On algebraic and non-algebraic neighborhoods of rational curves

Serge Lvovski National Research University Higher School of Economics, Moscow, Russia Federal State Institution ”Scientific-Research Institute for System Analysis of the Russian Academy of Sciences” (SRISA) [email protected]

Abstract.

We prove that for any $d>0$ there exists an embedding of the Riemann sphere $\mathbb{P}^{1}$ in a smooth complex surface, with self-intersection $d$ , such that the germ of this embedding cannot be extended to an embedding in an algebraic surface but the field of germs of meromorphic functions along $C$ has transcendence degree $2$ over $\mathbb{C}$ . We give two different constructions of such neighborhoods, either as blowdowns of a neighborhood of the smooth plane conic, or as ramified coverings of a neighborhood of a hyperplane section of a surface of minimal degree.

The proofs of non-algebraicity of these neighborhoods are based on a classification, up to isomorphism, of algebraic germs of embeddings of $\mathbb{P}^{1}$ , which is also obtained in the paper.

Key words and phrases:

Neighborhoods of rational curves, surfaces of minimal degree, blowup

1991 Mathematics Subject Classification:

32H99, 14J26

This study was partially supported by the HSE University Basic Research Program and by SRISA research project FNEF-2022-0007 (Reg. No 1021060909180-7-1.2.1).

1. Introduction

In this paper we study germs of embeddings of the Riemann sphere aka $\mathbb{P}^{1}$ in smooth complex surfaces (see precise definitions in Section 1.1 below). The structure of such germs for which the degree of the normal bundle, which is equal to the self-intersection index of the curve is question, is non-positive, is well known and simple: if $(C\cdot C)=d\leq 0$ , then for any such $d$ there exists only one germ up to isomorphism, and these germs are algebraic (see [6] for the negative degree case and [11] for the zero degree case). Germs of embeddings $(C,U)$ , where $C\cong\mathbb{P}^{1}$ , $U$ is a smooth complex surface, and $(C\cdot C)>0$ , are way more diverse.

Let us say that a germ of neighborhood of $C\cong\mathbb{P}^{1}$ is algebraic if it is isomorphic to the germ of embedding of $C$ in a smooth algebraic surface. M. Mishustin, in his paper [9], showed that the space of isomorphism classes of germs of embeddings of $C\cong\mathbb{P}^{1}$ , $(C\cdot C)>0$ , in smooth surfaces, is infinite-dimensional, so one would expect that “most” germs of such embeddings are not algebraic. In this paper we will construct explicit examples of non-algebraic germs of embeddings of $\mathbb{P}^{1}$ .

An interesting series of such examples was constructed in a recent paper by M. Falla Luza and F. Loray [4]. For each $d>0$ they construct an embedding of $C\cong\mathbb{P}^{1}$ is a smooth surface such that $(C\cdot C)=d$ and the field of germs of meromorphic functions along $C$ consists only of constants. For curves on an algebraic surface this is impossible, so the germs of neighborhoods constructed in [4] are examples of non-algebraic germs.

In Section 5.3 of the paper [5] the same authors give an example of a non-algebraic germ of neighborhood of $\mathbb{P}^{1}$ , with self-intersection $1$ , for which the field of germs of meromorphic functions is as big as possible: it has transcendence degree $2$ over $\mathbb{C}$ .

The aim of this paper is to construct two series of explicit examples of non-algebraic germs of neighborhoods of $\mathbb{P}^{1}$ for which the field of germs of meromorphic functions is as big as possible.

The first of these series is constructed in Section 4. Specifically, for any integer $m\geq 5$ we construct a non-algebraic germ of neighborhood $(C,U)$ such that $C\cong\mathbb{P}^{1}$ , $(C\cdot C)=m$ , and one can blow up $m-4$ points on $C$ to obtain the germ of neighborhood of the conic in the plane. The field of germs of meromorphic functions along $C$ has transcendence degree $2$ over $\mathbb{C}$ (Construction 4.10 and Proposition 4.11).

The second series of examples is constructed in Section 5. For any positive integer $n$ we construct a non-algebraic germ of neighborhood $(C,U)$ such that $C\cong\mathbb{P}^{1}$ , $(C\cdot C)=n$ , and the field of germs of meromorphic functions along $C$ has transcendence degree $2$ over $\mathbb{C}$ , as a ramified two-sheeted covering of the germ of a neighborhood of $\mathbb{P}^{1}$ with self-intersection $2n$ . This construction is a generalization of that from [5, Section 5.3] (and coincides with the latter for $n=1$ ), but the method of proof if non-algebraicity is different. See Construction 5.3 and Proposition 5.4.

I do not know whether the germs of neighborhoods constructed in Sections 4 and 5 are isomorphic.

The proofs of non-algebraicity of the neighborhoods constructed in Sections 4 and 5 are based on the classification of algebraic germs of neighborhoods of $\mathbb{P}^{1}$ . It turns out that any such germ with self-intersection $d>0$ is isomorphic to the germ of a hyperplane section of a surface of degree $d$ in $\mathbb{P}^{d+1}$ ; moreover, any isomorphism of germs of such embeddings $(C_{1},F_{1})$ and $(C_{2},F_{2})$ (where $F_{j}$ , $j=1,2$ , are surfaces of degree $d$ in $\mathbb{P}^{d+1}$ and $C_{j}$ a hyperplane section of $F_{j}$ ) is induced by a linear isomorphism between the surfaces $F_{1},F_{2}\subset\mathbb{P}^{d+1}$ (Propositions 3.3 and 3.4). The key role in the proofs is played by Lemma 3.6.

The paper is organized as follows. In Section 2 we recall the properties of surfaces of degree $d$ in $\mathbb{P}^{d+1}$ . In Section 3 we obtain a classification of algebraic neighborhoods of $\mathbb{P}^{1}$ . Finally, in Sections 4 and 5 we construct two series of examples of non-algebraic neighborhoods.

Acknowledgements

I am grateful to Frank Loray and Grigory Merzon for useful discussions.

1.1. Notation and conventions

All algebraic varieties are varieties over $\mathbb{C}$ . All topological terminology pertains to the classical (complex) topology.

Suppose we are given the pairs $(C_{1},S_{1})$ and $(C_{2},S_{2})$ , where $C_{j}$ , $j=1,2$ , are projective algebraic curves contained in smooth complex analytic surfaces $S_{j}$ . We will say that these two pairs are isomorphic as germs of neighborhoods if there exists an isomorphism $\varphi\colon V_{1}\to V_{2}$ , where $C_{1}\subset V_{1}\subset S_{1}$ , $C_{2}\subset V_{2}\subset S_{2}$ , $V_{1}$ and $V_{2}$ are open, such that $\varphi(C_{1})=C_{2}$ . In this paper we are mostly concerned with germs of neighborhoods, but sometimes, abusing the language, we will write “neighborhood” instead of “germ of neighborhoods”; this should not lead to confusion.

