Automorphisms of certain affine complements in the projective space

TL;DR

This paper characterizes automorphisms of certain affine complements in projective space, showing they extend to projective automorphisms under specific conditions on the hypersurface.

Contribution

It proves that automorphisms of affine complements of hypersurfaces with a unique singular point extend to projective automorphisms, revealing the structure of their automorphism groups.

Findings

Automorphisms of affine complements are restrictions of projective automorphisms.

For general hypersurfaces, the automorphism group of the complement is trivial.

The result applies to hypersurfaces with a specific singularity and degree conditions.

Abstract

We prove that every biregular automorphism of the affine algebraic variety ${\mathbb P}^M\setminus S$, $M\geqslant 3$, where $S\subset {\mathbb P}^M$ is a hypersurface of degree $m\geqslant M+1$ with a unique singular point of multiplicity $(m-1)$, resolved by one blow up, is a restriction of some automorphism of the projective space ${\mathbb P}^M$, preserving the hypersurface $S$; in particular, for a general hypersurface $S$ the group $\mathop{\rm Aut} ({\mathbb P}^M\setminus S)$ is trivial.

Equations

$$\mathrm{Aut}(\mathbb{P}\backslash S)$$

$$\chi :\mathbb{P}\backslash S\to \mathbb{P}\backslash S^{\prime }$$

$$f(z_{\ast })=q_{m-1}(z_{\ast })+q_m(z_{\ast })=0,$$

$$\chi :\mathbb{P}\backslash S\to \mathbb{P}\backslash S^{\prime }$$

$${\chi }_X\in \mathrm{Bir}X\backslash \mathrm{Aut}X,$$

$$\mathrm{Aut}(X\backslash Y)\text{mbox}and\mathrm{Aut}(X{)}_Y$$

$$\mathrm{Aut}(\mathbb{P}\backslash S)=\mathrm{Aut}(\mathbb{P}{)}_S$$

$$(\chi \circ \phi ){|}_E:E\to Y^{\prime }$$

$$Y=F_1\cap \cdots \cap F_k\subset {\mathbb{P}}^{M+k}$$

$$d_1+\cdots +d_k=M+k,$$

$$X_i=\bigcap _{j\ne i}{F}_j$$

$$\mathrm{Aut}(X_i\backslash Y)=\mathrm{Aut}(X_i{)}_Y$$

$$\chi :\mathbb{P}\backslash S\to \mathbb{P}\backslash S^{\prime }$$

$$\phi :\tilde{\mathbb{P}}\to \mathbb{P}$$

$$T=({\chi }_{\mathbb{P}}\circ \phi {)}_{\ast }^{-1}{S}^{\prime }\in {\mathcal{E}}_{\phi }.$$

$$b={\mathrm{ord}}_{T}S={\mathrm{ord}}_T{\phi }^{\ast }S.$$

$$\nu ={\mathrm{ord}}_T{\phi }^{\ast }\Sigma .$$

$$bn=\nu m+1,(M+1)b=am+(M+1).$$

$$\begin{array}[]{ccccccc}\widetilde{D}&\sim&nH&-&\nu T&-&\sum\limits_{E\in{\cal E}}\nu_{E}E,\\ &&&&&&\\ \widetilde{K}&=&-(M+1)H&+&aT&+&\sum\limits_{E\in{\cal E}}a_{E}E,\\ &&&&&&\\ \widetilde{S}&\sim&mH&-&bT&-&\sum\limits_{E\in{\cal E}}b_{E}E,\\ \end{array}$$

$$\bigcup _{E\in \mathcal{E}}\psi (E)\cup \psi (\tilde{S}),$$

$$(\tilde{L}\cdot \tilde{D})=1,(\tilde{L}\cdot \tilde{K})=-(M+1),(\tilde{L}\cdot \tilde{S})=0.$$

$$d=(\tilde{L}\cdot H).$$

$$dn-\nu m=1,$$

$$-(M+1)d+am=-(M+1),$$

$$dm-bm=0.$$

$$\nu =\frac{a}{M+1}n+\frac{n-1}{m}.$$

$$\nu >\frac{a}{M+1}n.$$

$$a_E=a(E,\mathbb{P})\text{mbox}{b}_E={\mathrm{ord}}_E{\phi }^{\ast }S,$$

$$\tilde{K}={\psi }^{\ast }{K}_{\mathbb{P}}+a^{\prime }\tilde{S}+\sum _{E\in \mathcal{E}}{a}_E^{\prime }E$$

$${\tilde{S}}^{\prime }=T\sim {\psi }^{\ast }(mH)-b^{\prime }\tilde{S}-\sum _{E\in \mathcal{E}}{b}_E^{\prime }E,$$

Full text

**Automorphisms of certain affine complements

in the projective space**

A.V.Pukhlikov

We prove that every biregular automorphism of the affine algebraic variety ${\mathbb{P}}^{M}\setminus S$, $M\geqslant 3$, where $S\subset{\mathbb{P}}^{M}$ is a hypersurface of degree $m\geqslant M+1$ with a unique singular point of multiplicity $(m-1)$, resolved by one blow up, is a restriction of some automorphism of the projective space ${\mathbb{P}}^{M}$, preserving the hypersurface $S$; in particular, for a general hypersurface $S$ the group $\mathop{\rm Aut}({\mathbb{P}}^{M}\setminus S)$ is trivial.

