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The log canonical threshold of homogeneous affine hypersurfaces

We prove that if Y is a hypersurface of degree d in P^n with isolated singularities, then the log canonical threshold of (P^n,Y) is at least min{n/d,1}. Moreover, if d is at least n+1, then we have equality if and only if Y is the projective cone over a (smooth) hypersurface in P^{n-1}. In the case when Y is a hyperplane section of a smooth hypersurface in P^{n+1}, Cheltsov and Park have proved that Y has isolated singularities and they have obtained the above lower bound for the log canonical threshold. Moreover they made the conjecture about the equality case (for d=n+1) and they proved that the conjecture follows from the Log Minimal Model Program. The purpose of this note is to give an easy proof of their conjecture using the description of the log canonical threshold in terms of jet schemes.


Introduction
In [CP], I. Cheltsov and J. Park studied the log canonical threshold of singular hyperplane sections of complex smooth, projective hypersurfaces. Let X ⊂ P n , n ≥ 4, be a complex smooth hypersurface of degree d and Y a hyperplane section of X (which has to be irreducible and reduced). I. Cheltsov and J. Park proved that Y has isolated singular points and they studied the log canonical threshold of the pair (X, Y ). They showed that c(X, Y ) ≥ min{(n − 1)/d, 1}, and they conjectured that if d = n, then equality holds if and only if Y is a cone over a (smooth) hypersurface in some P n−2 . Moreover, they showed that their conjecture follows from the Minimal Model Program.
The purpose of this note is to generalize their lower bound in the case of an arbitrary hypersurface in P n−1 and to give a direct proof of their conjecture in our more general setting. In fact, we get these results also for the affine cone over the hypersurface. The main ingredient is the description of the log canonical threshold in terms of the asymptotic growth of the jet schemes from [Mu2].
Here are our results. Let Y ⊂ P n−1 , n ≥ 2 be a complex hypersurface of degree d ≥ 1 and let Z ⊂ A n be the affine cone over Y . Suppose that dim Sing (Z) = r.
Theorem 1.1. With the above notation, we have the following lower bound for the log canonical threshold of (A n , Z): Theorem 1.2. If in addition d ≥ n − r + 1, then c(A n , Z) = (n − r)/d if and only if Z = T × A r , for some hypersurface T in an (n − r)dimensional affine space. Remark 1.3. Since we have it follows from Theorem 1.1 that c(P n−1 , Y ) ≥ min{(n−r)/d, 1}. Moreover, if d ≥ n − r + 1 and c(P n−1 , Y ) = (n − r)/d, we deduce from Theorem 1.2 that Y is the projective cone (with a P r−1 vertex) over a (smooth) hypersurface in some P n−r−1 .
The converse is well-known: if Y ⊂ P n−1 is the projective cone over a smooth hypersurface of degree d ≥ n − r + 1 in P n−r−1 , then c(P n−1 , Y ) = (n − r)/d. However, for completeness, we will include an argument for this assertion in the spirit of this paper in Proposition 2.4 below.
In order to make the connection between the way we stated our results and the results in [CP], we make the following Remark 1.4. Suppose we are in the situation in [CP]: X ⊂ P n is a smooth hypersurface and Y = X ∩ H, where H ⊂ P n is a hyperplane. We have the following equality: See, for example, Theorem 2.1 below for justification.
Since Cheltsov and Park proved that in this case r ≤ 1, Theorems 1.1 and 1.2 give in particular their lower bound and their conjectured characterization of equality.