If $C$ is a projective algebraic curve on a smooth complex analytic surface $S$ , then a germ of holomorphic (resp. meromorphic) function along $S$ is an equivalence class of pairs $(U,f)$ , where $U$ is a neighborhood of $C$ in $S$ , $f$ is a holomorphic (resp. meromorphic) function on $U$ , and $(U_{1},f_{1})\sim(U_{2},f_{2})$ if there exists a neighborhood $V\supset C$ , $V\subset U_{1}\cap U_{2}$ , on which $f_{1}$ and $f_{2}$ agree. Germs of holomorphic (resp. meromorphic) functions along $C\subset S$ form a ring (resp. a field). The field of germs of meromorphic functions along $C\subset S$ will be denoted, following the paper [4], by $\mathcal{M}(S,C)$ . If the self-intersection index $(C\cdot C)$ is positive, then, according to Theorem 2.1 from [10], the curve $C\subset S$ has a fundamental system of pseudoconcave neighborhoods, and it follows from [1, Théorème 5] that $\operatorname{tr.deg}_{\mathbb{C}}\mathcal{M}(S,C)\leq 2$ (here and below, $\operatorname{tr.deg}_{\mathbb{C}}$ means “transcendence degree over $\mathbb{C}$ ”).

A germ of neighborhood of a projective curve $C$ will be called algebraic if it is isomorphic, as a germ, to the germ of neighborhood of $C$ in $X$ , where $X\supset C$ is a smooth projective algebraic surface; since desingularization of algebraic surfaces over $\mathbb{C}$ exists, one may as well say that a germ of neighborhood of $C$ is algebraic if it is isomorphic to the germ of a neighborhood $C\subset X$ , where $X$ is a an arbitrary smooth algebraic surface, not necessarily projective. In this case, $\operatorname{tr.deg}_{\mathbb{C}}\mathcal{M}(X,C)=2$ since this field contains the field of rational functions on $X$ .

Remark 1.1.

A germ of neighborhood of a projective curve $C$ with positive self-intersection is algebraic if and only if it is isomorphic to the germ of embedding of $C$ into a compact complex surface. Indeed, any smooth complex surface containing a projective curve with positive self-intersection, can be embedded in $\mathbb{P}^{N}$ for some $N$ (see [2, Chapter IV, Theorem 6.2]).

A projective subvariety $X\subset\mathbb{P}^{N}$ is called non-degenerate if it does not lie in a hyperplane, and linearly normal if the natural homomorphism from $H^{0}(\mathbb{P}^{N},\mathcal{O}_{\mathbb{P}^{N}}(1))$ to $H^{0}(X,\mathcal{O}_{X}(1))$ is surjective. If $X$ is non-degenerate, the latter condition holds if and only if $X$ is not an isomorphic projection of a non-degenerate subvariety of $\mathbb{P}^{N+1}$ .

Non-degenerate projective subvarieties $X_{1}\subset\mathbb{P}^{N_{1}}$ and $X_{2}\subset\mathbb{P}^{N_{2}}$ will be called projectively isomorphic if $N_{1}=N_{2}$ and there exists a linear isomorphism $f\colon\mathbb{P}^{N_{1}}\to\mathbb{P}^{N_{2}}$ such that $f(X_{1})=X_{2}$ .

If $C_{1}$ and $C_{2}$ are two projective algebraic curves on a smooth complex surface, then $(C_{1}\cdot C_{2})$ is their intersection index.

The terms “vector bundle” and “locally free sheaf” will be used interchangeably.

If $C$ is a smooth curve on a smooth complex surface $X$ , then $\mathcal{N}_{X|C}$ is the normal bundle to $C$ in $X$ .

If $\mathcal{E}$ is a vector bundle on an algebraic variety, then $\mathbb{P}(\mathcal{E})$ (the projectivisation of $\mathcal{E}$ ) is the algebraic variety such that its points are lines in the fibers of $\mathcal{E}$ .

If $\mathcal{F}$ is a coherent sheaf on a complex space $X$ , we will sometimes write $h^{i}(\mathcal{F})$ instead of $\dim H^{i}(X,\mathcal{F})$ .

By definition, a projective surface $X$ has rational singularities if $p_{*}\mathcal{O}_{\bar{X}}=\mathcal{O}_{X}$ and $R^{1}p_{*}\mathcal{O}_{\bar{X}}=0$ for some (hence, any) desingularization $p\colon\bar{X}\to X$ .

If $f\colon S\to T$ is a dominant morphism of smooth projective algebraic surfaces, we will say that its critical locus is the set

R=f(\{x\in S\colon\text{$df_{x}$ is degenerate}\})\subset T,

and that its branch divisor $B\subset T$ is the union of one-dimensional components of $R$ .

2. Recap on surfaces of minimal degree

In this section we will recall, without proofs, some well-known results. Most of the details can be found in [3].

If $X\subset\mathbb{P}^{N}$ is a non-degenerate irreducible projective variety, then

(1)

\deg X\geq\operatorname{codim}X+1,

and there exists a classification of the varieties for which the lower bound (1) is attained. We reproduce this classification for the case $\dim X=2$ .

Notation 2.1.

For any integer $n\geq 0$ , put $\mathbb{F}_{n}=\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(n))$ .

The surface $\mathbb{F}_{0}$ is just the quadric $\mathbb{P}^{1}\times\mathbb{P}^{1}$ ; if $n>0$ , then the natural projection $\mathbb{F}_{n}\to\mathbb{P}^{1}$ has a unique section $E\subset\mathbb{F}_{n}$ such that $(E\cdot E)=-n$ . The section $E$ will be called the exceptional section of $\mathbb{F}_{n}$ . The divisor class group of $\mathbb{F}_{n}$ is generated by the class $e$ of the exceptional section and the class $f$ of the fiber (if $n=0$ , we denote by $e$ and $f$ the classes of lines of two rulings on $\mathbb{P}^{1}\times\mathbb{P}^{1}$ ). One has

(2)

(e\cdot e)=-n,\quad(e\cdot f)=1,\quad(f\cdot f)=0.

If $r\geqslant 0$ is an integer and $(n,r)\neq(0,0)$ , then the complete linear system $|e+(n+r)f|$ on $\mathbb{F}_{n}$ has no basepoints and defines a mapping $\mathbb{F}_{n}\to\mathbb{P}^{n+2r+1}$ .

Notation 2.2.

If $n,r$ are non-negative integers and $(n,r)\neq(0,0)$ , then by $F_{n,r}\subset\mathbb{P}^{n+2r+1}$ we will denote the image of the mapping $\mathbb{F}_{n}\to\mathbb{P}^{n+2r+1}$ defined by the complete linear system $|e+(n+r)f|$ .

It follows from (2) that the variety $F_{n,r}\subset\mathbb{P}^{n+2r+1}$ is a surface of degree $n+2r$ . If $r>0$ , the surface $F_{n,r}$ is smooth and isomorphic to $\mathbb{F}_{n}$ . The surface $F_{n,0}$ with $n\geq 2$ is the cone over the normal rational curve of degree $n$ in $\mathbb{P}^{n}$ (this cone is obtained by contracting the exceptional section on $\mathbb{F}_{n}$ ), and the surface $F_{1,0}$ is just the plane; the surfaces $F_{n,0}$ are normal and, moreover, have rational singularities. If $n>0$ and $r>0$ , then the exceptional section on $F_{n,r}$ is a rational curve of degree $r$ . Finally, the surface $F_{0,1}\subset\mathbb{P}^{3}$ is the smooth quadric.