Bibliography: 24 titles.

Key words: affine complement, birational map, maximal singularity.

1. Statement of the main result. Let ${\mathbb{P}}={\mathbb{P}}^{M}$ be the complex projective space of dimension $M\geqslant 3$ and $S\subset{\mathbb{P}}$ a hypersurface of degree $m\geqslant M+1$ with a unique singular point $o\in S$ of multiplicity $m-1$, that can be resolved by one blow up. More precisely, let $\sigma\colon{\mathbb{P}}^{+}\to{\mathbb{P}}$ be the blow up of the point $o$ with the exceptional divisor $E=\sigma^{-1}(o)\cong{\mathbb{P}}^{M-1}$. We assume that the strict transform $S^{+}\subset{\mathbb{P}}^{+}$ is a non-singular hypersurface and the projectivised tangent cone $S^{+}\cap E$ is a non-singular hypersurface of degree $m-1$ in $E\cong{\mathbb{P}}^{M-1}$. The main result of the present paper is the following claim.

Theorem 1. Every automorphism $\chi$ of the affine algebraic set ${\mathbb{P}}\backslash S$ is the restriction of some projective automorphism $\chi_{{\mathbb{P}}}\in\mathop{\rm Aut}{\mathbb{P}}$, preserving the hypersurface $S$. In particular, the group

[TABLE]

is finite and trivial for a Zariski general hypersurface $S$.

Due to certain well known facts about automorphisms of projective hypersurfaces (see, for instance, [1]) Theorem 1 is easily implied (see Sec. 6) by a somewhat more general fact. Let $S^{\prime}\subset{\mathbb{P}}$ be one more hypersurface of degree $m$ with a unique singular point $o^{\prime}\in S^{\prime}$ of multiplicity $m-1$, that is resolved by one blow up (in the sense specified above). Then the following claim is true.

Theorem 2. Every isomorphism of affine algebraic varieties

[TABLE]

is the restriction of some projective automorphism $\chi_{{\mathbb{P}}}\in\mathop{\rm Aut}{\mathbb{P}}$, transforming the hypersurface $S$ into hypersurface $S^{\prime}$.

Obviously, $\chi_{\mathbb{P}}(o)=o^{\prime}$. It is Theorem 2 that we prove below.

If $z_{1},\dots,z_{M}$ is a system of affine coordinates on ${\mathbb{P}}$ with the origin at the point $o$, then the hypersurface $S$ is defined by the equation

[TABLE]

where $q_{i}(z_{*})$ are homogeneous polynomials of degree $i$ in the coordinates $z_{*}$. An irreducible hypersurface of that type is rational and it is this property that makes the problem of describing the group of automorphisms $\mathop{\rm Aut}({\mathbb{P}}\backslash S)$ meaningful, see the discussion in Subsection 2 below.

The paper is organized in the following way. In Sec. 2 we discuss the general problem of describing the automorphisms of affine complements and what little is known in that direction (for non-trivial cases), and also some well known conjectures and non-completed projects. In Sec. 3 we start the proof of Theorem 2: for an arbitrary isomorphism of affine varieties

[TABLE]

we define the key numerical characteristics (such as the “degree) and obtain the standard relations between them (for instance, an analog of the “Noether-Fano inequality” for the affine case). In Sec. 4 we construct the resolution of the maximal singularity of the map $\chi$, which is now considered a birational map (Cremona transformation) $\chi_{{\mathbb{P}}}\colon{\mathbb{P}}\dashrightarrow{\mathbb{P}}$, the restriction of which onto the affine complement ${\mathbb{P}}\backslash S$ is an isomorphism onto ${\mathbb{P}}\backslash S^{\prime}$. Finally, in Sec. 5 we exclude the maximal singularity, which completes the proof of Theorem 2.

The author thanks the referee for useful comments and suggestions.

2. Automorphisms of affine complements. Let $X$ be a non-singular projective rationally connected variety, $Y$ and $Y^{\prime}$ irreducible ample divisors, so that their complements $X\backslash Y$ and $X\backslash Y^{\prime}$ are affine varieties. Two natural questions can be asked:

1. are the affine varieties $X\backslash Y$ and $X\backslash Y^{\prime}$ isomorphic,

2. if $Y^{\prime}=Y$, what is the group of biregular automorphisms $\mathop{\rm Aut}(X\backslash Y)$.

