Analytic Compactifications of C^2 part I - curvettes at infinity
aa r X i v : . [ m a t h . AG ] J u l ANALYTIC COMPACTIFICATIONS OF C PART I - CURVETTES ATINFINITY
PINAKI MONDAL
Abstract.
We study normal analytic compactifications of C and describe their singularitiesand configuration of curves at infinity, in particular improving and generalizing results of [Bre73].As a by product we give new proofs of Jung’s theorem on polynomial automorphisms of C andRemmert and Van de Ven’s result that P is the only smooth analytic compactification of C for which the curve at infinity is irreducible. We also give a complete answer to the question ofexistence of compactifications of C with prescribed divisorial valuations at infinity. In partic-ular, we show that a valuation on C ( x, y ) centered at infinity determines a compactification of C iff it is positively skewed in the sense of [FJ04].Nous ´etudions les compactifications analytiques normales de C et d´ecrire leurs singularit´es etla configuration des courbes `a l’infini, particulierment am´elioratant et g´en´eralisant les r´esultatsde [Bre73]. Comme un sous-produit, nous donnons de nouvelles preuves du th´eor`eme de Jung surautomorphismes polynomiaux de C et le r´esultat de Remmert et Van de Ven que P est le seulcompactifi´e analytique lisse C pour lequel la courbe `a l’infini est irr´eductible. Nous donnonsaussi une r´eponse compl`ete `a la question de l’existence de compactifications de C avec desvalorisations divisoriels prescrites `a l’infini. En particulier, nous montrons qu’une ´evaluation sur C ( x, y ) centr´ee `a l’infini d´etermine une compactification de C ssi il est positivement asym´etrique dans le sens de [FJ04]. Introduction
The topic of this article is compact normal analytic surfaces containing C , henceforth to becalled simply compactifications (of C ). Compactifications of C , being one of the most naturaland simplest classes of compact surfaces, have been the subject of numerous articles, see e.g.[RvdV60], [Mor72], [Bre73], [Bre80], [BDP81], [MZ88], [Fur97], [Oht01], [Koj01], [KT09], [FJ11].In particular, Kodaira (as part of his classification of surfaces), and independently Morrow [Mor72]showed that every nonsingular compactification of C is rational (i.e. bimeromorphic to P ) andcan be obtained from P or some Hirzebruch surface via a sequence of blow-ups and blow-downs.In this article we initiate a program to study these compactifications via studying the curvettes atinfinity - these are germs of curves which are transversal to a curve at infinity (i.e. a curve lyingon the complement of C ). We analyze parametrizations of images of these curvettes under thebimeromorphic correspondence to P and use them in two different ways: • To study singularities of the compactifications and of the curves at infinity (Sections 4, 5). • To study existence of a compactification such that the orders of vanishing along curves atinfinity is a prescribed collection of discrete valuations on C ( x, y ) (Section 6).In Part II [Mon13] of this article we use the tools developed here to completely classify compacti-fications of C with one (irreducible) curve at infinity. In a subsequent work we plan to emulatethis technique to study more general Moishezon surfaces (i.e. analytic surfaces which are bimero-morphic to algebraic surfaces).Our first main result is a description of singularities of compactifications of C and configurationof the curves at infinity. We call a compactification minimal if none of the irreducible componentsof the curve at infinity can be (analytically) contracted . Mathematics Subject Classification.
Key words and phrases.
Compactifications of C , curvettes, polynomial automorphisms, discreet valuations. Note that a minimal compactification of C may not be a minimal surface, see Example 5.2 Theorem 1.1.
Let ¯ X be a normal analytic compactification of C . Assume that ¯ X \ C has k irreducible components C , . . . , C k . Let Sing( ¯ X ) be the set of singular points of ¯ X .(1) | Sing( ¯ X ) | ≤ k .(2) ¯ X has at most one singular point which is not sandwiched .(3) (a) For each j , ≤ j ≤ k , C j has an open set isomorphic to C ; in particular, it has at mostone singular point.(b) There is at most one j such that C j has a singular point which is not in C i ∩ C j for some i = j . Moreover, if Q is such a point on C j , then ¯ X is also singular at Q and S i = j C i iscontractible; in particular, if in addition k ≥ , then ¯ X is not minimal.(4) Assume ¯ X is a minimal compactification of C . Then | Sing( ¯ X ) | ≤ k + 1 . Moreover, there isa point P ∈ ¯ X such that(a) C i ∩ C j = { P } for all i, j , ≤ i < j ≤ k .(b) C i \ { P } ∼ = C for each i .(c) (cid:12)(cid:12) Sing( ¯ X ) \ { P } (cid:12)(cid:12) ≤ k .(d) every point in Sing( ¯ X ) \ { P } is a cyclic quotient singularity. P Figure 1.
Configuration of curves at infinity on a minimal compactification
Remark 1.2. (a) Both of the upper bounds for | Sing( ¯ X ) | of Theorem 1.1 are sharp (see Examples 5.3 and 6.1).Example 5.2 shows that the lower bound for | Sing( ¯ X ) | in both cases is zero.(b) Let Q be a singular point of some C j . Assertion (3a) implies that C j has a totally extraor-dinary singularity at Q in the language of [Bre73]. Consequently, assertion (3) improves andgeneralizes the main result of [Bre73].We prove Theorem 1.1 essentially via combinatorial arguments stemming from a careful studyof the dual graphs of resolution of singularities of compactifications of C . The resolution ofsingularities of a compactification of C is on the other hand intimately related to the resolutionof singularities of generic curvettes at infinity associated to each irreducible curve at infinity. Astudy of this relation leads us to the second main result (Theorem 4.5) in which we give an ex-plicit description of the dual graph of minimal resolution of singularities of compactifications of C which are primitive , i.e. for which the curve at infinity is irreducible. As a by product of thisdescription we give new proofs of Jung’s theorem on polynomial automorphisms of C (Corollary4.8), and Remmert and Van de Ven’s result that P is the only smooth analytic compactificationof C for which the curve at infinity is irreducible (Corollary 4.6).A motivation for the work on this article was to understand divisorial valuations centered atinfinity on C [ x, y ] - each of these is the order of vanishing along some curve at infinity on somecompactification of C . However, these valuations can be explicitly described without resortingto any compactification, e.g. by a finite generating sequence [Spi90b] of polynomials, or a (finite)sequence of key polynomials [Mac36], or by a Puiseux polynomial (i.e. a Puiseux series with finitelymany terms) in x − or y − [FJ04, Chapter 4]. The most basic question in this context is: An analytic surface Y has a sandwiched singularity at a point P if there are proper bimeromorphic maps U ′′ → U → U ′ where U is a neighborhood of P in Y and U ′ , U ′′ are (open subsets of) non-singular surfaces[Spi90a, Remark 1.12]. Sandwiched singularities are rational [Lip69, Proposition 1.2]. Except for assertion (3b), the proof of all assertions of Theorem 1.1 requires only the background materialpresented in Section 2.4. The proof of assertion (3b) uses Corollary 4.9 which in turn uses Lemma 3.11.