Jet scheme dimension computations
For the standard definition of the log canonical threshold, as well as for equivalent definitions in singularity theory, we refer to [Ko]. We will take as definition the characterization from [Mu2] in terms of jet schemes.
Recall that for an arbitrary scheme W (of finite type over C), the mth jet scheme W m is a scheme of finite type over C characterized by , and in fact, we will be interested only in the dimensions of these spaces.
Theorem 2.1. ([Mu2] 3.4) If X is a smooth, connected variety of dimension n, and D ⊂ X is an effective divisor, then the log canonical threshold of (X, D) is given by It is easy to write down equations for jet schemes. We are interested in the jet schemes of a hypersurface Z ⊂ A n defined by a polynomial for all i and j, we take F (p) := D p (F ). The jet scheme Z m is defined by the ideal (F, F ′ , . . . , F (m) ).
For every m ≥ 1, there are canonical projections φ m : By composing these projections we get morphisms ρ m : W m −→ W .
If W is a smooth, connected variety, then W m is smooth, connected, and dim W m = (m + 1) dim W , for all m. It follows from the definition that taking jet schemes commutes with open immersions. In particular, if W has pure dimension n, then ρ −1 m (W reg ) is smooth, of pure dimension (m + 1)n.
For future reference, we record here two lemmas. We denote by [·] the integral part function. In fact, the only assertion we will need from Lemma 2.2 is that dim ρ −1 m (x) ≤ mn−1, if m ≥ q, which follows easily from the equations describing the jet schemes.
If we have a family of schemes π : W −→ S, we denote the fiber π −1 (s) by W s . The projection morphism (W s ) m −→ W s will be denoted by ρ Wm m . We give now the proofs of our results.
Proof of Theorem 1.1. If ρ m : Z m −→ Z is the canonical projection, then we have an isomorphism  (1).
Recall that we have dim Sing (Z) = r. An easy application of Lemma 2.3 shows that for every x ∈ Sing(Z), we have dim A recursive application of this inequality shows that for every p ≥ 1, we have dim Z pd−1 ≤ pd(n−1), if d ≤ n−r, and dim Z pd−1 ≤ p(nd−n+r), if d ≥ n − r. This implies dim Z pd−1 /pd ≤ max{n − 1, (nd − n + r)/d}. By Theorem 2.1, there is p ≥ 1 such that c(A n , Z) = n − dim Z pd−1 pd , and we get c(A n , Z) ≥ min{1, (n − r)/d}.
Proof of Theorem 1.2. We know that d ≥ n − r + 1 and c(A n , Z) = (n − r)/d. By Theorem 2.1, there is k ≥ 1 such that We first show that if k ≥ 2 and equation (2) holds for k, then it holds also for k − 1.
The above argument shows that equation (2) holds for k = 1, so that we have dim Z d−1 = dn − n + r > d(n − 1).
Equation (1) in the proof of Theorem 1.1 gives By a linear change of coordinates we may assume that x = (1, 0, . . . , 0) and we write the equation (1, 0, . . . , 0) = 0, for p < d, we deduce that f i = 0 for i < d, so that Z = T 0 × A 1 , where T 0 is the hypersurface defined by f d in A n−1 .
Since dim W = r, by applying inductively the above argument we get a homogeneous hypersurface T = T r−1 ⊂ A n−r such that Z = T × A r . Note that our assumption on the singular locus of Z implies that Sing (Z) = {0}.
The converse is standard, and in fact we will prove a slightly stronger statement in the next proposition.
The following proposition is well-known, but we include a proof for the benefit of the reader.
Proposition 2.4. Let T ⊂ A n−r be a hypersurface defined by a homogeneous polynomial of degree d ≥ n − r such that Sing (T ) = {0}. If Z = T × A r ⊂ A n and Y ⊂ P n−1 is the projectivization of Z, then c(P n−1 , Y ) = (n − r)/d.
Proof. An equivalent statement with that of the proposition is that c(A n \ {0}, Z \ {0}) = (n−r)/d. Since Z = T ×A r , we have Z m ≃ T m × A mr . This implies that every irreducible component of Z m dominates A r , so that c(A n \ {0}, Z \ {0}) = c(A n , Z) = c(A n−r , T ).
The fact that this number is (n − r)/d is well-known. To see this using jet schemes we can use the analogue of equation (1) in the proof of Theorem 1.1 together with the fact that dim T m = max{(m + 1)(n − r − 1), dim (ρ T m ) −1 (0)}, where ρ T m is the projection corresponding to T . By induction we get dim T kd−1 = k(n − r)(d − 1), for every k ≥ 1, and Theorem 2.1 gives c(A n−r , T ) = (n − r)/d.