It is natural to consider a biregular isomorphism $\chi\colon X\backslash Y\to X\backslash Y^{\prime}$ (if they exist) as a birational automorphism $\chi_{X}\in\mathop{\rm Bir}X$, regular on the affine open set $X\backslash Y$ and mapping it isomorphically onto $X\backslash Y^{\prime}$. The case when $\chi_{X}\in\mathop{\rm Aut}X$ is a biregular automorphism of the variety $X$ and the corresponding isomorphism $\chi$ of affine complements itself we will say to be trivial. We therefore consider the following problem: are there any non-trivial isomorphisms $\chi\colon X\backslash Y\to X\backslash Y^{\prime}$, when

[TABLE]

and, respectively, are the groups

[TABLE]

the same (the second symbol means the stabilizer of the divisor $Y$ in the group $\mathop{\rm Aut}(X)$). In particular, Theorem 1 claims that

[TABLE]

for hypersurfaces $S\subset{\mathbb{P}}$, described in Sec. 1.

Proposition 1. Let $\chi$ be a non-trivial isomorphism of affine complements $X\backslash Y$ and $X\backslash Y^{\prime}$. Then $Y$ and $Y^{\prime}$ are birationally ruled varieties, that is to say, for some irreducible varieties $Z$ and $Z^{\prime}$ of dimension $\mathop{\rm dim}X-2$ the varieties $Y$ and $Y^{\prime}$ are birational to the direct products $Z\times{\mathbb{P}}^{1}$ and $Z^{\prime}\times{\mathbb{P}}^{1}$, respectively.

Proof. The birational map $\chi^{-1}_{X}$ is regular at the generic point of the divisor $Y^{\prime}$, and its image can not be the generic point of the divisor $Y$: in such case $\chi_{X}$ would have been an isomorphism in codimension 1 and for that reason a biregular automorphism, contrary to our assumption. Therefore, $(\chi^{-1}_{X})_{*}Y^{\prime}\subset X$ is an irreducible subvariety of codimension at least 2 (which is, of course, contained in $Y$). Now let us consider a resolution of singularities $\varphi\colon\widetilde{X}\to X$ of the map $\chi_{X}$. By what we said above, there is an exceptional divisor $E\subset\widetilde{X}$ of this resolution, such that

[TABLE]

is a birational map. Therefore, $Y^{\prime}$ is a birationally ruled variety. For $Y$ we argue in a symmetric way. Q.E.D. for the proposition.

Remark 1. Assume in addition that $Y^{\prime}$ is a rationally connected variety. Then in the notations of the proof of Proposition 1 we conclude that the centre of the exceptional divisor $E$ on $X$, that is, the irreducible subvariety $\varphi(E)$, is a rationally connected variety. This is also true for the centres of the divisor $E$ on the “lower” stroreys of the resolution of $\varphi$.

Example 1. There are no non-trivial isomorphisms of affine complements ${\mathbb{P}}\backslash Y$ and ${\mathbb{P}}\backslash Y^{\prime}$, if $Y\subset{\mathbb{P}}$ is a non-singular hypersurface of degree at least $M+1$. Indeed, the hypersurface $Y$ is not a birationally ruled variety.

Assume now that $Y\subset X$ is a Fano variety. It is well known (see, for instance, [2, Chapter 2]), that birationally rigid Fano varieties are not birationally ruled. Therefore, if $Y$ is a birationally rigid variety, then every isomorphism of affine complements $X\backslash Y$ and $X\backslash Y^{\prime}$ (where $Y^{\prime}\subset X$ is an irreducible ample divisor) is trivial, that is, it extends to an automorphism of the variety $X$. In particular, the group $\mathop{\rm Aut}(X\backslash Y)$ is $\mathop{\rm Aut}(X)_{Y}$. This makes it possible to construct numerous examples of affine complements with no non-trivial isomorphisms and automorphisms. Below we give some of them.

Example 2. Let $V\subset{\mathbb{P}}^{4}$ be a smooth three-dimensional quartic. Because of its birational superrigidity ([3]) the affine complement ${\mathbb{P}}^{4}\backslash V$ has no non-trivial automorphisms and isomorphisms. The same is true for quartics with at most isolated double points, provided that the variety $V$ is factorial and its singularities are terminal, see [4, 5] and [6].

Example 3. Let $V\subset{\mathbb{P}}^{M}$ be a general smooth hypersurface of degree $M$, where $M\geqslant 5$. Because of its birational superrigidity ([7]) the affine complement ${\mathbb{P}}\backslash V$ has only trivial automorphisms and isomorphisms. The same is true if we allow $V$ to have quadratic singularities of rank at least 5 [8]. This example generalizes naturally for Fano complete intersections. Let $k\geqslant 2$,

[TABLE]

be a non-singular complete intersection of codimension $k$, where $F_{i}$ is a hypersurface of degree $d_{i}$, and

[TABLE]

$M\geqslant 4$, that is, $Y$ is a non-singular $M$-dimensional Fano variety of index 1. For $i\in\{1,\dots,k\}$ set