NALYTIC COMPACTIFICATIONS OF C PART I - CURVETTES AT INFINITY 3
Question 1.3.
Assume that we have explicit algebraic description (e.g. in one of the equivalentways mentioned above) of divisorial valuations ν , . . . , ν k on C [ x, y ]; in other words, assume thatfor all polynomials f ∈ C [ x, y ], we have explicit recipes to compute ν j ( f ), 1 ≤ j ≤ k . Determineif there exists a compactification ¯ X of C such that the ν j ’s are precisely the order of vanishingalong the curves at infinity on ¯ X .Question 1.3 is about the existence of a geometric ‘model’ underlying some algebraic data. Itfollows that the answer should involve interpretation of relevant geometric objects in terms of theinput data. Indeed, if ν is a divisorial valuation on C [ x, y ] associated to a curve C at infinityon some compactification ¯ X of C , then the key polynomials of ν can be used to define ‘natural’representatives of generic curvettes at infinity associated to C (see Remark 1.6). Combining thisobservation with Grauert’s characterization of contractible curves ([Gra62], see Theorem 2.12) wegive a complete and explicit answer to Question 1.3. Here we give a formulation of this answer interms of the sequence of key polynomials:Given ν j ’s as in Question 1.3, we may (by a generic linear change of coordinates) choosecoordinates ( x, y ) such that ν j ( x ) < ν j ( x ) ≤ ν j ( y ) for each j . Then set ( u, v ) := (1 /x, y/x ),so that each ν j is non-negative on C [ u, v ] (with ν j ( u ) > g j, = u, ˜ g j, = v, ˜ g j, , . . . , ˜ g j,l j ∈ C [ u, v ] be the sequenceof key polynomials of ν j (or a minimal generating sequence in the terminology of [Spi90b]) withrespect to ( u, v )-coordinates. Pick the smallest positive integer n j,l j such that n j,l j ν j (˜ g j,l j ) is inthe semigroup generated by ν j (˜ g j,s ), 1 ≤ s ≤ l j −
1. Then it follows from the property of keypolynomials that n j,l j ν j (˜ g j,l j ) = P l j − s =0 n j,s ν j (˜ g j,s ) where n j,s are non-negative integers such that n j,s < deg v (˜ g j,s +1 ) / deg v (˜ g j,s ) for 1 ≤ s ≤ l j −
1. Let M be the matrix with entries m ij = d j n j,l j ν i ( u ) − min n j,l j ν i (˜ g j,l j ) , l j − X s =0 n j,s ν i (˜ g j,s ) where d j = deg v (˜ g j,l j ). Theorem 1.4.
The answer to question 1.3 is affirmative iff det( −M ) < . In the special case that k = 1, Theorem 1.4 implies that a valuation ν (centered at infinity on C [ x, y ]) determines a compactification of C iff it is positively skewed in the sense of [FJ04]. As thefirst step to the proof of Theorem 1.4 we study a special case of Question 1.3, where the answeris affirmative and the resulting compactification dominates P : Theorem 1.5.
Assume ν = − deg , where deg is the degree in ( x, y ) coordinates. Also assume(w.l.o.g.) that ν i ’s are mutually non-proportional. Then(1) There exists a projective (in particular, algebraic) compactification ¯ X of C which affir-matively answers Question 1.3.(2) The singular points of ¯ X (if they exist) are sandwiched.(3) The matrix of intersection numbers of the curves at infinity on ¯ X is M − . Remark 1.6 (Interpretation of the matrix M ) . Let ξ be an indeterminate and define˜ g ν j := ˜ g n j,lj j,l j − ξ l j − Y s =0 ˜ g n j,s j,s ∈ C [ u, v, ξ ] ,g ν j := x deg v (˜ g νj ) ˜ g ν j (1 /x, y/x, ξ ) ∈ C [ x, x − , y, ξ ]Then it is straightforward to see that m ij = − ν i ( g ν j ( x, y, ˜ ξ )) for generic ˜ ξ ∈ C . Geometrically these g ν j ( x, y, ˜ ξ )’s define generic curvettes at infinity associated to ν j (see Definition 3.6 and Proposition3.7). PINAKI MONDAL
Remark 1.7.
Theorem 1.5 remains valid if − ν is any weighted degree corresponding to positiveweights for x and y , or even more generally, if ν is the divisorial valuation associated to the curveat infinity on any primitive compactification of C with at worst sandwiched singularities. Thisfollows from essentially the same arguments as in the proof of Theorem 1.5.1.1. Organization.