Recall that the quadratic Veronese surface $V\subset\mathbb{P}^{5}$ is the image of the mapping $v\colon\mathbb{P}^{2}\to\mathbb{P}^{5}$ defined by the formula

(3)

v\colon(x_{0}:x_{1}:x_{2})\mapsto(x_{0}^{2}:x_{1}^{2}:x_{2}^{2}:x_{0}x_{1}:x_{0}x_{2}:x_{1}x_{2});

one has $\deg V=4$ . The mapping $v$ induces an isomorphism from $\mathbb{P}^{2}$ to $V$ ; hyperplane sections of $V$ are images of conics in $\mathbb{P}^{2}$ .

Proposition 2.3.

If $X\subset\mathbb{P}^{N}$ is a non-degenerate irreducible projective surface, then $\deg X\geq N-1$ and the bound is attained if and only if either $X=F_{n,r}$ , where $(n,r)\neq(0,0)$ , or $X=V\subset\mathbb{P}^{5}$ , where $V$ is the quadratic Veronese surface.

If $(n,r)\neq(n^{\prime},r^{\prime})$ , then the surfaces $F_{n,r}$ and $F_{n^{\prime},r^{\prime}}$ are not projectively isomorphic, and none of them is projectively isomorphic to the Veronese surface $V$ . (Indeed, if $n\neq n^{\prime}$ then $F_{n,r}$ and $F_{n^{\prime},r^{\prime}}$ are not isomorphic even as abstract varieties, and if $n=n^{\prime}$ but $r\neq r^{\prime}$ then their degrees are different; speaking of the Veronese surface, it does not contain lines while each $F_{n,r}$ is swept by lines.)

Suppose now that $X\subset\mathbb{P}^{N+1}$ , $N\geq 2$ , is a surface of minimal degree (i.e., of degree $N$ ) and $p\in F$ is a smooth point (i.e., either $X\neq F_{n,0}$ and $p$ is arbitrary or $X=F_{n,0}$ and $p$ is not the vertex of the cone). Let $\pi_{p}$ denote the projection from $\mathbb{P}^{N+1}$ to $\mathbb{P}^{N}$ with the center $p$ .

Proposition 2.4.

In the above setting, the projection $\pi_{p}$ induces a birational mapping from $X$ onto its image. If $X^{\prime}\subset\mathbb{P}^{N}$ is (the closure of) the image of $\pi_{p}\colon X\dasharrow\mathbb{P}^{N}$ , then $X^{\prime}$ is also a surface of minimal degree.

If $X=F_{n,r}$ , where $n>0$ and $r>0$ , then $X^{\prime}=F_{n+1,r-1}$ if $p$ lies on the exceptional section and $X^{\prime}=F_{n-1,r}$ otherwise.

If $X=F_{0,r}$ , $r>0$ , then $X^{\prime}=F_{1,r-1}$ .

If $X=F_{n,0}$ , $n\geq 2$ , then $X^{\prime}=F_{n-1,0}$ .

Finally, if $X=V\subset\mathbb{P}^{5}$ , then $X^{\prime}=F_{1,1}\subset\mathbb{P}^{4}$ .

These projections for surfaces of minimal degree $\leq 6$ are depicted in Figure 1. Observe that no arrow points at the surface $V$ ; this fact will play a crucial role in the sequel.

Figure 1. Projections of surfaces of minimal degree

The birational transformations induced by the projections form Proposition 2.4 can be described explicitly.

Proposition 2.5.

Suppose that $F\subset\mathbb{P}^{N}$ is a surface of minimal degree, $p\in F$ is a non-singular point, and $\pi_{p}\colon F\dasharrow\mathbb{P}^{N-1}$ is the projection from $p$ . The projection $\pi_{p}$ acts on the surface $F$ as follows.

(1)

If $F=F_{n,r}\subset\mathbb{P}^{n+2r+1}$ , where $r>1$ , then the projection $\pi_{p}$ blows up the point $p$ and blows down the strict transform of the only line on $F$ passing through $p$ .
(2)

If $F=F_{n,1}\subset\mathbb{P}^{n+3}$ , where $n>0$ (in this case the exceptional section of $F\cong\mathbb{F}_{n}$ is a line), then two cases are possible.

If the point $p$ does not lie on the exceptional section, then the projection $\pi_{p}$ blows up the point $p$ and blows down the strict transform of the only line passing through $p$ , and the image of the projection is the surface $F_{n-1,1}\subset\mathbb{P}^{n+2}$ . If, on the other hand, $p$ lies on the exceptional section, then the projection $\pi_{p}$ blows up the point $p$ and blows down both the strict transform of the fiber passing through $p$ and the strict transform of the exceptional section; in this latter case the image of the projection is the cone $F_{n+1,0}\subset\mathbb{P}^{n+2}$ (the strict transform of the exceptional section is blown down to the vertex of the cone).
(3)

If $F=F_{0,1}\subset\mathbb{P}^{3}$ (that is, if $F\subset\mathbb{P}^{3}$ is the smooth quadric), then the projection $\pi_{p}$ blows up the point $p$ and blows down strict transforms of the two lines passing through $p$ ; the image of the projection is, of course, just the plane.
(4)

If $F=F_{n,0}\subset\mathbb{P}^{n+1}$ , where $n>1$ (that is, if $F$ is a cone over the rational normal curve of degree $n$ in $\mathbb{P}^{n}$ ), then the projection $\pi_{p}$ blows up the point $p$ and blows down the strict transform of the generatrix of the cone passing through $p$ ; the image of this projection is the cone $F_{n-1,0}\subset\mathbb{P}^{n}$ (if $n=2$ , this cone is just the plane).
(5)

Finally, if $F=V\subset\mathbb{P}^{5}$ is the Veronese surface, then the projection $\pi_{p}$ just blows up the point $p$ , and the image of this projection is $F_{1,1}\subset\mathbb{P}^{3}$ .

3. Rational curves with positive self-intersection and algebraic germs

Proposition 3.1.

Suppose that $X$ is a smooth projective surface and $C\subset X$ is a curve isomorphic to $\mathbb{P}^{1}$ . If $(C\cdot C)=m>0$ , then the complete linear system $|C|$ has no basepoints, $\dim|C|=m+1$ , the morphism $\varphi_{|C|}$ is a birational isomorphism between $X$ and $\varphi(X)\subset\mathbb{P}^{m+1}$ , and $\varphi(X)$ is a surface in $\mathbb{P}^{m+1}$ of minimal degree $m$ .

Proof.