[TABLE]

and assume that $X_{i}$ is also non-singular. Then $X_{i}\subset{\mathbb{P}}^{M+k}$ is a $(M+1)$-dimensional Fano variety of index $d_{i}+1$, containing $Y$ as a very ample divisor, so that the complement $X_{i}\backslash Y$ is an affine variety. If the set of integers $(d_{1},\dots,d_{k})$ satisfies the conditions of any of the papers [9, 10, 11], and the variety $Y$ is sufficiently general in its family, then due to its birational superrigidity the equality

[TABLE]

holds (and a similar claim for automorphisms). Of course, these arguments are non-trivial only for those cases, when $\mathop{\rm Aut}X_{i}\neq\mathop{\rm Bir}X_{i}$: for instance, for $k=2$ and $(d_{1},d_{2})=(2,M)$ the variety $X_{2}$ is a $(M+1)$-dimensional quadric and its group of birational automorphisms is the Cremona group of rank $M+1$. We get another non-trivial example for $k=2$ and $(d_{1},d_{2})=(3,M-1)$, where $X_{2}$ is a $(M+1)$-dimensional cubic hypersurface which has a huge group of birational automorphisms. Using other families of birationally superrigid or rigid Fano varieties, one can construct more non-trivial examples of affine complements, all automorphisms of which are trivial.

Example 4. In [13] it is shown that a very general hypersurface $V_{d}\subset{\mathbb{P}}$ for $d\geqslant\frac{2}{3}M$ is not birationally ruled. Therefore, for such hypersurfaces their affine complements ${\mathbb{P}}\backslash V_{d}$ have no non-trivial isomorphisms and automorphisms.

Example 5. In [12] it was shown that a Zariski general hypersurface $V_{M-1}\subset{\mathbb{P}}$ for $M\geqslant 16$ has no other structures of a rationally connected fibre space apart from pencils of hyperplane sections. In particular, it has no structures of a conic bundle and for that reason is not birationally ruled. It follows that for for those hypersurfaces the affine complements ${\mathbb{P}}\backslash V_{M-1}$ have no non-trivial isomorphisms and automorphisms.

Unfortunately, if the variety $Y$ is birationally ruled, then the problem of describing the isomorphisms of the affine complement $X\backslash Y$ becomes very hard (except for trivial cases, when, for instance, the variety $X$ itself satisfies the equality $\mathop{\rm Bir}X=\mathop{\rm Aut}X)$. The only complete result here is Theorem 2 of the present paper. As for the main objects of study today, they are particular classes of three-dimensional affine complements, such as the complement ${\mathbb{P}}^{3}\backslash S$ to a cubic surface (non-singular or with prescribed singularities) or the affine space ${\mathbb{A}}^{3}$ and certain similar affine varieties. In respect of complements to cubic surfaces there is a classical conjecture, stated by M.Kh.Gizatullin in [14, p. 6]: if the cubic surface $S$ is non-singular, then its complement ${\mathbb{P}}^{3}\backslash S$ has no non-trivial automorphisms. However, if the cubic surface has a double point, then non-trivial automorphisms do exist — they were discovered by S.Lamy and J.Blanc (as far as the author knows, those examples were not published). A similar conjecture was stated by A.Dubulouz for the case when $X$ is a Fano double cover of index 2, branched over a surface $W\subset{\mathbb{P}}^{3}$ of degree 4 and $S$ is the inverse image of a plane in ${\mathbb{P}}^{3}$.

In respect of the groups of automorphisms of affine varieties a huge material has been accumulated; there are a lot of results about special groups of automorphisms, dynamical properties of particular automorphisms etc. We only point out three recent papers [15, 16, 17], see also the bibliography in those papers.

The groups of automorphisms of affine algebraic surfaces are much better understood: here we have such fundamental results as the complete description of the groups of automorphisms of the plane $\mathop{\rm Aut}{\mathbb{A}}^{2}$, see [18, 19]. This direction is still being actively explored [20, 21, 22, 23].

3. Start of the proof of Theorem 2. Let

[TABLE]

be an isomorphism of affine varieties. Assume that $\chi$ is non-trivial, that is, the corresponding birational map $\chi_{\mathbb{P}}\colon{\mathbb{P}}\dashrightarrow{\mathbb{P}}$ is not a biregular isomorphism. Let

[TABLE]

be its resolution (a sequence of blow ups with non-singular centres), so that $\psi=\chi_{\mathbb{P}}\circ\varphi\colon\widetilde{\mathbb{P}}\to{\mathbb{P}}$ is aregular map. Furthermore, set ${\cal E}_{\varphi}$ to be the set of prime $\varphi$-exceptional divisors. By assumption, for the strict transform of $S^{\prime}$ we have

[TABLE]

