Research ContributionArnold Mathematical Journal

Received: 11 July 2016 / Revised: 25 October 2016 / Accepted: 17 November 2016

The $4n^2$-Inequality for Complete Intersection Singularities
Aleksandr V. Pukhlikov
###### Abstract
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 greater than $4n^2\mu$, where $\mu$ is the multiplicity of the singular point.
###### Keywords
Maximal singularity, Birational map, Linear system

## 1. Statement of the Result

Let $(X,o)$ be a germ of a complete intersection singularity of codimension $l$ and type $\underline{\mu}=(\mu_1,\dots,\mu_l)$, where \begin{eqnarray*} \text{dim} X=M\geqslant l+\mu_1+\dots+\mu_l+3. \end{eqnarray*} We will assume the singularity to be generic in the sense of Sect. 2 below. The aim of this note is to prove the following claim.

###### Theorem.
Let $\Sigma$ be a mobile linear system on X. Assume that for some positive $n\in{\mathbb Q}$ the pair $(X,\frac{1}{n}\Sigma)$ is not canonical at the point $o$ but canonical outside this point. Then the self-intersection $Z=(D_1\circ D_2)$ of the system $\Sigma$ satisfies the inequality

 $$\label{ccc} \mathop{\text{mult}}\nolimits_oZ>4n^2\mathop{\text{mult}}\nolimits_oX.$$ (1)
###### Remark 1.
1. (i) The assumption of the theorem means that the pair $(X,\frac{1}{n}\Sigma)$ 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 $a(R,X)$ is the discrepancy of $R$ with respect to $X$. point $o$, the Noether--Fano inequality \begin{eqnarray*} \mathop{\text{ord}}\nolimits_R\Sigma>n\cdot a(R,X) \end{eqnarray*} holds, where
2. (ii) The self-intersection $Z=(D_1\circ D_2)$ is the scheme-theoretic intersection of any two general divisors in $\Sigma$ which is well defined as $\Sigma$ is free from fixed components.
3. (iii) When $\mathop{\text{mult}_o}X=1$, we get the standard $4n^2$-inequality, see Pukhlikov ([Pukhlikov2013], 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 [Pukhlikov2000] on the basis of the technique developed in [Iskovskikh and Manin1971]. Later a different proof was found by [Corti2000] and various generalizations of the $4n^2$-inequality were investigated ([Cheltsov2005b], [Pukhlikov2010]), see Pukhlikov ([Pukhlikov2013], Chapter 2) for more details.

Note that in the smooth case (when $\mathop{\text{mult}}\nolimits_o X=1$) the $4n^2$- inequality holds for $\mathop{\text{dim}} X\geqslant 3$ 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 $\mathop{\text{dim}} X$ 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 \begin{eqnarray*} \begin{array}{ccccccc} 0 & = & q_{1,\mu_1} & + & q_{1,\mu_1+1} & + & \dots \\ & \dots & & & & & \\ 0 & = & q_{l,\mu_l} & + & q_{l,\mu_l+1} & + & \dots \\ \end{array} \end{eqnarray*} in ${\mathbb C}^{M+l}$, where $2\leqslant\mu_1\leqslant\dots\leqslant\mu_l$, $l\geqslant 1$ and the polynomials $q_{j,i}$ are homogeneous of degree $i$ in the coordinates $z_1,\dots,z_{M+l}$; the point $o=(0,\dots,0)$ is the origin. We denote by \begin{eqnarray*} \underline{\mu}=(\mu_1,\dots,\mu_l) \end{eqnarray*} the type of the singularity $o\in X$ and set \begin{eqnarray*} \mu=\mu_1\cdots\mu_l=\mathop{\text{mult}}\nolimits_o X \end{eqnarray*} to be the multiplicity of the point $o$ (assuming the conditions of general position for the first polynomials $q_{1,\mu_1},q_{2,\mu_2},\dots,q_{l,\mu_l}$, stated below). Set also \begin{eqnarray*} |\underline{\mu}|=\mu_1+\dots +\mu_l. \end{eqnarray*} Recall that by assumption $M\geqslant l+|\underline{\mu}|+3$. Let $P\ni o$ be a linear subspace in ${\mathbb C}^{M+l}$ of dimension $2l+|\underline{\mu}|+3$. Denote by $X_P$ the intersection $X\cap P$.