After presenting some background material in Section 2, we introduce inSection 3 the notion of generic curvettes at infinity on C associated to (irreducible) curves atinfinity on compactifications on C . In Section 4 we describe the dual graph of minimal resolutionof singularities of primitive compactifications of C and as corollaries prove Jung’s theorem onpolynomial automorphisms of C (Corollary 4.8), and Remmert and Van de Ven’s result that P is the only smooth primitive compactification of C (Corollary 4.6). Section 5 contains the proofof Theorem 1.1 and Section 6 contains the proof of Theorems 1.4 and 1.5.1.2. (Un)convention. In this article we make the unconventional choice to parametrize analyticcurves as the parameter approaches infinity (as opposed to zero). We do this because it is moreconvenient for studying the behaviour of analytic curves on C as they approach infinity, andstudying how the ‘order of the growth’ of these parametrizations is affected by change of coordi-nates on C . E.g. if f ∈ C [ x, y ] and L is the line y = ax , in order to measure the order of growthof f | L near infinity, we could say • either parametrize L as t ( t, at ) as t → ∞ and compute the degree in t of f ( t, at ), • or parametrize L as t ( t − , at − ) as t →
0, compute the order in t of f ( t − , at − ), andtake its negative.In this article we chose to adopt the first approach. A consequence of this choice is that instead ofusing the usual Puiseux series (Definition 2.1) in t where terms appear with increasing order in t ,we have to use series in t in which terms appear with decreasing order in t ; we call these descendingPuiseux series (Definition 2.3). As a justification of our choice, we invite the reader to formulateLemma 3.11 (which is a crucial tool in our proof of the results of Section 4) using parametrizationfrom a neighborhood of zero and usual Puiseux series, and to compare the resulting formulationwith ours.1.3. Acknowledgements.
I heartily thank Professor Pierre Milman. This work was done whileI was his post-doc at University of Toronto. It was essentially an attempt to understand some ofhis questions in a simple case and the exposition profited enormously from speaking in his weeklyseminar and from his questions. Very special thanks also go to Dmitry Kerner - his questions forced me to think and formulate the results in geometric and much more understandable terms.Some of the results of this article were announced in [Mon12].2.
Background
Puiseux series.Definition 2.1 (Meromorphic Puiseux series) . A meromorphic Puiseux series in a variable u isa fractional power series of the form P m ≥ M a m u m/p for some m, M ∈ Z , p ≥ a m ∈ C forall m ∈ Z . If all exponents of u appearing in a meromorphic Puiseux series are positive, then it issimply called a Puiseux series (in u ). Given a meromorphic Puiseux series φ ( u ) in u , write it inthe following form: φ ( u ) = · · · + a u q p + · · · + a u q p p + · · · + a l u qlp p ··· pl + · · · where q /p is the smallest non-integer exponent, and for each k , 1 ≤ k ≤ l , we have that a k = 0, p k ≥
2, gcd( p k , q k ) = 1, and the exponents of all terms with order between q k p ··· p k and q k p ··· p k +1 (or,if k = l , then all terms of order > p ··· p l ) belong to p ··· p k Z . Then the pairs ( q , p ) , . . . , ( q l , p l ),are called the Puiseux pairs of φ and the exponents q k p ··· p k , 1 ≤ k ≤ l , are called characteristicexponents of φ . The polydromy order [CA00, Chapter 1] of φ is p := p · · · p l , i.e. the polydromy NALYTIC COMPACTIFICATIONS OF C PART I - CURVETTES AT INFINITY 5 order of φ is the smallest p such that φ ∈ C (( u /p )). Let ζ be a primitive p -th root of unity. Thenthe conjugates of φ are φ j ( u ) := · · · + a ζ jq p ··· p l u q p + · · · + a ζ jq p ··· p l u q p p + · · · + a l ζ jq l u qlp p ··· pl + · · · for 1 ≤ j ≤ p (i.e. φ j is constructed by multiplying the coefficients of terms of φ with order n/p by ζ jn ).We use the standard fact that the field of meromorphic Puiseux series in u is the algebraicclosure of C (( u )): Theorem 2.2.
Let f ∈ C (( u ))[ v ] be an irreducible monic polynomial in v of degree d . Then thereexists a meromorphic Puiseux series φ ( u ) in u of polydromy order d such that f = d Y i =1 ( v − φ i ( u )) , where φ i ’s are conjugates of φ . Definition 2.3 (descending Puiseux Series) . A descending Puiseux series in x is a meromorphicPuiseux series in x − . The notions regarding meromorphic Puiseux series defined in Definition 2.1extend naturally to the setting of descending Puiseux series. In particular, if φ ( x ) is a descendingPuiseux series and the Puiseux pairs of φ (1 /x ) are ( q , p ) , . . . , ( q l , p l ), then φ has Puiseux pairs( − q , p ) , . . . , ( − q l , p l ), polydromy order p := p · · · p l , and characteristic exponents − q k / ( p · · · p k )for 1 ≤ k ≤ l .We use descending Puiseux series via the following result, which is an immediate corollary ofTheorem 2.2. Corollary 2.4.
Let ( x, y ) be a system of (polynomial) coordinates on X = C . Embed X ֒ → P via the map ( x, y ) [1 : x : y ] . Let P = [0 : a : b ] be a point at infinity and γ be the germof an analytic curve at P . Assume a = 0 and γ is not the germ of the line at infinity. Thenin ( x, y ) -coordinates γ has a parametrization of the form t ( t, φ ( t )) , | t | ≫ , where φ ( t ) is adescending Puiseux series in t . Divisorial discrete valuations.
Let σ : Y ′ Y be a bimeromorphic correspondence ofnormal complex algebraic surfaces and C be an irreducible analytic curve on Y ′ . Then the localring O Y ′ ,C of C on Y ′ is a discrete valuation ring. Let ν be the associated valuation on the field K of meromorphic functions on Y ′ ; in other words ν is the order of vanishing along C . We saythat ν is a divisorial discrete valuation on K ; the center of ν on Y is σ ( C \ S ), where S is theset of points of indeterminacy of σ (the normality of Y ensures that S is a discrete set, so that C \ S = ∅ ). Moreover, if U is an open subset of Y , we say that ν is centered at infinity with respectto U iff σ ( C \ S ) ⊆ Y \ U . The following result, which connects Puiseux series and divisorialdiscrete valuations, is a reformulation of [FJ04, Proposition 4.1]. Theorem 2.5.