Ser $B=\varphi(T)$ to be the centre of the exceptional divisor $T$ on ${\mathbb{P}}$, an irreducible subvariety of codimension at least 2, and moreover $B\subset S$. Therefore we get the positive integers $a=a(T,{\mathbb{P}})$ (the discrepancy of the divisor $T$ with respect to ${\mathbb{P}}$) and

[TABLE]

Furthermore, let $\Sigma$ be the strict transform of the linear system of hyperplanes with respect to $\chi_{\mathbb{P}}$. This is a mobile linear system $\Sigma\subset|nH|$, where $H$ is a hyperplane in ${\mathbb{P}}$, and $n\geqslant 2$. Set

[TABLE]

Proposition 2. The following equalities are true:

[TABLE]

Proof. Write down ${\cal E}={\cal E}_{\varphi}\setminus\{T\}$, so that ${\cal E}_{\varphi}={\cal E}\coprod\{T\}$. Let $D\in\Sigma$ be a general divisor, $\widetilde{D}\in\widetilde{\Sigma}$ its strict transform on $\widetilde{\mathbb{P}}$, where $\widetilde{\Sigma}$ is the strict transform of the linear system $\Sigma$ on $\widetilde{\mathbb{P}}$ with respect to $\varphi$. Let $\widetilde{S}\subset\widetilde{\mathbb{P}}$ be the strict transform of the hypersurface $S$. By the symbol $\widetilde{K}$ we denote the canonical class of the variety $\widetilde{\mathbb{P}}$. We obtain the following presentations:

[TABLE]

where the coefficients $\nu_{E},a_{E},b_{E}$ have the obvious meaning (in order to simplify the formulas we write $H$ in stead of $\varphi^{*}H$). Consider the family of lines ${\cal L}$ on ${\mathbb{P}}$. Obviously, a general line $L\in{\cal L}$ does not meet the set

[TABLE]

since it is of codimension at least 2 (recall that $\widetilde{S}\subset\widetilde{\mathbb{P}}$ is a $\psi$-exceptional divisor). Therefore, the strict transform $\widetilde{L}\subset\widetilde{\mathbb{P}}$ satisfies the equalities

[TABLE]

Besides, $(\widetilde{L}\cdot T)=(L\cdot S^{\prime})=m$. Set

[TABLE]

Obviously, $d$ is the degree of the curve $\varphi(\widetilde{L})\subset\widetilde{\mathbb{P}}$ in the usual sense. Finally, we have the equality $(\widetilde{L}\cdot E)=0$ for every exceptional divisor $E\in{\cal E}$. Therefore the equalities (3) imply the relations

[TABLE]

[TABLE]

[TABLE]

The last equality implies that $d=b$. Now the equalities (2) follow in a straightforward way. Q.E.D. for the proposition.

Remark 2. The relations (2) imply the equality

[TABLE]

Since $n\geqslant 2$, we obtain the inequality

[TABLE]

This is the usual Noether-Fano inequality for the birational map $\chi_{\mathbb{P}}$. Therefore, the prime divisor $T$ (the strict transform of the hypersurface $S^{\prime}$ on $\widetilde{\mathbb{P}}$) is a maximal singularity of the linear system $\Sigma$ (see, for instance, Definition 1.4 in [2, Chapter 2]).

Although the relations (2) are sufficient for the proof of Theorem 2, we will show similar relations for every infinitely near divisor $E\in{\cal E}$. Recall that we defined the integers

[TABLE]

where the discrepancy is understood with respect to the birational morphism $\varphi$. Let $a^{\prime}_{E}$ be the discrepancy of the divisor $E$ with respect to the birational morphism $\psi$ and $b^{\prime}_{E}=\mathop{\rm ord}_{E}\psi^{*}S^{\prime}$, so that we get the equality

[TABLE]

and the presentation

[TABLE]

where $a^{\prime}>0$ and $b^{\prime}>0$ have the same sense in respect of the image of the map $\chi$ as $a$ and $b$ in respect of the original projective space ${\mathbb{P}}$.

Proposition 3. For every divisor $E\in{\cal E}$ the following equalities hold:

[TABLE]

and

[TABLE]

Proof. Consider a mobile family of curves ${\cal C}$ on ${\mathbb{P}}$ with the following properties:

1. every curve $C\in{\cal C}$ is an irreducible rational curve of degree $l\geqslant 2$,

2. the strict transform $\widetilde{C}$ of a general curve $C\in{\cal C}$ on $\widetilde{\mathbb{P}}$ with respect to the birational morphism $\psi$ meets $E$ transversally at a unique point $p_{C}$ of general position on $E$ and does not meet other $\psi$-exceptional divisors, in particular $\widetilde{C}\cap\widetilde{S}=\emptyset$,

3. the curves of the family ${\cal C}$ sweep out a Zariski open subset of the space ${\mathbb{P}}$.

