multiple of it, the value of ϕ never changes. Therefore ϕ factors through the quotient space inducing the continuous function ϕ : S → Q.
The set of points represented by those w's satisfying the assumptions in Step 1 is dense in S. Hence, by the continuity of ϕ, we see that holds for arbitrary s ∈ S. We finish the proof.
Step 2 in the proof above is inevitable, for (A n k , ∆ ′ ) need not be lc if g is inseparable. For example, consider the case n = 2, w(x 1 ) = w(x 2 ) = p and f (x 1 , x 2 ) = x 1 − x 2 . In this case lct 0 (A 2 k , div(f )) = 1. On the other hand is not lc at the origin. Note also that even in this case a(E; A 2 k , ∆ ′ ) ≥ −1 holds, because of Proposition 4.9. 4.2. Proof of Theorem 4.1.
Proof of Theorem 4.1. We only discuss stable case. Semi-stable case can be proven exactly in the same way. We have only to confirm the inequality (5) of Lemma 2.7. On the other hand, by the assumption and Proposition 4.9, the inequality clearly holds.
Remark 4.11. Theorem 4.1 has the following direct corollary, which is slightly weaker: , then X is Chow stable (resp. Chow semi-stable).
Above Fpt denotes the F-pure threshold of the pair (G, Z(X)). Corollary 4.12 is deduced from Theorem 4.1 via [HW,Theorem 3.3], which says: F-pure ⇒ log canonical.
We can also show Corollary 4.12 by directly proving the Fpt version of Proposition 4.9, using the Fedder-type criterion for F-purity due to [HW].
Appendix A. Chow stability of the sum In this section we show that the sum of two Chow semi-stable cycles of the same dimension are again Chow semi-stable. Moreover if one of them is stable, it follows that the sum also becomes stable.
Proposition A.1. Let Y, Z be Chow semi-stable cycles of the same dimension in a projective space P n k . Then Y + Z is again Chow semistable. Furthermore if Y is Chow stable, so is Y + Z.
In the proof we freely use the notation like lim t→0 λ(t)·F , as in [GIT], since the idea becomes clearer. To be logically complete, we of course need to replace the argument suitably. It is a routine work, so we omit the detail.
Proof. Let d,e be the degrees of Y and Z respectively. Let F ∈ B d , G ∈ B e be the Chow forms of Y and Z, respectively. Then the Chow form of Y + Z is given by F · G ∈ B d+e .
Choose a non-trivial 1-parameter subgroup (1-PS) λ : G m → SL(n+ 1, k). Via λ we pull back the canonical actions of SL(n + 1, k) onto B d , B e and B d+e to G m . Now consider the natural multiplication map µ : B d × B e → B d+e , given by (F, G) → F · G. If we pose the diagonal action of G m on the source, µ becomes equivariant.
Remark A.2. We can not expect the converse of Proposition A.1 at all. There exists a semi-stable cycle such that all of its subcycles are unstable: Example A.3. Take the union of three lines on a plane which are in a general position. The union itself is Chow semi-stable (see [GIT,), but lines and reducible conics on a plane are Chow unstable.
However, the following holds: Proposition A.4. Let Z be a cycle of P n k . Then the followings are equivalent: (1) Z is Chow (semi-)stable.
Proof. We have only to prove (3)⇒(1). Let G be the Chow form of Z.
Then G m gives the Chow form of mZ. Assume that mZ is Chow semistable. Take any 1-PS λ as in the proof of the Proposition A.1. Then 0 = lim t→0 λ(t) · G m = (lim t→0 λ(t) · G) m , hence lim t→0 λ(t) · G = 0. Therefore Z is semi-stable. Stable case can also be shown via a similar argument.
Example A.5. Let Y ⊂ P n k be a non-singular hypersurface of degree three, which is Chow stable by Theorem 3.1. By Proposition A.1, mY is also Chow stable for all the positive integers m. On the other hand, lct(P n k , mY ) = 1 m and hence 1 m < n+1 3m if n ≥ 3. Thus we obtain a sequence of examples of Chow stable hypersurfaces whose stability can not be detected by Theorem 4.1.