Let P ∈ σ ( C \ S ) . Assume Y is non-singular at P . Let ( u, v ) be an analyticsystem of coordinates on a neighborhood U of P such that ν ( u ) > . Then there is a Puiseuxpolynomial (i.e. a Puiseux series with finitely many terms) φ ν ( u ) (unique up to conjugacy) in u and a (unique) rational number r ν > deg u ( φ ν ) such that for every f ∈ C [[ u, v ]] , ν ( f ( u, v )) = ν ( u ) ord u ( f ( u, φ ν ( u ) + ξu r ν )) , (1) where ξ is an indeterminate. Remark 2.6 (Geometric interpretation of φ ν ( u ) + ξu r ν ) . If Q is a generic point of C ∩ σ − ( U )such that both Y ′ and C are non-singular at Q , and D is an irreducible analytic curve on Y ′ whichintersects C transversally at Q , then near σ ( Q ) the (possibly singular) curve σ ( D ) has a Puiseuxparametrization of the form v = φ ν ( u ) + ξ ′ u r ν + h.o.t., where ξ ′ ∈ C is generic, and h.o.t. denotes‘higher order terms’ (in u ). See Proposition 2.10, assertion 3c for a more precise statement.Combining Theorem 2.5 with Corollary 2.4 yields: PINAKI MONDAL
Corollary 2.7.
Retain the notations and assumptions of Theorem 2.5. Assume moreover thatthere exists an open subset U of Y such that(1) ν is centered at infinity with respect to U .(2) there are analytic coordinates ( x, y ) on U such that ( u, v ) = (1 /x, y/x ) .Then there is a descending Puiseux polynomial (i.e. a descending Puiseux series with finitely manyterms) φ ν ( x ) (unique up to conjugacy) in x and a (unique) rational number r ν < ord x ( φ ν ) suchthat for every f ∈ C [ x, y ] , ν ( f ( x, y )) = ν ( x ) deg x ( f ( x, φ ν ( x ) + ξx r ν )) , (2) where ξ is an indeterminate. Definition 2.8.
In the situation of Corollary 2.7, we say that ψ ν ( x, ξ ) := φ ν ( x ) + ξx r ν is the generic descending Puiseux series of ν . Moreover, if Y ′ is a surface bimeromorphic to Y and C ⊆ Y ′ is a curve such that ν is the order of vanishing along C , then we also say that ψ ν ( x, ξ ) isthe generic descending Puiseux series associated to C . Definition 2.9 (Formal Puiseux pairs of generic descending Puiseux series) . Let ν and ψ ν ( x, ξ ) = φ ν ( x ) + ξx r ν be as in Definition 2.8. Let the Puiseux pairs of φ ν be ( q , p ) , . . . , ( q l , p l ). Express r ν as q l +1 / ( p · · · p l p l +1 ), where p l +1 ≥ q l +1 , p l +1 ) = 1. Then the formal Puiseux pairs of ψ ν are ( q , p ) , . . . , ( q l +1 , p l +1 ), with ( q l +1 , p l +1 ) being the generic formal Puiseux pair. The formal polydromy order of ψ ν is p := p · · · p l +1 .2.3. Key polynomials (and generating sequences).
In addition to Puiseux series, divisorialdiscrete valuations centered at a non-singular point on a surface can also be described in terms of a(finite) generating sequence (in the terminology of [Spi90b]) or a (finite) sequence of key polynomi-als (in the terminology of [Mac36]). In this article we use key polynomials; regarding generatingsequences, we only point out that every sequence of key polynomials contains a generating se-quence [FJ04, Remark 2.31].Consider the setting of Theorem 2.5. The key polynomials of ν with respect to ( u, v )-coordinatesis a finite sequence of polynomials ˜ g = u, ˜ g = v, ˜ g , . . . , ˜ g l ∈ C [ u, v ]. We refer to [FJ04, Section2.1] or [Mac36] for their defining properties. The following proposition is the compilation of allproperties of key polynomials that we use. Proposition 2.10.
Let U be an open neighborhood of P such that ( u, v ) defines a system ofcoordinates on U .(1) For each j ≥ , ˜ g j is of the form ˜ g j ( u, v ) = ( v − a ) d j + u ˜ h j ( u, v ) where a ∈ C and ˜ h j ∈ C [ u, v ] with deg v (˜ h j ) < d j (where deg v denotes the degree in v ).In particular, ˜ g j is monic in v of degree d j . Moreover, d j +1 /d j is an integer for each j , ≤ j ≤ l − .(2) For each j ≥ , ˜ g j is irreducible as an element in C [[ u ]][ v ] .(3) Let n l be the smallest positive integer such that n l ν (˜ g l ) is in the semigroup generated by ν (˜ g ) , . . . , ν (˜ g l − ) . Then(a) There exist (unique) non-negative integers n , . . . , n l − such that n j < d j +1 /d j for ≤ j ≤ l − and n l ν (˜ g l ) = P l − j =0 n j ν (˜ g j ) .(b) Let ξ be an indeterminate. Define ˜ g ν ( u, v, ξ ) := ˜ g n l l − ξ Q l − j =0 ˜ g n j j ∈ C [ u, v, ξ ] . Thenthere exists a non-empty open disc ˜∆ ⊆ C such that for all ˜ ξ ∈ ˜∆ , the strict transformof the curve { ˜ g ν ( u, v, ˜ ξ ) = 0 } ⊆ U on σ − ( U ) intersects C transversally at a singlepoint.(c) Let φ ν ( u ) + ξu r ν be as in (1) . Then for all ˜ ξ ∈ ˜∆ , ˜ g ν ( u, v, ˜ ξ ) is irreducible in C [[ u ]][ v ] and has a root v = ˜ φ ( u ) where ˜ φ ( u ) is a Puiseux series in u of the form ˜ φ ( u ) = φ ν ( u ) + ˜ ξ /n l u r ν + h.o.t. NALYTIC COMPACTIFICATIONS OF C PART I - CURVETTES AT INFINITY 7
Example 2.11.