Such a family of rational curves is easy to construct using the methods of elementary algebraic geometry, see [2, Chapter 2, Section 3]. Let $p\in E$ be a point of general position, $q=\psi(p)\in{\mathbb{P}}$ its image on ${\mathbb{P}}$ and $(v_{1},\dots,v_{M})$ a system of affine coordinates on ${\mathbb{P}}$ with the origian at that point. We construct the curve $C$ in the parametric form:

[TABLE]

where $l$ is sufficiently large. In [24], see also [2, Chapter 2, Theorem 3.1] it was shown that there is a set of coefficients $\alpha_{i,j},i=1,\dots,M$, $j=1,\dots,a^{\prime}_{E}$ (in fact, in stead of $a^{\prime}_{E}$ one can take an essentially smaller number, but we do not need that), such that for any coefficients $\alpha_{i,j}$ for $j\geqslant a^{\prime}_{E}+1$ the strict transform of such curve meets $E$ transversally at the point $p$ when $t=0$. Varying the coefficients $\alpha_{i,j}$ for $j\geqslant a^{\prime}_{E}+1$, one can ensure that the curve $C$ goes through the point $q$ only when $t=0$ and intersects the closed subset of codimension $\geqslant 2$

[TABLE]

only at the point $q$. Such curves satisfy the properties 1)-3).

Now we argue in exactly the same way as in the proof of Proposition 2. We have the equality

[TABLE]

where $d=\mathop{\rm deg}\varphi(\widetilde{C})$. Multiplying $\widetilde{C}$ by the canonical class $\widetilde{K}$ and using the presentation (4), we obtain the equality

[TABLE]

Finally, multiplying the curve $\widetilde{C}$ by $\widetilde{S}$, we get:

[TABLE]

(the expression in brackets $lm-b^{\prime}_{E}$ is the “residual intersection” $(\widetilde{C}\cdot T$)). From here, using the equalities (2), by means of easy computations we get the equalities (5) and (6). Q.E.D. for the proposition.

4. The resolution of the maximal singularity. Let

[TABLE]

$i=1,\dots,N$, be the resolution of the maximal singularity $T$ of the linear system $\Sigma$ in the sense of [2, Chapter 2], that is, $X_{0}={\mathbb{P}}$, each map $\varphi_{i,i-1}$ is the blow up of the (possibly singular) irreducible subvariety $B_{i-1}\subset X_{i-1}$, which is the centre of the exceptional divisor $T$ on $X_{i-1}$. Set $E_{i}=\varphi^{-1}_{i,i-1}(B_{i-1})$. The strict transform of the subvariety $R\subset X_{i}$ on a higher storey of the resolution $X_{j}$, where $j>i$, is denoted by adding the upper index $j$: we write $R^{j}$.

For $j>i$ we set

[TABLE]

The exceptional divisor $E_{N}\subset X_{N}$ of the last blow up realizes the maximal singularity $T$: the birational map

[TABLE]

is regular at the general point of the divisor $E_{N}$ and maps it onto $T$.

On the set $\{1,\dots,N\}$ there is a natural structure of an oriented graph: $i\to j$, if and only if $i>j$ and the inclusion

[TABLE]

holds. If the vertices $i$ and $j$ are not joined by an orinted edge, we write $i\nrightarrow j$.

For $i\neq j$ we denote by the symbol $p_{ij}$ the number of paths in that graph from the vertex $i$ to the vertex $j$ (so that $p_{ij}=0$ for $i<j$ and $p_{ij}\geqslant 1$ for $i>j$). For convenience we set $p_{ii}=1$ for $i=1,\dots,N$. Finally, in order to simplify our notations, we write $p_{i}$ in stead of $p_{Ni}$. Let

[TABLE]

be the elementary discrepancies. Then the following equality holds:

[TABLE]

Let us also introduce the elementary multiplicities

[TABLE]

$i=1,\dots,N$ (where, in accordance with the general principle of notations, $\Sigma^{i-1}$ means the strict transform of the mobile linear system $\Sigma$ on $X_{i-1}$) and

[TABLE]

$i=1,\dots,N$. Obviously,

[TABLE]

Note that for some $k\leqslant N$ the strict transform $S^{k-1}$ contains $B_{k-1}$, but $S^{k}$ no longer contains $B_{k}$, so that $\mu_{k+1}=\dots=\mu_{N}=0$, and for that reason

[TABLE]

If $B_{0}\neq o$ is not the unique singular point of the hypersurface $S$, then obviously

[TABLE]

so that $b=p_{1}+\dots+p_{k}$. If $B_{0}=o$, then $\mu_{1}=m-1$ and by the assumption about the singularities of the divisor $S$ the strict transform $S^{1}$ is smooth, so that $\mu_{i}=1$ for $2\leqslant i\leqslant k$. Therefore if $B_{0}=o$, then

[TABLE]

Finally, let us point out one property of the numbers $p_{i}$. Since by construction we have $\varphi_{i,i-1}(B_{i})=B_{i-1}$ ($B_{i}$ is the centre of the exceptional divisor $T$ on $X_{i}$, and $B_{i-1}$ is its centre on $X_{i-1}$), the dimensions $\mathop{\rm dim}B_{i}$ do not decrease when $i$ is growing. Accordingly, the codimensions $\mathop{\rm codim}B_{i}$ do not increase when $i$ is growing, so that $\delta_{1}\geqslant\delta_{2}\geqslant\dots\geqslant\delta_{N}$. Assume that for some $k_{1}<k$ the centres of the blow ups $B_{i-1}$, $i=k_{1}+1,\dots,k$, have the maximal dimension $M-2$.

