The $4n^2$-Inequality for Complete Intersection Singularities

To Askol’d Georgievich Khovanskii

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.

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

$$\begin{aligned} \text {dim} X=M\geqslant l+\mu _1+\dots +\mu _l+3. \end{aligned}$$

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

$$\begin{aligned} \mathop {\text {mult}}\nolimits _oZ>4n^2\mathop {\text {mult}}\nolimits _oX. \end{aligned}$$

(1)

Remark 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 point o, the Noether–Fano inequality

  $$\begin{aligned} \mathop {\text {ord}}\nolimits _R\Sigma >n\cdot a(R,X) \end{aligned}$$

  holds, where a(R, X) is the discrepancy of R with respect to X.

• (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.

• (iii) When $\mathop {\text {mult}_o}X=1$, we get the standard $4n^2$-inequality, see Pukhlikov (2013, 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 Pukhlikov (2000) on the basis of the technique developed in Iskovskikh and Manin (1971). Later a different proof was found by Corti (2000) and various generalizations of the $4n^2$-inequality were investigated (Cheltsov 2005b; Pukhlikov 2010), see Pukhlikov (2013, 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

$$\begin{aligned} \begin{array}{ccccccc} 0 &{} = &{} q_{1,\mu _1} &{} + &{} q_{1,\mu _1+1} &{} + &{} \dots \\ &{} \dots &{} &{} &{} &{} &{} \\ 0 &{} = &{} q_{l,\mu _l} &{} + &{} q_{l,\mu _l+1} &{} + &{} \dots \\ \end{array} \end{aligned}$$

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

$$\begin{aligned} \underline{\mu }=(\mu _1,\dots ,\mu _l) \end{aligned}$$

the type of the singularity $o\in X$ and set

$$\begin{aligned} \mu =\mu _1\cdots \mu _l=\mathop {\text {mult}}\nolimits _o X \end{aligned}$$

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

$$\begin{aligned} |\underline{\mu }|=\mu _1+\dots +\mu _l. \end{aligned}$$

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

$$\begin{aligned} \varphi _P:X^+_P\rightarrow X_P \end{aligned}$$

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

$$\begin{aligned} Q_P=\{q_{1,\mu _1}=q_{2,\mu _2}=\dots =q_{l,\mu _l}=0\}\subset {\mathbb P}^{2l+|\underline{\mu }|+2} \end{aligned}$$

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 Lyubeznik 1994), 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 (2013, 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 (2013, 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 (Shokurov 1993; Kollár 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,

$$\begin{aligned} Z_P=Z|_P=(Z\circ X_P) \end{aligned}$$

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

$$\begin{aligned} \varphi \,\,:X^+\rightarrow X \end{aligned}$$

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 $\varphi _{\Pi }\,\,:X^+_{\Pi }\rightarrow X_{\Pi }$ 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

$$\begin{aligned} D^+\sim -\nu \, Q \end{aligned}$$

for some positive integer $\nu$ (recall that we consider a local situation: $o\in X$ is a germ). Obviously, if $\nu >2n$, then

$$\begin{aligned} \mathop {\text {mult}}\nolimits _oZ\geqslant \nu ^2\mu >4n^2\mu \end{aligned}$$

and the $4n^2$-inequality holds. For that reason, from now on we assume that

$$\begin{aligned} \nu \leqslant 2n. \end{aligned}$$

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

$$\begin{aligned} \mathop {\text {ord}}\nolimits _{E_{\Pi }}\Sigma _{\Pi }>n a(E_{\Pi },X_{\Pi }) \end{aligned}$$

is satisfied. As $\nu \leqslant 2n$ and $a(Q_{\Pi },X_{\Pi })=2$, we see that $E_{\Pi }\ne Q_{\Pi }$ and $E_{\Pi }$ is a non log canonical (and so not canonical) singularity of the pair

$$\begin{aligned} \left( X^+_{\Pi },\frac{1}{n}D^+_{\Pi }+\frac{(\nu -2n)}{n}Q_{\Pi }\right) \end{aligned}$$

(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

$$\begin{aligned} \mathop {\text {mult}}\nolimits _o Z\geqslant 8n^2\mu \end{aligned}$$

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 (2013, Chapter 2) [see also Cheltsov (2005b, Section 1.7)], we get the following estimate:

$$\begin{aligned} \mathop {\text {mult}}\nolimits _oZ_{\Pi }\geqslant \nu ^2\mu + 4\left( 3-\frac{\nu }{n}\right) n^2\mu , \end{aligned}$$