Assume σ : Y ′ → Y is the minimal resolution of the singularity of the germ of v − u = 0 at the origin, and C ⊂ Y ′ is the last exceptional curve. Then key polynomials are u, v .Moreover, ν ( u ) = 3 and ν ( v ) = 2. Proposition 2.10 in this case simply says that for generic ˜ ξ ∈ C ,the strict transform of the germ of v − ˜ ξu = 0 at the origin is transversal to C . Similarly, assume σ is the minimal resolution of the singularity at the origin of the curve ( v − u ) − u v = 0, and C ⊂ Y ′ is the last exceptional curve. Then key polynomials are u, v, v − u . Moreover, ν ( u ) = 6, ν ( v ) = 4, ν ( v − u ) = 13, and Proposition 2.10 says that for generic ˜ ξ ∈ C , the strict transformof the germ of ( v − u ) − ˜ ξu v = 0 at the origin is transversal to C .2.4. Theory of surfaces.
In this section we compile some facts from bimeromorphic geometryof analytic surfaces. We start with Grauert’s criterion for (analytic) contractibility of curves:
Theorem 2.12 ([Gra62]) . Let Y be a smooth complex analytic surface. Let C , . . . , C n be irre-ducible curves on Y and C := C ∪ · · · ∪ C n . The following are equivalent:(1) The matrix of intersection numbers ( C i , C j ) is negative definite.(2) There exists a morphism f : Y → Z such that Z is a normal complex analytic surface, f ( C ) is a finite set of points and f | Y \ C : Y \ C → Z \ f ( C ) is an isomorphism. It is a standard fact that singularities of complex analytic surfaces can be resolved. The singularsurfaces Y ′ we encounter in this article are normal and they come equipped with a bimeromorphiccorrespondence σ : Y ′ Y , where Y is a non-singular projective surface. In this case theresolution of singularities of Y is easy to describe: Theorem 2.13.
Let σ := σ and Y := Y . Algorithm 2.14 stops after finitely many steps with abimeromorphic correspondence σ k : Y ′ Y k . Moreover, σ − k : Y k → Y ′ is a holomorphic mapand is a resolution of singularities of Y ′ . Algorithm 2.14 (Resolution of singularities of Y ′ ) . Assume σ i : Y ′ Y i has been defined for i ≥
0. If σ i does not contract any curve of Y ′ , then stop. Otherwise pick an irreducible curve C ′ on Y ′ which gets contracted to a point P ∈ Y i . Let Y i +1 be the blow up of Y i at P and σ i +1 : Y ′ Y i +1 be the induced bimeromorphic correspondence. Now repeat.We also use the well known fact that every compactification of C is an algebraic space , i.e. ananalytic surface for which the field of meromorphic functions has transcendence degree 2: Theorem 2.15 ([Mor72]) . Let ¯ X be a normal analytic compactification of C . Then ¯ X is an algebraic space . In particular, the identity map between C and one of the affine coordinate chartsof P extends to a bimeromorphic correspondence of analytic varieties. Dual graph of the resolution of curve singularities.Definition 2.16.
Let E , . . . , E k be non-singular curves on a (non-singular) surface such thatfor each i = j , either E i ∩ E j = ∅ , or E i and E j intersect transversally at a single point. Then E = E ∪ · · · ∪ E k is called a simple normal crossing curve . The (weighted) dual graph of E is aweighted graph with k vertices V , . . . , V k such that • there is an edge between V i and V j iff E i ∩ E j = ∅ , • the weight of V i is the self intersection number of E i .Usually we will abuse the notation, and label V i ’s also by E i .We recall the description of the dual graph of the exceptional divisor of the resolution of anirreducible plane curve singularity following [BK86, Section 8.4]. Assume that we are given ananalytically irreducible curve singularity (at a non-singular point of a surface) with Puiseux pairs(˜ q , ˜ p ) , . . . , (˜ q m , ˜ p m ). Then the dual weighted graph for the minimal resolution of the singularityis as in figure 2, where we denoted the ‘last exceptional divisor’ by E ∗ and the ‘left-most’ t verticesby E , . . . , E t (and left all other vertices untitled). The weights u ji and v ji satisfy: u i , v i ≥ u ji , v ji ≥ j >
0, and are uniquely determined from the continued fractions (see, e.g. [MN05,
PINAKI MONDAL − u E − u t − u − − u − v r − v − v E t − u t m − m − − u m − − u m − v r m − m − − v m − − v m − − u t m m − E ∗ − v r m m − v m − v m Figure 2.
Dual graph for the minimal resolution of singularities of an irreducibleplane curve-germSection 2.2]):˜ p i q ′ i = u i − u i −
1. . . − u tii , q ′ i ˜ p i = v i − v i −
1. . . − v rii , where q ′ i := ( ˜ q if i = 1˜ q i − ˜ q i − ˜ p i otherwise.(3)Note that ( q ′ , ˜ p ) , . . . , ( q ′ l , ˜ p l ) are called the Newton pairs of the curve branch, and the Puiseuxseries of the branch can be expressed in the following form: ψ ( u ) = · · · + u q ′ p ( a ′ + · · · + u q ′ p p ( a ′ + · · · + u q ′ p p p ( · · · ))) . Generic curvettes at infinity
Notation 3.1.