Proposition 4. Under the assumptions above for $k-k_{1}\geqslant 3$ the subgraph with the vertices $k_{1}+1,\dots,k$ is a chain:

[TABLE]

that is, between the vertices of the subgraph there are no other arrows apart from the consecutive ones $i\leftarrow i+1$. Moreover,

[TABLE]

Proof. By the definition of the number $k$, for $i\leqslant k$ we have $B_{i-1}\subset S^{i-1}$, where the divisor $S^{i-1}$ is non-singular at the general point $B_{i-1}$ for $i\geqslant k_{1}+1$. Therefore, for $k_{1}+1\leqslant i\leqslant k-2$ we have

[TABLE]

(since $B_{i}$ is contained in both $E_{i}$ and $S^{i}$ and has codimension 2, and the same is true for $B_{i+1}$), and $E_{i}$ and $S^{i}$ (respectively, $E_{i+1}$ and $S^{i+1}$) meet transversally at the general point of $B_{i}$ (respectively, of $B_{i+1}$), so that $E^{i+1}_{i}$ and $S^{i+1}$ do not meet over a point of general position in $B_{i}$. Therefore, $B_{i+1}\not\subset E^{i+1}_{i}$ and the first claim of the proposition is shown.

In particular, $k\nrightarrow k-2$. But then for any vertex $e\geqslant k+1$ we have $e\nrightarrow k-2$, either, so that every path from the vertex $N$ to the vertex $i\leqslant k-2$ must go through the vertex $k-1$. This proves the second claim of Proposition 4. Q.E.D.

Now we are ready to complete the proof of Theorem 2.

5. Exclusion of the maximal singularity. Let us write down the second of the equalities (2) in terms of elementary multiplicities and discrepancies:

[TABLE]

We conclude immediately that $B_{0}=o$ is the singular point of the hypersurface $S$. Otherwise all multiplicities $\mu_{i}=1$, so that from the formula (7) we would have obtained

[TABLE]

which is impossible, since $m\geqslant M+1$ and $\delta_{i}\geqslant 1$, so that all three components in the right hand side of the last formula are non-negative and at least one of them is positive.

So $B_{0}=o$. Here $\mu_{1}=m-1$ and $\delta_{1}=M-1$, so we get the equality

[TABLE]

all components in which both in the right and left hand side are non-negative. By Remark 1, all centres $B_{i}$ of the blow ups $\varphi_{i+1,i}$ are rationally connected varieties. In particular, $B_{1}\neq S^{1}\cap E_{1}$, since $S^{1}\cap E_{1}\subset E_{1}\cong{\mathbb{P}}^{M-1}$ is a non-singular hypersurface of degree $m-1\geqslant M$, which is not rationally connected. Thus if $k\geqslant 2$, then $B_{1}$ is a subvariety of codimension at least 3 in $X_{1}$, so that $\delta_{2}\geqslant 2$ and the coefficient at $p_{2}$ is not smaller than $2m-M-1$. If also $3\nrightarrow 1$, then $p_{1}=p_{2}$ (every path from the vertex $N$ to the vertex 1 must go through the vertex 2) and we ob tain a contradiction: in the equality (8) the right hand side is strictly higher than the left hand side. If $N=2$, then $p_{1}=p_{2}=1$ and we obtain a contradiction again: in this case the equality (8) takes the form

[TABLE]

with $\delta_{2}\geqslant 2$, which is also impossible. We conclude that for $k\geqslant 2$ with necessity $N\geqslant 3$ and $3\to 1$.

Proposition 5. The case $k=1$ is impossible.

Proof. Assume the converse: $k=1$. Then $b=(m-1)p_{1}$. Let $Q\subset{\mathbb{P}}$ be a general hypersurface of degree $m$ with the point $o$ as a singular point of multiplicity $m-1$. Since it is general, $B_{1}\not\subset Q^{1}$, so that

[TABLE]

and it follows that for the strict transform $\widetilde{Q}\subset\widetilde{\mathbb{P}}$ we get the presentation

[TABLE]

Therefore, $(\widetilde{L}\cdot\widetilde{Q})=0$, where $\widetilde{L}$ is the strict $\psi$-transform of a general line $L\subset{\mathbb{P}}$ (see the proof of Proposition 2). But the curves $\widetilde{L}$ sweep out a Zariski open subset of the space ${\mathbb{P}}$, and the hypersurfaces $Q$ sweep out $\mathbb{P}$. This contradiction proves Proposition 5.

