The $4n^2$-inequality for complete intersection singularities

The famous $4n^2$-inequality is extended to generic complete intersection singularities: it is shown that the multiplicity of the self-intersection of a mobile linear system with a maximal singularity is higher than $4n^2\mu$, where $\mu$ is the multiplicity of the singular point.

We will assume the singularity to be generic in the sense of Sec. 2 below. The aim of this note is to prove the following claim.
Theorem. Let Σ be a mobile linear system on X. Assume that for some positive n ∈ Q the pair (X, 1 n Σ) is not canonical at the point o but canonical outside this point. Then the self-intersection Z = (D 1 • D 2 ) of the system Σ satisfies the inequality mult o Z > 4n 2 mult o X. (1) The assumption of the theorem means that the pair (X, 1 n Σ) has a non-canonical singularity with the centre at the point o. Explicitly, for some exceptional divisor R over X, the centre of which is the point o, the Noether-Fano inequality ord R Σ > n · a(R, X) holds, where a(R, X) is the discrepancy of R with respect to X.
(ii) The self-intersection Z = (D 1 •D 2 ) is the scheme-theoretic intersection of any two general divisors in Σ which is well defined as Σ is free from fixed components.
(iii) When mult o X = 1, we get the standard 4n 2 -inequality, see [14,Chapter 2]. For that reason, we call the inequality (1) the 4n 2 -inequality as well. The standard 4n 2 -inequality (for the non-singular case) was first shown in [9] on the basis of the technique developed in [7]. Later a different proof was found by Corti [4] and various generalizations of the 4n 2 -inequality were investigated [3,13], see [14, Chapter 2] for more details.
Note that in the smooth case (when mult o X = 1) the 4n 2 -inequality holds for dim X 3 without any additional assumptions. This is because the exceptional divisor of the blow up of the point o on X is just the projective space, and in the projective space it is very easy to bound multiplicities in terms of degrees. Unfortunately, it is not so easy to do so (in the way we need) for hypersurfaces and complete intersections, which generate the need for additional assumptions.
The author thanks the referees for a number of useful suggestions, especially for spotting the insufficient lower bound for dim X in the first version of the paper.
2. Generic complete intersection singularities. The germ (X, o) is given by a system of l analytic equations in C M +l , where 2 µ 1 . . . µ l , l 1 and the polynomials q j,i are homogeneous of degree i in the coordinates z 1 , . . . , z M +l ; the point o = (0, . . . , 0) is the origin. We denote by µ = (µ 1 , . . . , µ l ) the type of the singularity o ∈ X and set µ = µ 1 · · · µ l = mult o X to be the multiplicity of the point o (assuming the conditions of general position for the first polynomials q 1,µ 1 , q 2,µ 2 , . . . , q l,µ l , stated below). Set also Recall that by assumption M l + |µ| + 3. Let P ∋ o be a linear subspace in C M +l of dimension 2l + |µ| + 3. Denote by X P the intersection X ∩ P . Definition 1. We say that the complete intersection singularity (X, o) is generic, if for a general subspace P of dimension 2l + |µ| + 3 the singularity o ∈ X P is an isolated singularity, dim X P = l + |µ| + 3 and for the blow up ϕ P : X + P → X P of the point o, the variety X + P is non-singular in neighborhood of the exceptional divisor Q P = ϕ −1 P (o), which is a non-singular complete intersection of codimension l and type µ = (µ 1 , . . . , µ l ). From now on, we assume that the singularity o ∈ X is generic. In particular, by Grothendieck's theorem on factoriality [1], X is a factorial variety near the point o.
3. Start of the proof. The idea of the proof is as follows. We use as a model the proof of the standard 4n 2 -inequality by means of the technique of counting multiplicities as it is given in [14, Chapter 2, Section 2.2]. First, we observe that by inversion of adjunction, the existence of a non-canonical singularity R implies the existence of another singularity E of the same pair (X, 1 n Σ) which satisfies a Noether-Fano type inequality. The latter is somewhat weaker (but sufficient for our purposes). However, the new singularity E has the crucial advantage that its centre on the blow up X + of the point o has a high dimension. This is done in the present section.