###### Definition 1.
We say that the complete intersection singularity $(X,o)$ is generic , if for a general subspace $P$ of dimension $2l+|\underline{\mu}|+3$ the singularity $o\in X_P$ is an isolated singularity, $\mathop{\text{dim}}X_P=l+|\underline{\mu}|+3$ and for the blow up \begin{eqnarray*} \varphi_P\colon X^+_P\to X_P \end{eqnarray*} of the point $o$, the variety $X^+_P$ is non-singular in neighborhood of the exceptional divisor $Q_P=\varphi^{-1}_P(o)$, which is a non-singular complete intersection \begin{eqnarray*} Q_P=\{q_{1,\mu_1}=q_{2,\mu_2}=\dots=q_{l,\mu_l}=0\}\subset{\mathbb P}^{2l+|\underline{\mu}|+2} \end{eqnarray*} of codimension $l$ and type $\underline{\mu}=(\mu_1,\dots,\mu_l)$.

From now on, we assume that the singularity $o\in X$ is generic. In particular, by Grothendieck's theorem on factoriality ([Call and Lyubeznik1994]), $X$ 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 Pukhlikov ([Pukhlikov2013], 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 $(X,\frac{1}{n}\Sigma)$ 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 Sect. 4 we resolve the singularity $E$ and use the assumptions on the singular point $o\in X$ to relate the multiplicities of the system $\Sigma$ and its self-intersection at the point $o$ with the multiplicities of the strict transforms of $\Sigma$ 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 Pukhlikov ([Pukhlikov2013], Chapter 2, Section 2.2) and complete the proof.

Let us realize this programme.

For a general $(2l+|\underline{\mu}|+3)$-subspace $P$ set $\Sigma_P=\Sigma|_P$ to be the restriction of $\Sigma$ onto $P$. By inversion of adjunction ([Shokurov1993], [Koll[U+00B4]ar et al.1993]), the pair $(X_P,\frac{1}{n}\Sigma_P)$ is not canonical (for $M>l+|\underline{\mu}|+3$, even non-log canonical, but we do not need that.) Obviously, \begin{eqnarray*} Z_P=Z|_P=(Z\circ X_P) \end{eqnarray*} is the self-intersection of the system $\Sigma_P$ and $\mathop{\text{ mult}_o}Z=\mathop{\text{mult}_o}Z_P$. Therefore, we may (and will) assume from the beginning that $M=l+|\underline{\mu}|+3$ and so $P={\mathbb C}^{M+l}$, so that already the original singularity $o\in X$ is isolated. Now we omit the index $P$ and write \begin{eqnarray*} \varphi\,\,\colon X^+\to X \end{eqnarray*} for the blow up of the point $o$ and $Q=\varphi^{-1}(o)$ for the exceptional divisor, which is a non-singular complete intersection of type $\underline{\mu}$ in ${\mathbb P}^{2l+|\underline{\mu}|+2}$.

Now let $\Pi\ni o$ be a general linear subspace of dimension $|\underline{\mu}|+3$. By the symbol $X_{\Pi}$ we denote the intersection $X\cap\Pi$. Clearly, $o\in X_{\Pi}\subset\Pi={\mathbb C}^{|\underline{\mu}|+3}$ is an isolated complete intersection singularity of codimension $l$. Let $\varphi_{\Pi}\,\,\colon X^+_{\Pi}\to X_{\Pi}$ be the blow up of the point $o$ and $Q_{\Pi}=\varphi^{-1}_{\Pi}(o)$ the exceptional divisor. Clearly $Q_{\Pi}\subset{\mathbb P}^{|\underline{\mu}|+2}$ is a non-singular complete intersection of type $\underline{\mu}$ (and codimension $l$).

Note that by the adjunction formula for the discrepancy we have the equality $a(Q_{\Pi},X_{\Pi})=2$.