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

$$\begin{aligned} \mathop {\text {ord}}\nolimits _E\Sigma >n(2\mathop {\text { ord}}\nolimits _EQ+a(E,X^+)) \end{aligned}$$

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

$$\begin{aligned} X_0=X\leftarrow X_1=X^+\leftarrow X_2\leftarrow \dots \leftarrow X_K, \end{aligned}$$

where $\varphi _{i,i-1}:X_i\rightarrow X_{i-1}$ 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 (2013, Chapter 2, Section 2.2), we set

$$\begin{aligned} E_i=\varphi ^{-1}_{i,i-1}(B_{i-1})\subset X_i \end{aligned}$$

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 (2013, 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:

$$\begin{aligned} \nu _i=\mathop {\text {mult}}\nolimits _{B_{i-1}}\Sigma ^i \end{aligned}$$

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 (2013, 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

$$\begin{aligned} \sum ^K_{i=1}p_i\nu _i>\left( 2p_1+\sum ^K_{i=2}p_i\delta _i\right) , \end{aligned}$$

(2)

where $\nu _1=\nu$ and $\delta _i=\mathop {\text {codim}}(B_{i-1}\subset X_{i-1})$ 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 $\nu _K>n$ (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 $Y\subset {\mathbb P}^N$ be a non-singular complete intersection of codimension $l\geqslant 1$, $S\subset Y$ an irreducible subvariety of codimension $a\geqslant 1$ and $B\subset Y$ an irreducible subvariety of dimension al, where the estimate $N\geqslant (l+1)(a+1)$ is satisfied. Then the inequality

$$\begin{aligned} \mathop {\text {mult}}\nolimits _BS\leqslant m \end{aligned}$$

holds, where $m\geqslant 1$ is defined by the condition $S\sim mH^a_Y$ and $H_Y\in A^1Y$ is the class of a hyperplane section of Y.

Proof

for the case $l=1$ was given in Pukhlikov (2002b). The argument extends directly to the general case of an arbitrary l, see Suzuki (2015) (also Pukhlikov 2006; Cheltsov 2005a). $\square$

Applying Proposition 2 to a divisor in the linear system $\Sigma ^1|_Q$, we conclude that $\nu _1\geqslant \nu _2$, since $\mathop {\text {dim}}B_1=\mathop {\text {dim}}\Delta \geqslant 2l$. The inequalities

$$\begin{aligned} \nu _2\geqslant \nu _3\geqslant \dots \geqslant \nu _K \end{aligned}$$

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 $\Gamma$ is a chain:

$$\begin{aligned} L\leftarrow (L+1)\leftarrow \dots \leftarrow K; \end{aligned}$$

moreover, there are no arrows connecting either of the vertices $L+1,\dots ,K$ with any of vertices $1,2,\dots ,L-1$. [These are the standard consequences of inequalities $\nu _K>n$ and $\nu _1\leqslant 2n$, see Pukhlikov (2013, 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 $D_1,D_2\in \Sigma$ and set

$$\begin{aligned} Z=Z_0=(D_1\circ D_2) \end{aligned}$$

to be their scheme-theoretic intersection, the self-intersection of the mobile linear system $\Sigma$. 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 $X_j$, $j\leqslant i$, of the resolution and that the operation of taking the strict transform is well defined for that particular subvariety etc.) For $i\geqslant 1$ write

$$\begin{aligned} (D^i_1\circ D^i_2)=(D^{i-1}_1\circ D^{i-1}_2)^i+Z_i, \end{aligned}$$

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 $i\leqslant L$ we obtain the presentation

$$\begin{aligned} (D^i_1\circ D^i_2)=Z^i_0+Z^i_1+\dots +Z^i_{i-1}+Z_i. \end{aligned}$$

For any $j>i$, where $j\leqslant L$, set

$$\begin{aligned} m_{i,j}=\mathop {\text {mult}}\nolimits _{B_{j-1}}Z^{j-1}_i \end{aligned}$$

and for $i=2,\dots ,L$ set $d_i=\mathop {\text {deg}}Z_i$ in the same sense as in Pukhlikov (2013, Chapter 2, Section 2.2). For the effective divisor $Z_1$ on $E_1=Q$ we have the relation

$$\begin{aligned} Z_1\sim d_1H_Q \end{aligned}$$

for some $d_1\in {\mathbb Z}_+$, where $H_Q$ is the class of a hyperplane section of the complete intersection $Q\subset {\mathbb P}^{4l+2}$. Following the procedure of Pukhlikov (2013, Chapter 2), we obtain the system of equalities

$$\begin{aligned} \begin{array}{rccccccc} \mu \!\!&{} (\nu _1^2+d_1) &{} = &{} m_{0,1}, &{} &{} &{} &{} \\ &{} \nu _2^2+d_2 &{} = &{} m_{0,2} &{} + &{} m_{1,2}, &{} &{} \\ &{} &{} \dots &{} &{} &{} &{} &{} \\ &{} \nu _i^2+d_i &{} = &{} m_{0,i} &{} + &{} \dots &{} + &{} m_{i-1,i}, \\ &{} &{} \dots &{} &{} &{} &{} &{} \end{array} \end{aligned}$$