Since the normal bundle $\mathcal{N}_{X|C}$ is isomorphic to $\mathcal{O}_{\mathbb{P}^{1}}(m)$ , where $m>0$ , one has $h^{0}(\mathcal{N}_{X|C})=m+1$ , $h^{1}(\mathcal{N}_{X|C})=0$ , so the Hilbert scheme of the curve $C\subset X$ is smooth and has dimension $m+1$ at the point corresponding to $C$ . A general curve from this $(m+1)$ -dimensional family is isomorphic to $\mathbb{P}^{1}$ ; since $m>0$ , through a general point $p\in X$ there passes a positive-dimensional family of rational curves. Therefore, the Albanese mapping of $X$ is constant, whence $H^{1}(X,\mathcal{O}_{X})=0$ . Now it follows from the exact sequence

(4)

0\to\mathcal{O}_{X}\to\mathcal{O}_{X}(C)\xrightarrow{\alpha}\mathcal{O}_{C}(C)\to 0

that the homomorphism $\alpha_{*}\colon H^{0}(X,\mathcal{O}_{X}(C))\to H^{0}(C,\mathcal{O}_{C}(C))$ is surjective. Since the linear system $|\mathcal{O}_{C}(C)|=|\mathcal{O}_{\mathbb{P}^{1}}(m)|$ has no basepoints, the linear system $|C|$ has no basepoints either, and it follows from (4) and the vanishing of $H^{1}(\mathcal{O}_{X})$ that $\dim|C|=m+1$ . If $\varphi=\varphi_{|C|}\colon X\to\mathbb{P}^{m+1}$ and $Y=\varphi(X)$ , then $\dim Y=2$ and $\deg Y\cdot\deg\varphi=(C\cdot C)=m$ . Since $Y\subset\mathbb{P}^{m+1}$ is non-degenerate, one has $\deg Y\geq m$ (see (1)), whence $\deg Y=m$ and $\deg\varphi=1$ , so $\varphi$ is birational onto its image. ∎

Corollary 3.2.

If $X$ is a projective surface with rational singularities and $C\subset X_{\mathrm{sm}}$ is a curve that is isomorphic to $\mathbb{P}^{1}$ and such that $(C\cdot C)>0$ , then $h^{1}(\mathcal{O}_{X})=0$ and $h^{0}(\mathcal{O}_{X}(C))=(C\cdot C)+1$ .

Proof.

Let $\bar{X}$ be a desingularization of $X$ . Arguing as in the proof of Proposition 3.1, we conclude that through a general point of $\bar{X}$ there passes a positive-dimensional family of rational curves, whence $H^{1}(\bar{X},\mathcal{O}_{\bar{X}})=0$ . Since the singularities of the surface $X$ are rational, $H^{1}(\bar{X},\mathcal{O}_{\bar{X}})\cong H^{1}(X,\mathcal{O}_{X})$ , so $H^{1}(X,\mathcal{O}_{X})=0$ . Now the result follows from the exact sequence (4). ∎

Proposition 3.1 implies the following characterisation of algebraic neighborhoods of rational curves.

Proposition 3.3.

If $(C,U)$ is an algebraic neighborhood of the curve $C\cong\mathbb{P}^{1}$ and if $(C\cdot C)=d>0$ , then the germ of this neighborhood is isomorphic to the germ of neighborhood of a smooth hyperplane section in a surface of minimal degree $d$ in $\mathbb{P}^{d+1}$ .

Proof.

Passing to desingularization, one may without loss of generality assume that the neighborhood in question is a neighborhood of a curve $C\subset X$ , where $C\cong\mathbb{P}^{1}$ , $X$ is a smooth projective surface, and $(C\cdot C)=d>0$ . If $\varphi\colon X\to Y$ , where $Y$ is a surface of minimal degree, is the birational morphism the existence of which is asserted by Proposition 3.1, then $\varphi$ is an isomorphism in a neighborhood of $C$ and $\varphi(C)$ is a hyperplane section of $Y$ , whence the result. ∎

Proposition 3.3 may be regarded as a generalization of Proposition 4.7 from [8], which asserts that any algebraic germ of neighborhood of $\mathbb{P}^{1}$ with self-intersection $1$ is isomorphic to the germ of neighborhood of a line in $\mathbb{P}^{2}$ .

Now we show that not only all germs of algebraic neighborhoods of $\mathbb{P}^{1}$ can be obtained from surfaces of minimal degree, but that their isomorphisms are induced by isomorphisms of surfaces of minimal degree.

Proposition 3.4.

Suppose that $X_{1}\subset\mathbb{P}^{N_{1}}$ and $X_{2}\subset\mathbb{P}^{N_{2}}$ are linearly normal projective surfaces with rational singularities, $C_{1}\subset X_{1}$ and $C_{2}\subset X_{2}$ are their smooth hyperplane sections, and that $C_{1}$ and $C_{2}$ are isomorphic to $\mathbb{P}^{1}$ .

If there exist analytic neighborhoods $U_{j}\supset C_{j}$ , $j=1,2$ , and a holomorphic isomorphism $\varphi\colon U_{1}\to U_{2}$ such that $\varphi(C_{1})=C_{2}$ , then $\varphi$ extends to a projective isomorphism $\Phi\colon X_{1}\to X_{2}$ .

To prove this proposition we need two lemmas. The first of them is well known.

Lemma 3.5.

If $X$ is a projective surface with isolated singularities and $C\subset X_{\mathrm{sm}}$ is an ample irreducible curve, then the ring of germs of holomorphic functions along $C$ coincides with $\mathbb{C}$ .

Sketch of proof.

This follows immediately from the fact that $H^{0}(\hat{X},\mathcal{O}_{\hat{X}})=\mathbb{C}$ , where $\hat{X}$ is the formal completion of $X$ along $C$ (see [7, Chapter V, Proposition 1.1 and Corollary 2.3]).

Here is a more elementary argument. Since $C$ is an ample divisor in $X$ , there exists and embedding of $X$ in $\mathbb{P}^{N}$ such that $rC$ , for some $r>0$ , is a hyperplane section of $X$ . Suppose that $U\supset C$ , $U\subset X$ is a connected neighborhood of $C$ . There exists a family of hyperplane sections $\{H_{\alpha}\}$ , close to the one corresponding to $rC$ , such that $H_{\alpha}\subset U$ for each $\alpha$ and $\bigcup_{\alpha}H_{\alpha}$ contains a non-empty open subset $V\subset U$ . If $f$ is a holomorphic function on $U$ , then $f$ is constant on each $H_{\alpha}$ ; since $H_{\alpha}\cap H_{\alpha^{\prime}}\neq\varnothing$ for each $\alpha$ and $\alpha^{\prime}$ , $f$ is constant on the union of all the $H_{\alpha}$ ’s, hence on $V$ , hence on $U$ . ∎

Lemma 3.6.

Suppose that $X$ is a projective surface such that $H^{1}(X,\mathcal{O}_{X})=0$ , $D\subset X_{\mathrm{sm}}$ is an ample irreducible projective curve, and $r$ is a positive integer. Then any germ of meromorphic function along $C$ , with possibly a pole of order $\leq r$ along $C$ and no other poles, is induced by a rational function on $X$ with possibly a pole of order $\leq r$ along $C$ and no other poles.

Proof.

Let $\mathcal{I}_{C}\subset\mathcal{O}_{X}$ be the ideal sheaf of $C$ ; put $\mathcal{I}_{rC}=\mathcal{I}_{C}^{m}$ , $\mathcal{O}_{rC}=\mathcal{O}_{X}/\mathcal{I}_{C}^{m}$ , and

\mathcal{N}_{X|rC}=\underline{\mathrm{Hom}}_{\mathcal{O}_{rC}}(\mathcal{I}_{rC}/\mathcal{I}_{rC}^{2},\mathcal{O}_{rC})=\underline{\mathrm{Hom}}_{\mathcal{O}_{X}}(\mathcal{I}_{rC},\mathcal{O}_{rC}).