Throughout the rest of the article we use X to denote C with coordinate ring C [ x, y ] and ¯ X ( x,y ) to denote copy of P such that X is embedded into ¯ X ( x,y ) via the map ( x, y ) [1 : x : y ]. We also denote by L ∞ the line at infinity ¯ X ( x,y ) \ X , and by Q y the point of intersectionof L ∞ and (closure of) the y -axis. Definition 3.2. An irreducible analytic curve germ at infinity on X is the image γ of an analyticmap η from a punctured neighborhood ∆ ′ of the origin in C to X such that | η ( s ) | → ∞ as | s | → η is analytic on ∆ ′ and has a pole at the origin). Let ¯ X be an analyticcompactification of X . Theorem 2.15 implies that there is a unique point P ∈ ¯ X \ X such that | η ( s ) | → P as | s | →
0. We call P the center of γ on ¯ X , and write P = lim ¯ X γ . Let ¯ X ( x,y ) be asin Notation 3.1. Assume lim ¯ X ( x,y ) γ = Q y . Then Corollary 2.4 implies that for | t | ≫ γ has aparametrization of the form θ : t ( t, φ ( t )), where φ ( t ) is a descending Puiseux series in t . Wecall θ a descending Puiseux parametrization of γ . Example 3.3.
Note that if lim ¯ X ( x,y ) γ = Q y , then γ might not have descending Puiseux parametriza-tion. Indeed, let γ be the curve-germ at infinity on X corresponding to the germ of the (closureof the) y -axis at Q y . Then there is no descending Puiseux series φ ( t ) in t such that γ has aparametrization of the form t ( t, φ ( t )) for | t | ≫ X be a normal analytic compactification of X and C be an irreducible componentof the curve at ¯ X ∞ := ¯ X \ X at infinity on ¯ X . Theorem 2.15 implies that the identity map of X induces a bimeromorphic correspondence σ ( x,y ) : ¯ X ¯ X ( x,y ) . Let S be the set of points ofindeterminacy of σ ( x,y ) . Since ¯ X is normal, it follows that S is a finite set. After a linear changeof coordinates of C [ x, y ], we may ensure that ¯ X satisfies the property ( C ( x,y ) )for every irreduciblecurve C ⊆ ¯ X \ X : σ ( x,y ) ( C \ S ) = { Q y } (i.e. either σ ( x,y ) does not contract C , or it contracts C to some point other than Q y ).( C ( x,y ) ) NALYTIC COMPACTIFICATIONS OF C PART I - CURVETTES AT INFINITY 9
Remark-Notation 3.4.
Note that if C is an irreducible curve in ¯ X \ X and ν is the order ofvanishing along C , then¯ X satisfies ( C ( x,y ) ) ⇐⇒ σ ( x,y ) ( C \ S ) = { Q y }⇐⇒ y/x restricts to a regular function on a non-empty open set of C ⇐⇒ ν ( y/x ) ≥ ⇐⇒ ν ( x ) ≤ ν ( y ) . Pick P ∈ σ ( x,y ) ( C \ S ) \ { Q y } ⊆ L ∞ . Let γ be an irreducible curve-germ at infinity on X withlim ¯ X ( x,y ) γ = P . Let P γ := lim ¯ X γ ∈ ¯ X and ¯ γ ¯ X := γ ∪ { P γ } be the closure of γ in ¯ X . We say that γ is a curvette at infinity associated to C iff P γ ∈ C and ¯ γ ¯ X intersects C transversally at P γ (inparticular, P γ is a non-singular point of both C and ¯ γ ¯ X ). We say that γ is a generic curvette atinfinity associated to C if furthermore P γ is a generic point of C . Proposition 3.5 (Parametrizations of generic curvettes at infinity) . Let γ be a generic curvetteat infinity associated to C and let t ( t, φ ( t )) be a descending Puiseux parametrization of γ .(1) There is a unique rational number r and a finite set E ⊆ C such that if ˜ γ is a curvette atinfinity on X , then lim ¯ X ˜ γ ∈ C \ E iff ˜ γ has a descending Puiseux parametrization of theform t ( t, ˜ φ ( t )) such that deg t ( ˜ φ ( t ) − φ ( t )) = r .(2) Let ˜ γ be a curvette at infinity on X with a descending Puiseux parametrization of the form t ( t, ˜ φ ( t )) such that deg t ( ˜ φ ( t ) − φ ( t )) ≤ r . Write φ − ˜ φ = ˜ ξx r + l.o.t. where ˜ ξ ∈ C . Then(a) lim ¯ X ˜ γ depends only on ˜ ξ . In particular, for generic values of ˜ ξ , lim ¯ X ˜ γ is a genericelement of C .(b) lim ¯ X ˜ γ is a non-singular point of ¯˜ γ ¯ X iff there are no characteristic exponents of ˜ φ smaller than r .(c) for all but finitely many values of ˜ ξ , ˜ γ is a curvette at infinity associated to C iffeither (and therefore, both!) of the properties of assertion 2b is satisfied.(3) Let [ φ ] >r ( x ) be the descending Puiseux polynomial in x obtained by removing from φ ( x ) allterms with degree ≤ r and define ψ ( x, ξ ) := [ φ ] >r ( x ) + ξx r , where ξ is an indeterminate.Then ψ ( x, ξ ) is precisely the generic descending Puiseux series of ν .Proof. The relation between (generic) descending Puiseux series and key polynomials of a valuationis given by assertion 3c of Proposition 2.10. Proposition 3.5 follows from interpreting the propertiesof key polynomials compiled in Proposition 2.10 in terms of the associated descending Puiseuxseries. (cid:3)
Set ( u, v ) := (1 /x, y/x ) and let U be the coordinate chart of ¯ X ( x,y ) with coordinates ( u, v ).Consider the situation of Corollary 2.7 with σ = σ ( x,y ) . Definition 3.6.