$i=2,\dots ,L$, where the estimate

$$\begin{aligned} d_L\geqslant \sum ^K_{i=L+1}\nu ^2_i \end{aligned}$$

holds as usual, see Pukhlikov (2013, p. 53).

Proposition 3

• (i) The inequality

  $$\begin{aligned} d_1\geqslant m_{1,2} \end{aligned}$$

  holds.

• (ii) The inequality

  $$\begin{aligned} m_{0,1}\geqslant \mu m_{0,2} \end{aligned}$$

  holds.

Proof

Part (i) follows from Proposition 2 as $Z_1\sim d_1 H_Q$ and $\mathop {\text {dim}}B_1\geqslant 2l$. In order to show part (ii), we note that (numerically)

$$\begin{aligned} (Z^1\circ E_1)\sim \frac{1}{\mu }m_{0,1}H^2_Q \end{aligned}$$

as $m_{0,1}=\mathop {\text {deg}}(Z^1\circ E_1)$, the cycle $(Z^1\circ E_1)=(Z^1\circ Q)$ being of pure codimension 2 on Q. Applying Proposition 2 to the cycle $(Z^1\circ Q)$, we get the inequality

$$\begin{aligned} m_{0,2}\leqslant \mathop {\text {mult}}\nolimits _{\Delta }(Z^1\circ Q)\leqslant \frac{1}{\mu }m_{0,1}, \end{aligned}$$

which completes the proof of the proposition. $\square$

The more so, $m_{0,1}\geqslant \mu m_{0,i}$ for $i\geqslant 3$ as $m_{0,2}\geqslant m_{0,3}\geqslant \dots \geqslant m_{0,L}$.

Now set

$$\begin{aligned} m^*_{i,j}=\mu m_{i,j} \end{aligned}$$

for $(i,j)\ne (0,1)$ and $m^*_{0,1}=m_{0,1}$. Also set

$$\begin{aligned} d^*_i=\mu d_i \end{aligned}$$

for $i=1,\dots ,L$. We obtain the following system of equalities:

$$\begin{aligned} \begin{array}{ccccccc} \mu \nu _1^2+d^*_1 &{} = &{} m^*_{0,1}, &{} &{} &{} &{} \\ \mu \nu _2^2+d^*_2 &{} = &{} m^*_{0,2} &{} + &{} m^*_{1,2}, &{} &{} \\ &{} \dots &{} &{} &{} &{} &{} \\ \mu \nu _i^2+d^*_i &{} = &{} m^*_{0,i} &{} + &{} \dots &{} + &{} m^*_{i-1,i}, \\ &{} \dots &{} &{} &{} &{} &{} \end{array} \end{aligned}$$

where $i=1,\dots ,L$, and

$$\begin{aligned} d^*_L\geqslant \mu \sum ^K_{i=L+1}\nu ^2_i, \end{aligned}$$

where the integers $m^*_{i,j}$ and $d^*_i$ satisfy precisely the same properties, as the integers $m_{i,j}$ and $d_i$ in the non-singular case considered in Pukhlikov (2013, Chapter 2, p. 52–53). Now repeating the arguments of Pukhlikov (2013, Chapter 2, p. 52–53) word for word, we obtain the inequality

$$\begin{aligned} \left( \sum ^L_{i=1}p_i\right) \mathop {\text { mult}}\nolimits _oZ\geqslant \mu \sum ^K_{i=1}p_i\nu ^2_i, \end{aligned}$$

which in the standard way implies the desired estimate

$$\begin{aligned} \mathop {\text {mult}}\nolimits _oZ>\mu \cdot 4n^2. \end{aligned}$$

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 Pukhlikov (2002a). The cases of singular points of multiplicity $\mu =3$ and 4 in that paper are really hard. The inequality (1) gives for the multiplicity $\mathop {\text {mult}_o}Z$ 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 Eckl and Pukhlikov (2014), Evans and Pukhlikov (2016) will be given separately.