Identifying $\mathcal{O}_{X}(rC)$ with the sheaf of meromorphic functions having at worst a pole of order $\leq r$ along $C$ , one has the exact sequence

(5)

0\to\mathcal{O}_{X}\to\mathcal{O}_{X}(rC)\xrightarrow{\alpha}\mathcal{N}_{X|rC}\to 0,

in which the homomorphism $\alpha$ has the form

g\mapsto(s\mapsto gs\bmod\mathcal{I}_{rC}),

where $g$ is a meromorphic function on an open subset of $V\subset X$ , with at worst a pole of order $\leq r$ along $C$ and $s$ is a section of $\mathcal{I}_{rC}$ over $V$ . In particular, if $U\supset C$ is a neighborhood, then any section of $g\in H^{0}(U,\mathcal{O}_{X}(rC))$ induces a global section of $\mathcal{N}_{X|rC}$ . Since $H^{1}(X,\mathcal{O}_{X})=0$ , the homomorphism $\alpha$ from (5) induces a surjection on global sections, so there exists a section $f\in H^{0}(X,\mathcal{O}_{X}(rC))$ such that $f$ and $g$ induce the same global section of $\mathcal{N}_{X|rC}$ . Hence, the meromorphic function $(f|_{U})-g$ has no pole in $U$ , so by virtue of Lemma 3.5 this function is equal to a constant $c$ on a (possibly smaller) neighborhood of $C$ . Thus, the germ of $f-c$ along $C$ equals that of $g$ . ∎

Proof of Proposition 3.4.

It follows from the hypothesis that $(C_{1}\cdot C_{1})=(C_{2}\cdot C_{2})$ . If we denote these intersection indices by $m$ , then Corollary 3.2 implies that $\dim|C_{1}|=\dim|C_{2}|=m+1$ . Let $f_{0},\dots,f_{m+1}$ be a basis of $H^{0}(\mathcal{O}_{X_{1}}(C_{1}))$ (i.e., the basis of space of meromorphic functions on $X_{1}$ with at worst a simple pole along $C_{1}$ ), and similarly let $(g_{0},\dots,g_{m+1})$ be a basis of $H^{0}(\mathcal{O}_{X_{2}}(C_{2}))$ . Embed $X_{1}$ (resp. $X_{2}$ ) into $\mathbb{P}^{m+1}$ with the linear system $|C_{1}|$ (resp. $|C_{2}|$ ), that is, with the mappings

x\mapsto(f_{0}(x):\dots:f_{n+1}(x))\quad\text{and}\quad y\mapsto(g_{0}(y):\dots:g_{n+1}(y)).

If $\gamma_{i}$ , $0\leq i\leq m+1$ , is the germ along $C_{1}$ of the meromorphic function $g_{i}\circ\varphi$ , then by virtue of Lemma 3.6, which we apply in the case $r=1$ , each $\gamma_{i}$ is the germ along $C_{1}$ of a meromorphic function $h_{i}\in H^{0}(\mathcal{O}_{X_{1}}(C_{1}))$ . If $h_{i}=\sum a_{ij}f_{j}$ , then the matrix $\|a_{ij}\|$ defines a linear automorphism $\Phi\colon\mathbb{P}^{m+1}\to\mathbb{P}^{m+1}$ such that its restriction to a neighborhood of $C_{1}$ coincides with $\Phi$ . Hence, $\Phi$ maps $X_{1}$ isomorphically onto $X_{2}$ . ∎

Corollary 3.7.

Suppose that $F\subset\mathbb{P}^{m}$ (resp. $F^{\prime}\subset\mathbb{P}^{m^{\prime}}$ ) is a surface of minimal degree and $C$ (resp. $C^{\prime}$ ) is its smooth hyperplane section. Then the germs of neighborhoods of $C$ in $F$ and of $C^{\prime}$ in $F^{\prime}$ are isomorphic if and only if there exists a linear isomorphism $\Phi\colon\mathbb{P}^{m}\to\mathbb{P}^{m^{\prime}}$ such that $\Phi(F)=F^{\prime}$ and $\Phi(C)=C^{\prime}$ .

Proof.

Immediate from Proposition 3.4 if one takes into account that all surfaces of minimal degree have rational singularities. ∎

Remark 3.8.

We see that any algebraic neighborhood of $C\cong\mathbb{P}^{1}$ has one more discrete invariant, besides the self-intersection $d=(C\cdot C)$ : if this neighborhood is isomorphic to the germ of neighborhood of the minimal surface $F_{n,r}$ , where $d=n+2r$ , this is the integer $n\geq 0$ (and if the surface is not $F_{n,r}$ but the Veronese surface $V\subset\mathbb{P}^{5}$ , we assign to our neighborhood the tag $V$ instead). It should be noted however that, as a rule, the pair $(d,n)$ does not determine the germ of neighborhood up to isomorphism. Indeed, the dimension of the group of automorphisms of the surface $\mathbb{F}_{n}$ is $n+5$ if $n>0$ and $6$ if $n=0$ . In most cases this is less that the dimension of the space of hyperplanes in $\mathbb{P}^{n+2r+1}$ , in which $F_{n,r}$ is embedded. On the other hand, linear automorphisms of $F_{n,0}\subset\mathbb{P}^{n+1}$ act transitively on the set of smooth hyperplane sections of $F_{n,0}$ , and ditto for $V\subset\mathbb{P}^{5}$ . Thus, tags $(n,0)$ or $V$ do determine an algebraic germ of neighborhood of $\mathbb{P}^{1}$ up to isomorphism.

4. Blowups and blowdowns

Suppose that a smooth projective curve $C$ lies on a smooth complex analytic surface $S$ and that $p\in C$ . Let $\sigma\colon\tilde{S}\to S$ be the blowup of $S$ at $p$ , and let $\tilde{C}\subset\tilde{S}$ be the strict transform of $C$ . It is clear that the germ of neighborhood $\tilde{C}\subset\tilde{S}$ depends only on the germ of neighborhood $C\subset S$ and on the point $p\in C$ .

Definition 4.1.

In the above setting, the germ of neighborhood $(\tilde{C},\tilde{U})$ will be called the blowup of the germ $(C,U)$ at the point $p$ .

For future reference we state the following obvious properties of blowups of neighborhoods.

Proposition 4.2.

Suppose that the germ of neighborhoods $(\tilde{C},\tilde{U})$ is a blowup of $(C,U)$ . Then

(1)

$\tilde{C}$ is isomorphic to $C$ ;
(2)

$(\tilde{C}\cdot\tilde{C})=(C\cdot C)-1$ ;
(3)

if $(C,U)$ is algebraic then $(\tilde{C},\tilde{U})$ is algebraic.

For algebraic neighborhoods of $\mathbb{P}^{1}$ the assertion (3) of Proposition 4.2 can be made more explicit.

Proposition 4.3.

If an algebraic germ $(C,U)$ is isomorphic to the germ of neighborhood of a hyperplane section of a surface of minimal degree $F\subset\mathbb{P}^{N}$ , then the blowup of this germ at a point $p\in C$ is isomorphic to the germ of neighborhood of a hyperplane section of the surface $F^{\prime}\subset\mathbb{P}^{N-1}$ that is obtained from $F$ by projection from the point $p$ .