Let ˜ g , . . . , ˜ g l ∈ C [ u, v ] be the sequence of key polynomials of ν with respect to( u, v )-coordinates. Set g i := ( x if i = 0 ,x deg v (˜ g i ) ˜ g i (1 /x, y/x ) othewise.For each i ≥ g i ∈ C [ x, x − , y ] and it is monic in y . We call g i ’s the sequence of key forms of ν with respect to ( x, y )-coordinates. Finally, let n , . . . , n l be as in Proposition 2.10. Then define g ν ( x, y, ξ ) := x deg v (˜ g ν ) ˜ g ν (1 /x, y/x, ξ ) = g n l l − ξx n ′ l − Y j =1 g n j j ∈ C [ x, x − , y, ξ ]where n ′ = n l deg v ( g l ) − n − P l − j =1 n j deg v ( g j ). We call g ν ( x, y, ξ ) the generic key form of ν in( x, y )-coordinates. The use of the term ‘curvette’ to denote germs of transversal curves at smooth points of a given curve is dueto Deligne [sga73].
Proposition 3.7 (Affine equations of generic curvettes at infinity) . Pick n ≥ such that x n g ν ∈ C [ x, y, ξ ] . For all ˜ ξ ∈ C , let Z ˜ ξ be the closure in ¯ X ( x,y ) of the curve { x n g ν ( x, y, ˜ ξ ) = 0 } ⊆ X .(1) For each ˜ ξ ∈ C , Z ˜ ξ intersects L ∞ \ { Q y } at a single point Q ˜ ξ .(2) For generic ˜ ξ ∈ C , the germ of Z ˜ ξ in a punctured neighborhood of Q ˜ ξ is a curvette atinfinity associated to C .(3) Z ˜ ξ intersects L ∞ at Q y with intersection multiplicity n .Proof. Assertions (1) and (3) follow from assertion 1 of Proposition 2.10, and assertion 2 followsfrom assertion (3b) of Proposition 2.10. (cid:3)
Example 3.8.
Let ¯ X = ¯ X ( x,y ) and C = L ∞ (so that ν is the negative of degree in ( x, y )-coordinates). Then the key forms are x, y , and the generic key form is y − ξx . Propostion 3.7 inthis case simply states (the obvious fact) that for y − ˜ ξx intersects the line L ∞ transversally forgeneric ˜ ξ .3.1. Effect of automorphisms of C on parametrizations of generic curvettes at infinity. Let γ be a curve-germ at infinity on X with a descending Puiseux parametrization t ( t, φ ( t )).In this section we study the effect on deg t ( φ ( t )) of two ‘simple’ types of automorphisms of theplane described below; the (simple) observations made in this section will be crucial in our proofof Jung’s theorem that these automorphisms generate the full group of polynomial automorphismsof C . Definition 3.9. let F : C [ x, y ] → C [ x, y ] be an automorphism. We call F a Type I automorphism ifit is of the form ( x, y ) ( y, x ) and a Type II automorphism if it is of the form ( x, y ) ( x, y + ax n ),where a ∈ C and n ≥ Lemma 3.10.
Let γ be a curve-germ at infinity on X with a descending Puiseux parametrization t ( t, φ ( t )) and ω := deg t ( φ ( t )) , i.e. φ ( t ) = at ω + l.o.t.for some a ∈ C . Assume ω > .(1) (a) After the type I automorphism ( x, y ) ( y, x ) , γ has a descending Puiseux parametriza-tion t ( t, ˜ φ ( t )) where deg t ( ˜ φ ( t )) = 1 /ω .(b) Moreover, if ω = 1 /n for some integer n ≥ , then the number of Puiseux pairs of ˜ φ ( t ) is one less than the number of Puiseux pairs of φ ( t ) .(2) If ω is a non-negative integer, then after the type II automorphism ( x, y ) ( x, y − ax ω ) , γ has a descending Puiseux parametrization of the form t ( t, φ ( t ) − at ω ) .Proof. Assertions (1a) and (2) are easy to see. Assertion (1b) follows from a straightforwardinduction on the number of Puiseux pairs of φ . (cid:3) Let ¯ X be a compactification of X and C be an irreducible component of the curve at infinityon ¯ X . The following lemma shows that after a composition of finitely many Type I and IIautomorphisms, we can ensure that generic curvettes associated to C have descending Puiseuxparametrizations, and the initial term of these parametrizations has a ‘normal form’. Lemma 3.11.
Let ¯ X and C be as above, and γ be a generic curvette at infinity on X associatedto C . After a finite sequence of Type I and Type II automorphisms of C [ x, y ] , we can ensure that γ has a descending Puiseux parametrization t ( t, φ ( t )) , where φ ( t ) is of the following form: φ ( t ) = ( ˜ ξt r , r ∈ Q , r ≤ , ˜ ξ ∈ C is generic, or a t ω + l.o.t. , a ∈ C \ { } , ω ∈ Q \ ( Z ≥ ∪ { /n : n ∈ N } ) , ω < . (4) Proof.
Since any linear change of coordinates of C [ x, y ] is a composition of Type I and II auto-morphisms, it follows that after composition of finitely many Type I and II automorphisms, wecan ensure that ¯ X satisfies ( C ( x,y ) ), which implies in particular that γ has a descending Puiseuxparametrization t ( t, φ ( t )). Assertion (1a) of Lemma 3.10 then implies that it suffices to prove NALYTIC COMPACTIFICATIONS OF C PART I - CURVETTES AT INFINITY 11 the following statement: after a a finite sequence of automorphisms of C [ x, y ] of types I and II,we can ensure that φ ( t ) is not of the following form: a t ω + l.o.t. , where a ∈ C \ { } , and ω ∈ Z ≥ ∪ { /n : n ∈ Z ≥ } . (!)Indeed, assume φ ( t ) is of the form (!). Then either φ ( t ) = at n + l.o.t. for some polynomial f ( x ) ∈ C [ x ], or φ ( t ) = at /n + l.o.t. for some a = 0 and a positive integer n >
1. In the firstcase apply Type II automorphism ( x, y ) ( x, y − ax n ) and in the second case apply the Type Iautomorphism ( x, y ) ( y, x ). Note that(1) in the second case the number of Puiseux pairs of φ ( t ) decreases by one (assertion (1b) ofLemma 3.10),(2) in the first case the number of Puiseux pairs of φ ( t ) does not change, but deg t ( φ ( t ))decreases (assertion (2) of Lemma 3.10).The above observations imply that this process ends after finitely many steps, as required tocomplete the proof of the lemma. (cid:3) Remark-Definition 3.12.