For a general divisor $D\in\Sigma$ and its strict transform $D^+\in\Sigma^+$ on $X^+$ we have \begin{eqnarray*} D^+\sim -\nu\, Q \end{eqnarray*} for some positive integer $\nu$ (recall that we consider a local situation: $o\in X$ is a germ). Obviously, if $\nu>2n$, then \begin{eqnarray*} \mathop{\text{mult}}\nolimits_oZ\geqslant\nu^2\mu>4n^2\mu \end{eqnarray*} and the $4n^2$-inequality holds. For that reason, from now on we assume that \begin{eqnarray*} \nu\leqslant 2n. \end{eqnarray*} Setting $D_{\Pi}=D|_{X_\Pi}$, we get $D^+_{\Pi}\sim -\nu\, Q_{\Pi}$. By the inversion of adjunction the pair $\left(X_{\Pi},\frac{1}{n}D_{\Pi}\right)$ is not log canonical at the point $o$, the more so not canonical, so for some exceptional divisor $E_{\Pi}$ over $X_{\Pi}$ the Noether--Fano inequality \begin{eqnarray*} \mathop{\text{ord}}\nolimits_{E_{\Pi}}\Sigma_{\Pi}>n a(E_{\Pi},X_{\Pi}) \end{eqnarray*} is satisfied. As $\nu\leqslant 2n$ and $a(Q_{\Pi},X_{\Pi})=2$, we see that $E_{\Pi}\neq Q_{\Pi}$ and $E_{\Pi}$ is a non log canonical (and so not canonical) singularity of the pair \begin{eqnarray*} \left(X^+_{\Pi},\frac{1}{n}D^+_{\Pi}+\frac{(\nu-2n)}{n}Q_{\Pi}\right) \end{eqnarray*} (the more so, of the pair $\left(X^+_{\Pi},\frac{1}{n}D^+_{\Pi}\right))$. Denote by $\Delta_{\Pi}\subset Q_{\Pi}$ the centre of $E_{\Pi}$ on $X^+_{\Pi}$, an irreducible subvariety in $Q_{\Pi}$.

###### Proposition 1.
If $\mathop{\text{codim}}\,(\Delta_{\Pi}\subset Q_{\Pi})=1$, then the estimate \begin{eqnarray*} \mathop{\text{mult}}\nolimits_o Z\geqslant 8n^2\mu \end{eqnarray*} holds.

###### Proof.
We note that $\mathop{\text{mult}_o}Z=\mathop{\text{mult}}_oZ_{\Pi}$. Arguing as in the proof of Proposition 4.1 in Pukhlikov ([Pukhlikov2013], Chapter 2) [see also Cheltsov ([Cheltsov2005b], Section 1.7)], we get the following estimate: \begin{eqnarray*} \mathop{\text{mult}}\nolimits_oZ_{\Pi}\geqslant\nu^2\mu+ 4\left(3-\frac{\nu}{n}\right)n^2\mu, \end{eqnarray*} and easy calculations complete the proof. $\square$

Therefore we may assume that $\mathop{\text{codim}}(\Delta_{\Pi}\subset Q_{\Pi})\geqslant 2$.

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 \begin{eqnarray*} \mathop{\text{ord}}\nolimits_E\Sigma>n(2\mathop{\text{ ord}}\nolimits_EQ+a(E,X^+)) \end{eqnarray*} is satisfied. Moreover, the centre $\Delta\subset Q$ of $E$ on $X$ has codimension at least 2 and dimension at least $2l$.

## 4. Resolution of the Singularity $E$

Consider the sequence of blow ups \begin{eqnarray*} X_0=X\leftarrow X_1=X^+\leftarrow X_2\leftarrow\dots\leftarrow X_K, \end{eqnarray*} where $\varphi_{i,i-1}\colon X_i\to X_{i-1}$ is the blow up of the centre $B_{i-1}\subset X_{i-1}$ of the exceptional divisor $E$ on $X_{i-1}$. In particular, $B_0=o$ and $B_1=\Delta$. Using the notations, identical to those in Pukhlikov ([Pukhlikov2013], Chapter 2, Section 2.2), we set \begin{eqnarray*} E_i=\varphi^{-1}_{i,i-1}(B_{i-1})\subset X_i \end{eqnarray*} 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 Pukhlikov ([Pukhlikov2013], Chapter 2, Section 2.2) work automatically for the blow ups $\varphi_{i,i-1}$ with $i\geqslant 2$.

The last exceptional divisor $E_K$ defines the discrete valuation $\mathop{\text{ord}}_E$.

We divide the sequence $\varphi_{i,i-1}$, $i=1,\dots,K$, of blow ups into the lower part , $i=1,\dots,L\leqslant K$, corresponding to the centres $B_{i-1}$ of codimensions at least 3, and the upper part , $i=L+1,\dots,K$, 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: \begin{eqnarray*} \nu_i=\mathop{\text{mult}}\nolimits_{B_{i-1}}\Sigma^i \end{eqnarray*} for any $i=2,\dots,K$ to be the elementary multiplicities. Let $\Gamma$ 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 Pukhlikov ([Pukhlikov2013], Chapter 2, Section 2.2) for the standard details]. We also set $p_i=p_{Ki}$, $i=1,\dots,K$. Now the Noether--Fano type inequality takes the form

 $$\label{aaa} \sum^K_{i=1}p_i\nu_i>\left(2p_1+\sum^K_{i=2}p_i\delta_i\right),$$ (2)

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