Proof.

Immediate from Proposition 2.5. ∎

Remark 4.4.

Even though $\mathbb{P}^{1}$ is homogeneous, which means that its points are indistinguishable, germs of blowups of a given neighborhood of $\mathbb{P}^{1}$ at different points are not necessarily isomorphic. Indeed, suppose that $C\cong\mathbb{P}^{1}$ is a smooth hyperplane section of the surface $F_{n,r}$ , where $n>0$ and $r>0$ . If $p\in C$ does not lie on the exceptional section $E\subset F_{n,r}$ , then Proposition 2.4 implies that the blowup at $p$ of the germ of neighborhood of $C$ is isomorphic to a neighborhood of a hyperplane section of $F_{n-1,r}$ , and if $p$ does lie on $E$ , Proposition 2.4 implies that the blowup in question is isomorphic to a neighborhood of a hyperplane section of $F_{n+1,r-1}$ (observe that $C\neq E$ and $C\cap E\neq\varnothing$ , so points of both kinds are present). If the blowups at such points were isomorphic as germs of neighborhoods, then, by virtue of Proposition 3.4, this isomorphism would be induced by a linear isomorphism between $F_{n-1,r}$ and $F_{n+1,r-1}$ , which does not exist.

Proposition 4.5.

Suppose that a curve $D\cong\mathbb{P}^{1}$ is embedded in a surface $U$ . If, blowing up $s>0$ points of the germ of neighborhood $(D,U)$ , one obtains a germ of neighborhood $(C,W)$ that is isomorphic to the germ of neighborhood of a non-degenerate conic in $\mathbb{P}^{2}$ , then the original germ $(D,U)$ is not algebraic.

Proof.

Without loss of generality we may and will assume that $C$ is a conic in $\mathbb{P}^{2}$ and $W\supset C$ is an open subset of $\mathbb{P}^{2}$ . The Veronese mapping $v\colon\mathbb{P}^{2}\to V$ (see (3)) identifies $\mathbb{P}^{2}$ with the Veronese surface $V\subset\mathbb{P}^{5}$ and $C\subset\mathbb{P}^{2}$ with a smooth hyperplane section of $V$ .

Arguing by contradiction, suppose that the germ $(D,U)$ is algebraic. Then Proposition 3.3 implies that this germ is isomorphic to the germ of neighborhood of a hyperplane section of a surface of minimal degree $m=4+s>4$ . Hence, this surface is of the form $F_{n,r}$ , where $n+2r=m$ (see Proposition 2.3).

By construction, the germ of $(C,W)$ can be obtained from the germ $(D,U)$ by blowing up $s$ points on $D$ . Hence, by Proposition 4.3, the germ of $(C,W)$ is isomorphic to the germ of a hyperplane section of a surface that can be obtained from $F_{n,r}$ by $s$ consecutive projections. However, Proposition 2.4 shows that the resulting surface cannot be projectively isomorphic to $V\subset\mathbb{P}^{5}$ (cf. Figure 1). On the other hand, Proposition 3.4 implies that if germs of hyperplane sections of two surfaces of minimal degree are isomorphic then the surfaces are projectively isomorphic. This contradiction shows that the germ $(D,U)$ is not algebraic. ∎

Now we can construct our first series of non-algebraic examples.

Lemma 4.6.

Suppose that $C\subset\mathbb{P}^{2}$ is a non-degenerate conic and $s$ is a positive integer. Then there exist $s$ lines $L_{1},\ldots,L_{s}\subset\mathbb{P}^{2}$ and neighborhoods $W\supset C$ , $W_{j}\supset L_{j}$ , $1\leq j\leq s$ , in $\mathbb{P}^{2}$ having the following properties.

(1)

each $L_{j}$ intersects the conic $C$ at precisely two points, $p_{j}$ and $q_{j}$ ;
(2)

all the points $p_{1},\ldots,p_{s}$ are distinct, all the points $q_{1},\ldots,q_{s}$ are distinct, and $p_{i}\neq q_{j}$ for any $i,j$ .
(3)

$W_{j}\cap W\cap\{p_{1},\ldots,p_{s},q_{1},\ldots,q_{s}\}=\{p_{j},q_{j}\}$ for each $j$ .
(4)

The open subset $W_{j}\cap W$ has precisely two connected components $P_{j}\ni p_{j}$ and $Q_{j}\ni q_{j}$ .
(5)

${P_{i}}\cap{P_{j}}=\varnothing$ whenever $i\neq j$ , ${Q_{i}}\cap{Q_{j}}=\varnothing$ whenever $i\neq j$ , and ${P_{i}}\cap{Q_{j}}=\varnothing$ for any $i,j$ .

Proof.

Only the assertions (3) and (4) deserve a sketch of proof. To justify them choose a Hermitian metric on $\mathbb{P}^{2}$ and let $W$ be a small enough tubular neighborhood of $C$ and $W_{1},\ldots,W_{s}$ be small enough tubular neighborhoods of $L_{1},\ldots,L_{s}$ . ∎

The proof of the following lemma is left to the reader.

Lemma 4.7.

Suppose that $X_{1}$ and $X_{2}$ are Hausdorff topological spaces, $O_{1}\subset X_{1}$ and $O_{2}\subset X_{2}$ are open subsets, and $X$ is the topological space obtained by gluing $X_{1}$ and $X_{2}$ via a homeomorphism $\Phi\colon O_{1}\to O_{2}$ .

If for any $x_{1}\in\operatorname{bd}(O_{1})\subset X_{1}$ , $x_{2}\in\operatorname{bd}(O_{2})\subset X_{2}$ , where $\operatorname{bd}$ means “boundary”, there exist open neighborhoods $V_{1}\ni x_{1}$ in $X_{1}$ and $V_{2}\ni x_{2}$ in $X_{2}$ such that

\Phi(V_{1}\cap O_{1})\cap V_{2}=\varnothing,

then $X$ is Hausdorff.∎

Construction 4.8.

Suppose that $C$ , $L_{1},\ldots,L_{s}$ , $W_{1},\ldots,W_{s}$ , and $W$ are as in Lemma 4.6.

Put

U_{0}=W\sqcup W_{1}\dots\sqcup W_{s}

(disjoint sum), and let $\pi_{0}\colon U_{0}\to\mathbb{P}^{2}$ be the natural projection. Define the equivalence relation $\sim$ on $U_{0}$ as follows: if $x,y\in U_{0}$ and $x\neq y$ , then $x\sim y$ if and only if $\pi_{0}(x)=\pi_{0}(y)\in P_{1}\cup\dots\cup P_{s}$ .

Let $U_{1}$ be the quotient of $U_{0}$ by the equivalence relation $\sim$ , and let $\pi_{1}\colon U_{1}\to\mathbb{P}^{2}$ be the natural projection.

Denote the images of $C\subset W$ and $L_{j}\subset W_{j}$ in $U_{1}$ by $C_{1}$ and $L_{j}^{\prime}$ .

Lemma 4.9.

In the above setting, $U_{1}$ is a Hausdorff and connected complex surface and $\pi_{1}\colon U_{1}\to\mathbb{P}^{2}$ is a local holomorphic isomorphism.