Set $l=\mathop{\rm max}\{i\,|\,i\to 1,1\leqslant i\leqslant N\}$.

Proposition 6. The case $l\leqslant k$ is impossible.

Proof. Assume the converse: $l\leqslant k$. We could see above that $N\geqslant 3$ and $3\to 1$, so that $l\geqslant 3$. For any $i\leqslant l$, $i\geqslant 2$, we have

[TABLE]

so that $\mathop{\rm codim}B_{i-1}\geqslant 3$ and $\delta_{i}\geqslant 2$. Let us re-write the right hand side of the equality (8) in the form

[TABLE]

The first component in this sum is not smaller than

[TABLE]

For that reason the equality (8) is impossible. Q.E.D. for the proposition.

The last step in the proof of Theorem 2 is the following

Proposition 7. The case $l>k$ is impossible.

Proof. Assume the converse: $l>k$. As in the proof of the previous proposition, for any $i\leqslant k,i\geqslant 2$ we have

[TABLE]

Set $\Delta=S^{1}\cap E_{1}$. Consider the hypersurface $Q\subset{\mathbb{P}}$, containing the point $o$, which in the affine coordinates $z_{1},\dots,z_{M}$ is defined by the equation

[TABLE]

where $q_{m-1}(z_{*})$ is the same polynomial as in the equation (1) of the hypersurface $S$, and $\widetilde{q}_{m}(z_{*})$ is a generic homogeneous polynomial of degree $m$. Obviously, $Q^{1}\cap E_{1}=\Delta$, the hypersurface $Q^{1}\subset X_{1}$ is non-singular and the intersection of $Q^{1}$ with $E_{1}$ is everywhere transversal. Therefore for every $i\leqslant k$, $i\geqslant 2$, we have

[TABLE]

On the other hand, by the definition of the number $k$ we have $B_{k}\not\subset S^{k}$, so that, because of the polynomial $\widetilde{q}_{m}$ being general, we have $B_{k}\not\subset Q^{k}$. Thus

[TABLE]

Now we argue ia the word for word the same way as in the proof of Proposition 5: for the strict transform $\widetilde{Q}\subset\widetilde{\mathbb{P}}$ we get the presentation (9), which immediately implies that $(\widetilde{L}\cdot\widetilde{Q})=0$ for a general line $L\subset{\mathbb{P}}$, which is impossible since the polynomial $\widetilde{q}_{m}$ is a general one. Q.E.D. for the proposition.

Proof of Theorem 2 is complete.

6. Automorphisms of the hypersurface $S$. Let us prove Theorem 1. By Theorem 2 we need to show only the claim that the group $\mathop{\rm Aut}({\mathbb{P}}\setminus S)=\mathop{\rm Aut}({\mathbb{P}})_{S}$ is finite, generically trivial. First of all, every projective automorphism $\chi_{\mathbb{P}}$, preserving the hypersurface $S$, maps the point $o$ to itself. Let $\mathop{\rm Aut}({\mathbb{P}})_{o}\subset\mathop{\rm Aut}({\mathbb{P}})$ be the stabilizer of the point $o$, and

[TABLE]

the natural projection, sending a projective automorphism $\xi\in\mathop{\rm Aut}({\mathbb{P}})_{o}$ to the corresponding automorphism of the projectivized tangent space ${\mathbb{P}}(T_{o}{\mathbb{P}})\cong E$. Obviously, for every $\chi_{\mathbb{P}}\in\mathop{\rm Aut}({\mathbb{P}})_{S}$ its image $\pi(\chi_{\mathbb{P}})$ preserves the hypersurface $S^{+}\cap E$ (that is, the hypersurface $\{q_{m-1}=0\}$ in the sense of the equation (1)). By [1], the group $\pi(\mathop{\rm Aut}({\mathbb{P}})_{S})$ is finite, and for a Zariski general hypersurface $S$, trivial. Setting

[TABLE]

we see that it is sufficient to show that the kernel $\mathop{\rm Ker}\pi_{S}$ is trivial. This is really easy.

Every projective automorphism $\xi\in\mathop{\rm Ker}\pi$ in a system of homogeneous coordinates $(x_{0}:x_{1}:\dots:x_{M})$, such that $o=(1:0:\dots:0)$, has the form

[TABLE]

where $a_{0}\neq 0$. The hypersurface $S$ in such a system of coordinates is given by the equation $\Phi(x_{*})=0$, where

[TABLE]

see the equation (1). If $\xi\in\mathop{\rm Ker}\pi_{S}$, then the homogeneous polynomial

[TABLE]

is proportional to $\Phi(x_{*})$. It is easy to see that this is possible in one case only, when $a_{0}=1$ and $a_{1}=\dots=a_{M}=0$, that is, $\xi=\mathop{\rm id}\nolimits_{\mathbb{P}}$. This completes the proof of Theorem 1.