After that, in Section 4 we resolve the singularity E and use the assumptions on the singular point o ∈ X to relate the multiplicities of the system Σ and its self-intersection at the point o with the multiplicities of the strict transforms of Σ and the self-intersection at the "higher storeys" of the resolution, at the centres of the singularity E on those "higher storeys".
This done, we apply the technique of counting multiplicities in word for word the same way as in [14, Chapter 2, Section 2.2] and complete the proof.
Let us realize this programme. For a general (2l + |µ| + 3)-subspace P set Σ P = Σ| P to be the restriction of Σ onto P . By inversion of adjunction [15,8], the pair (X P , 1 n Σ P ) is not canonical (for M > l + |µ| + 3, even non-log canonical, but we do not need that.) Obviously, is the self-intersection of the system Σ P and mult o Z = mult o Z P . Therefore, we may (and will) assume from the beginning that M = l + |µ| + 3 and so P = C M +l , so that already the original singularity o ∈ X is isolated. Now we omit the index P and write ϕ : X + → X for the blow up of the point o and Q = ϕ −1 (o) for the exceptional divisor, which is a non-singular complete intersection of type µ in P 2l+|µ|+2 . Now let Π ∋ o be a general linear subspace of dimension |µ| + 3. By the symbol is a non-singular complete intersection of type µ (and codimension l).
Note that by the adjunction formula for the discrepancy we have the equality a(Q Π , X Π ) = 2.
For a general divisor D ∈ Σ and its strict transform D + ∈ Σ + on X + we have for some positive integer ν (recall that we consider a local situation: o ∈ X is a germ). Obviously, if ν > 2n, then mult o Z ν 2 µ > 4n 2 µ and the 4n 2 -inequality holds. For that reason, from now on we assume that ν 2n.
Setting D Π = D| X Π , we get D + Π ∼ −ν Q Π . By the inversion of adjunction the pair X Π , 1 n D Π is not log canonical at the point o, the more so not canonical, so for some exceptional divisor E Π over X Π the Noether-Fano inequality is satisfied. As ν 2n and a(Q Π , X Π ) = 2, we see that E Π = Q Π and E Π is a non log canonical (and so not canonical) singularity of the pair Therefore we may assume that codim(∆ Π ⊂ Q Π ) 2.
Coming back to the variety X, we conclude that for some exceptional divisor E over X with the centre at o the Noether-Fano type inequality ord E Σ > n(2 ord E Q + a(E, X + )) is satisfied. Moreover, the centre ∆ ⊂ Q of E on X has codimension at least 2 and dimension at least 2l.

Resolution of the singularity E. Consider the sequence of blow ups
where ϕ i,i−1 : X i → X i−1 is the blow up of the centre B i−1 ⊂ X i−1 of the exceptional divisor E on X i−1 . In particular, B 0 = o and B 1 = ∆. Using the notations, identical to those in [14, Chapter 2, Section 2.2], we set to be the exceptional divisor, so that E 1 = Q. As X 1 = X + is non-singular in a neighborhood of E 1 , all subsequent varieties X i are non-singular at the generic point of B i and all constructions of [14, Chapter 2, Section 2.2] work automatically for the blow ups ϕ i,i−1 with i 2.