The curves $C_{1}$ and $L_{j}^{\prime}$ are isomorphic to $\mathbb{P}^{1}$ , $(L^{\prime}_{j}\cdot L^{\prime}_{j})=(L_{j}\cdot L_{j})=1$ , $(L^{\prime}_{j}\cdot C_{1})=1$ for each $j$ , and $(C_{1}\cdot C_{1})=(C\cdot C)=4$ .

Proof.

If we put, in Lemma 4.7, $X_{1}=W$ , $X_{2}=W_{1}\sqcup\dots\sqcup W_{s}$ , $O_{1}=P_{1}\cup\ldots\cup P_{s}\subset W$ , $O_{2}=P_{1}\sqcup\dots\sqcup P_{s}\subset W_{1}\sqcup\dots\sqcup W_{s}$ , $\Phi=\mathrm{Id}$ , then the hypothesis of this lemma is satisfied if, putting $P=P_{1}\cup\dots\cup P_{s}$ and $Q=Q_{1}\cup\dots\cup Q_{s}$ , one has

(6)

((\bar{P}\cap W)\setminus P)\cap(\bar{P}\cap(W_{1}\cup\dots\cup W_{s})\setminus P)=\varnothing.

The left-hand side of (6) is equal to

(\bar{P}\cap W\cap(W_{1}\cup\dots\cup W_{s}))\setminus P=(\bar{P}\setminus P)\cap(P\cup Q)\subset\bar{P}\cap Q,

which is empty by Lemma 4.6(5). Hence, $U_{1}$ is Hausdorff.

The rest is obvious. ∎

Construction 4.10.

In the above setting, for each $j$ , $1\leq j\leq s$ , choose two distinct points $u_{j},v_{j}\in L^{\prime}_{j}$ , different from the intersection point of $L^{\prime}_{j}$ and $C_{1}$ . Let $U_{2}$ be the blowup of the surface $U_{1}$ at the points $u_{1},\ldots,u_{s},v_{1},\ldots,v_{s}$ . If $\sigma\colon U_{2}\to U_{1}$ is the natural morphism, put $C_{2}=\sigma^{-1}(C^{\prime})\subset U_{2}$ ; it is clear that $\sigma$ is an isomorphism on a neighborhood of $C_{2}$ ; in particular, $C_{2}\cong C_{1}\cong\mathbb{P}^{1}$ and $(C_{2}\cdot C_{2})=4$ .

For each $j$ , let $\tilde{L}_{j}\subset U_{2}$ be the strict transform of $L^{\prime}_{j}$ with respect to the blowup $\sigma\colon U_{2}\to U_{1}$ . By construction, $\tilde{L}_{j}\cong\mathbb{P}^{1}$ , $(\tilde{L}_{j}\cdot\tilde{L}_{j})=-1$ for each $j$ , and the curves $\tilde{L}_{j}$ are pairwise disjoint. Hence, one can blow down the curves $\tilde{L}_{1},\ldots,\tilde{L}_{s}$ to obtain a smooth complex surface $U_{3}$ and a curve $C_{3}\subset U_{3}$ , which is the image of $C_{2}$ ; one has $C_{3}\cong\mathbb{P}^{1}$ and $(C_{3}\cdot C_{3})=(C_{2}\cdot C_{2})+s=4+s>4$ .

Proposition 4.11.

If $(C_{3},U_{3})$ is the germ of neighborhood from Construction 4.10, then this germ is not algebraic and $\operatorname{tr.deg}_{\mathbb{C}}\mathcal{M}(U_{3},C_{3})=2$ .

Proof.

It follows from the construction that the blowup of the germ of neighborhood of $C_{3}$ in $U_{3}$ at the $s$ points to which $\tilde{L}_{1},\ldots,\tilde{L}_{s}$ were blown down, is isomorphic to the germ of neighborhood of $C_{2}$ in $U_{2}$ , which is isomorphic to the of neighborhood of the conic $C$ in $\mathbb{P}^{2}$ . Now Proposition 4.5 implies that the germ $(C_{3},U_{3})$ is not algebraic.

Since $\pi_{1}\colon U_{1}\to\mathbb{P}^{2}$ is a local isomorphism, the filed of meromorphic functions on $\mathbb{P}^{2}$ , which is isomorphic to the field $\mathbb{C}(X,Y)$ of rational functions in two variables, can be embedded in the field of meromorphic functions on $U_{1}$ . Since the surface $U_{3}$ is obtained from $U_{1}$ by a sequence of blowups and blowdowns, the fields of meromorphic functions on $U_{1}$ and $U_{3}$ are isomorphic. Hence, $\mathbb{C}(X,Y)$ can be embedded in the field of meromorphic functions on $U_{3}$ , which can be embedded in $\mathcal{M}(U_{3},C_{3})$ . Thus, $\operatorname{tr.deg}_{\mathbb{C}}\mathcal{M}(U_{3},C_{3})\geq 2$ , whence $\operatorname{tr.deg}_{\mathbb{C}}\mathcal{M}(U_{3},C_{3})=2$ . This completes the proof. ∎

5. Ramified coverings

In this section we construct another series of examples of non-algebraic neighborhoods $(C,U)$ , where $C\cong\mathbb{P}^{1}$ and $\operatorname{tr.deg}_{\mathbb{C}}\mathcal{M}(U,C)=2$ . In these examples, the self-intersection $(C\cdot C)$ may be an arbitrary positive integer. We begin with two simple lemmas.

Lemma 5.1.

Suppose that $X$ is a smooth complex surface, $C\subset X$ is a projective curve, $C\cong\mathbb{P}^{1}$ , and $U\subset X$ is a tubular neighborhood of $C$ . Then $\pi_{1}(U\setminus C)\cong\mathbb{Z}/m\mathbb{Z}$ , where $m=(C\cdot C)$ .

Proof.

Immediate from the homotopy exact sequence of the fiber bundle $U\setminus C\to C$ . ∎

Lemma 5.2.

Suppose that $f\colon S\to T$ is a dominant morphism of smooth projective algebraic surfaces and that $B\subset T$ is the branch divisor of $f$ (see the definition in Section 1.1). If $T\setminus B$ is simply connected, then $\deg f=1$ .

Proof.

The critical locus $R\subset T$ of $f$ is of the form $B\cup E$ , where $B$ is the branch divisor and $E$ is a finite set. The mapping

f|_{S\setminus f^{-1}(R)}\colon S\setminus f^{-1}(R)\to T\setminus R

is a topological covering. Since the subset $E\subset T\setminus B$ is finite and $T\setminus B$ is smooth, fundamental groups of $T\setminus R$ and $T\setminus B$ are isomorphic, so $T\setminus R$ is also simply connected, whence the result. ∎

Construction 5.3.

Fix an integer $n>0$ . We are going to construct a certain neighborhood of $\mathbb{P}^{1}$ with self-intersection $n$ .

To that end, suppose that $X\subset\mathbb{P}^{2n+1}$ is a non-degenerate surface of degree $2n$ which is not the cone $F_{2n,0}$ and, if $n=2$ , not the Veronese surface $V$ . Let $C\subset X$ be a smooth hyperplane section, and let $U\supset C$ , $U\subset X$ be a tubular neighborhood of $C$ . By virtue of Lemma 5.1 one has $\pi_{1}(U\setminus C)=\mathbb{Z}/2n\mathbb{Z}$ . Hence, there exists a two-sheeted ramified covering $\pi\colon V\to U$ that is ramified along $C$ with index $2$ . If $C^{\prime}=\pi^{-1}(C)$ , then $C^{\prime}\cong\mathbb{P}^{1}$ and $(C^{\prime}\cdot C^{\prime})=n$ .

Proposition 5.4.

If $(C^{\prime},V)$ is the neighborhood from Construction 5.3, then $\operatorname{tr.deg}_{\mathbb{C}}\mathcal{M}(V,C^{\prime})=2$ and the neighborhood $(C^{\prime},V)$ is not algebraic.

Proof.

The neighborhood $(C,U)$ is algebraic, so $\operatorname{tr.deg}_{\mathbb{C}}\mathcal{M}(U,C)=2$ , and the morphism $\pi\colon V\to U$ induces an embedding of $\mathcal{M}(U,C)$ in $\mathcal{M}(V,C^{\prime})$ , hence $\operatorname{tr.deg}_{\mathbb{C}}\mathcal{M}(V,C^{\prime})\geq 2$ , hence this transcendence degree equals $2$ .

To prove the non-algebraicity of $(C^{\prime},V)$ , assume the converse. Then, by virtue of Proposition 3.3, the germ of the neighborhood $(C^{\prime},V)$ is isomorphic to the germ of neighborhood of a smooth hyperplane section of a non-degenerate surface $X^{\prime}\subset\mathbb{P}^{n+1}$ , $\deg X^{\prime}=n$ ; we will identify $C^{\prime}$ with this hyperplane section and $V$ with a neighborhood of $C^{\prime}$ in $X^{\prime}$ .

Let $f_{0},\ldots,f_{2n}$ be a basis of the space $H^{0}(X,\mathcal{O}_{X}(C))$ (i.e., of the space of meromorphic functions on $X$ with at worst a simple pole along $C$ ). For each $j$ , the function $(f_{j}|_{U})\circ\pi$ is a meromorphic function on $V$ with at worst a pole of order $2$ along $C^{\prime}$ . Using Lemma 3.6 (in which one puts $r=2$ ), one sees that there exist meromorphic functions $g_{0},\ldots,g_{2n}\in H^{0}(X^{\prime},\mathcal{O}_{X^{\prime}}(2C^{\prime}))$ such that, for each $j$ , the germ of $g_{j}$ along $C$ is the same as that of $(f_{j}|_{U})\circ\pi$ .

Choose a basis $h_{0},\ldots,h_{N}$ is of $H^{0}(X^{\prime},\mathcal{O}_{X^{\prime}}(2C^{\prime}))$ , and let $X_{1}\subset\mathbb{P}^{N}$ be the image of $X^{\prime}$ under the embedding

x\mapsto(h_{0}(x):\dots:h_{N}(x))

(this is the embedding defined by the complete linear system $|2C|=|\mathcal{O}_{X^{\prime}}(2)|$ ). If $g_{j}=\sum a_{jk}h_{k}$ for each $j$ , $0\leq j\leq 2n$ , then the matrix $\|a_{jk}\|$ defines a rational mapping $p\colon X_{1}\dasharrow X$ , induced by a linear projection $\bar{p}\colon\mathbb{P}^{N}\dasharrow\mathbb{P}^{2n}$ . Hence,

(7)

\deg X\leq\frac{\deg X_{1}}{\deg p},

and the equality is attained if and only if the rational mapping $p$ is regular.

On the other hand, $\deg X_{1}=4\deg X^{\prime}=4n$ , $\deg X=2n$ , and $\deg p\geq 2$ since the restriction of $p$ to $V\subset X^{\prime}\cong X_{1}$ coincides with our ramified covering $\pi\colon V\to U$ . Now it follows from (7) that the projection $p$ is regular (i.e., its center does not intersect $X_{1}$ ) and $\deg p=2$ .

If $X^{\prime}\subset\mathbb{P}^{n+1}$ is not a cone (i.e., if $X^{\prime}\neq F_{n,0}$ ), put $X_{2}=X_{1}$ and $q=p$ , and if $X^{\prime}$ is the cone $F_{n,0}$ , put $X_{2}=\mathbb{F}_{n}$ and $q=p\circ\sigma$ , where $\sigma\colon\mathbb{F}_{n}\to F_{n,0}=X^{\prime}$ is the standard resolution. So, we have a holomorphic mapping $q\colon X_{2}\to X$ , $\deg q=2$ . One has either $X_{2}\cong\mathbb{F}_{n}$ , or $X_{2}\cong\mathbb{P}^{2}$ (the latter case is possible only if $n=4$ and $X^{\prime}\subset\mathbb{P}^{5}$ is the Veronese surface).

Since $q$ agrees with $\pi$ on a neighborhood of the curve $C^{\prime}\subset X_{2}$ , the curve $C\subset X$ is contained in the branch divisor of $q$ . Let us show that the branch divisor of $q\colon X_{2}\to X$ coincides with $C$ .

To that end, denote this branch divisor by $B\subset X$ . Let $D\subset X$ be a general hyperplane section, and put $D_{2}=q^{-1}(D)\subset X_{2}$ . For a general $D$ , one has $D\cong\mathbb{P}^{1}$ , $D_{2}$ is a smooth and connected projective curve, and the morphism $q|_{D_{2}}\colon D_{2}\to\mathbb{P}^{1}\cong D$ is ramified over $\deg B$ points.

If $n=4$ and $X^{\prime}$ is the Veronese surface $V\subset\mathbb{P}^{5}$ , then $(X_{2},\mathcal{O}_{X_{2}}(D))\cong(\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{2}}(4))$ , so the curve $D_{2}$ is isomorphic to a smooth plane quartic. Such a curve does not admit a mapping to $\mathbb{P}^{1}$ of degree $2$ , so this case is impossible.

Thus, $X^{\prime}=F_{k,l}$ , where $k+2l=n$ . In the notation of Section 2, the divisor $D_{2}\subset X_{2}$ is equivalent to $2(e+(k+l)f)$ ; since the canonical class $K_{X_{2}}$ of the surface $X_{2}\cong\mathbb{F}_{k}$ is equivalent to $-2e-(k+2)f$ , one has, denoting the genus of $D_{2}$ by $g$ ,

2g-2=(D_{2}\cdot D_{2}+K_{X_{2}})=(2e+2(k+l)f\cdot(k+2l-2)f)=2(n-2),

whence $g=n-1$ . Applying Riemann–Hurwitz formula to the degree $2$ morphism $q|_{D_{2}}\colon D_{2}\to D$ , one sees that the number of its branch points equals $2n$ . So, $\deg B=2n$ . Since $B\supset C$ and $\deg C=2n$ , one has $B=C$ .

Observe now that $X\setminus C$ is simply connected since $X\setminus C$ , as a topological space, is a fiber bundle over $\mathbb{P}^{1}$ with the fiber $\mathbb{C}$ , and the complement $X_{2}\setminus q^{-1}(C)$ is connected. Applying Lemma 5.2 to the mapping $q\colon X_{2}\to X$ , one sees that $\deg q=1$ . We arrived at a contradiction. ∎

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