The last exceptional divisor E K defines the discrete valuation ord E . We divide the sequence ϕ i,i−1 , i = 1, . . . , K, of blow ups into the lower part, i = 1, . . . , L K, corresponding to the centres B i−1 of codimensions at least 3, and the upper part, i = L + 1, . . . , K, corresponding to the centres B i−1 of codimension two. As usual, we denote the strict transform of any geometric object on X i by adding the upper index i and set: for any i = 2, . . . , K to be the elementary multiplicities. Let Γ be the oriented graph of the resolution of the singularity E and p ij the number of paths from the vertex i to the vertex j, p ii = 1 by definition (see [14, Chapter 2, Section 2.2] for the standard details). We also set p i = p Ki , i = 1, . . . , K. Now the Noether-Fano type inequality takes the form where ν 1 = ν and δ i = codim(B i−1 ⊂ X i−1 ) are the elementary discrepancies. By the linearity of the Noether-Fano type inequality (2) and the standard properties of the numbers p ij we may assume that ν K > n (replacing, if necessary, E K by a lower singularity E j for some j < K). In order to proceed, we need the following known fact. Proposition 2. Let Y ⊂ P N be a non-singular complete intersection of codimension l 1, S ⊂ Y an irreducible subvariety of codimension a 1 and B ⊂ Y an irreducible subvariety of dimension al, where the estimate N (l + 1)(a + 1) is satisfied. Then the inequality mult B S m holds, where m 1 is defined by the condition S ∼ mH a Y and H Y ∈ A 1 Y is the class of a hyperplane section of Y .
Proof for the case l = 1 was given in [11]. The argument extends directly to the general case of an arbitrary l, see [16] (also [12,2]). Q.E.D.
Applying Proposition 2 to a divisor in the linear system Σ 1 | Q , we conclude that ν 1 ν 2 , since dim B 1 = dim ∆ 2l. The inequalities ν 2 ν 3 . . . ν K are standard. We deduce that the upper part of the resolution of E is non-empty (that is to say, L < K) and the upper part of the graph Γ is a chain: moreover, there are no arrows connecting either of the vertices L + 1, . . . , K with any of vertices 1, 2, . . . , L − 1. (These are the standard consequences of inequalities ν K > n and ν 1 2n, see [14, Chapter 2, Section 2.2].) We do not need this additional information for the proof of our theorem, but in particular geometric problems it might be useful.
5. The technique of counting multiplicities. Now everything is ready for the proof of the desired inequality (1). Take a general pair of divisors D 1 , D 2 ∈ Σ and set to be their scheme-theoretic intersection, the self-intersection of the mobile linear system Σ. Recall that the strict transform of an irreducible subvariety or an effective cycle, or a linear system on some X i is denoted by adding the upper index i. (This notation silently implies that the irreducible subvariety or the effective cycle etc. is sitting on a lower storey X j , j i, of the resolution and that the operation of taking the strict transform is well defined for that particular subvariety etc.) For where the effective cycle Z i of codimension 2 is supported on E i and so may be viewed as an effective divisor on E i . Thus for any i L we obtain the presentation For any j > i, where j L, set holds.

holds.
Proof. Part (i) follows from Proposition 2 as Z 1 ∼ d 1 H Q and dim B 1 2l. In order to show part (ii), we note that (numerically) being of pure codimension 2 on Q. Applying Proposition 2 to the cycle (Z 1 • Q), we get the inequality which completes the proof of the proposition. Q.E.D. The more so, m 0,1 µm 0,i for i 3 as m 0,2 m 0,3 . . . m 0,L . Now set m * i,j = µm i,j for (i, j) = (0, 1) and m * 0,1 = m 0,1 . Also set d * i = µd i for i = 1, . . . , L. We obtain the following system of equalities: Proof of the theorem is completed. Remark 2. The inequality (1) essentially simplifies the proof of birational superrigidity of Fano hypersurfaces with isolated singularities of general position given in [10]. The cases of singular points of multiplicity µ = 3 and 4 in that paper are really hard. The inequality (1) gives for the multiplicity mult o Z at such points the lower bound 12n 2 and 16n 2 , respectively, which is more than enough to exclude the maximal singularities over such points by the standard (in fact, relaxed) technique of hypertangent divisors. More applications of the inequality (1) in the spirit of [5,6] will be given separately.