We say that the initial exponent of φ ( t ) is in the normal form if φ ( t ) is as in (4). Note that φ ( t ) is in the normal form iff either σ ( x,y ) maps C generically on to L ∞ ⊆ ¯ X ( x,y ) (in which case r = 1), or contracts C to the point of intersection of L ∞ and x -axis. Remark 3.13.
With a bit of more work than the proof of Lemma 3.11, it can be shown that thereis a ‘normal form’ for φ ( t ) itself (i.e. not only the initial exponent). In [Mon13] we use this normalform to compute the moduli spaces and groups of automorphisms of algebraic compactificationsof C with one irreducible curve at infinity.4. Primitive compactifications and resolution of their singularities
Definition 4.1.
Let π : ˜ X → ¯ X be a resolution of singularities of a compactification ¯ X of X ∼ = C such that ˜ X \ X is a simple normal crossing curve. The augmented dual graph of π is the dualgraph (Definition 2.16) of ˜ X \ X .Let ¯ X be a (normal analytic) compactification of C which is primitive , i.e. the curve C atinfinity on ¯ X is irreducible. In this section we show that the minimal resolution of singularitiesof ¯ X satisfies the properties of Definition 4.1, and describe its augmented dual graph. As a con-sequence, we derive a new proof of Remmert and Van de Ven’s characterization of P as the onlynon-singular primitive compactification of C and Jung’s theorem on polynomial automorphisms.We continue to adopt Notation 3.1 and assume that ¯ X satisfies ( C ( x,y ) ), i.e. there exists P ∈ σ ( x,y ) ( C \ S ) \ { Q y } . Let the generic descending Puiseux series for C be ψ ( x, ξ ) = φ ( x ) + ξx r = · · · + a x q p + · · · + a x q p p + · · · + a l x qlp p ··· pl + · · · + ξx qlp p ··· pl +1 where ( q , p ) , . . . , ( q l +1 , p l +1 ) are the formal Puiseux pairs (Definition 2.9) of ψ . Then ( u, v ) =(1 /x, y/x ) is a system of coordinate near P , and Proposition 3.5 implies that generic curvettes atinfinity associated to C have Puiseux parametrizations of the form v = · · · + a u ˜ q p + · · · + a u ˜ q p p + · · · + a l u ˜ ql ˜ p p ··· ˜ pl + · · · + ˜ ξu ˜ ql ˜ p p ··· ˜ pl +1 + h.o.t.(5)where (˜ q i , ˜ p i ) = ( p · · · p i − q i , p i ), 1 ≤ i ≤ l + 1, and ˜ ξ is a generic element of C . ApplyAlgorithm 2.14 with σ = σ ( x,y ) to construct a resolution of singularities ˜ σ : ˜ X → ¯ X . Let Γ be thecorresponding augmented dual graph. The following proposition gives a description of Γ in termsof the dual graph of the minimal resolution of the plane curve singularity of the curve germ withPuiseux parametrization (5). Proposition 4.2.
Let E be the strict transform of C . Assume the initial exponent of ψ is in thenormal form (Definition 3.12). Then (1) If ψ ( x, ξ ) = ξx , then ¯ X σ ( x,y ) ∼ = ¯ X ( x,y ) ∼ = P (in particular, ¯ X is non-singular), and Γ consistsof a single vertex E .(2) Otherwise if p l +1 > , then Γ is as in figure 3(a), where Γ ′ is as in figure 2 with m = l + 1 .In particular, ¯ X has at most two singular points, one of them is at worst a cyclic quotientsingularity.(3) Otherwise ( p l +1 = 1 and) Γ is as in figure 3(b), where Γ ′′ is the graph of Figure 2 with m = l and one change - namely the self-intersection number of E ∗ in Γ ′′ is − . Inparticular, ¯ X has at most one singular point. − u E − u E − u − − v − E ∗ = E − v l +1 Γ ′ (a) Case p l +1 > − u E − u E − u − − v − E ∗ − v l Γ ′′ − − − Eq l − q l +1 − vertices (b) Case p l +1 = 1 Figure 3.
Augmented dual graph for the resolution of Algorithm 2.14
Remark 4.3.
Note that the resolution of Proposition 4.2 is not minimal if (and only if) u = 2. Proof of Proposition 4.2.
The first assertion is straightforward. The other assertions follow fromthe discussion in Section 2.5 and the following observations:1. In the scenario of assertion 2, the Puiseux pairs of generic curvettes at infinity associated to C are (˜ q , ˜ p ) , . . . , (˜ q l +1 , ˜ p l +1 ) and Algorithm 2.14 corresponds precisely to resolution of singular-ities of these curvettes at infinity.2. In the scenario of assertion 3, the Puiseux pairs of generic curvettes at infinity associated to C are (˜ q , ˜ p ) , . . . , (˜ q l , ˜ p l ) and Algorithm 2.14 corresponds to at first resolving the singularities ofthese curvettes at infinity, and then q l − q l +1 additional blow-ups.3. The vertex e in Figures 3(a) and 3(b) corresponds to E , which is the strict transform of L ∞ ⊆ ¯ X ( x,y ) . The equation of L ∞ near P is u = 0. On the other hand, the normal form of ψ implies that the order (in u ) of the right hand side of (5) is ˜ q / ˜ p . It follows that strict transformof L ∞ contains the center of precisely the first u -blow ups (where u is defined in (3)). (cid:3) Remark 4.4.
More generally, if ¯ X is an arbitrary normal analytic compactification of C and C is an irreducible curve at infinity on ¯ X , then the arguments from the proof of Proposition 4.2imply that there is a non-singular compactification ˜ X of C dominating ¯ X ( x,y ) ∼ = P such thatthe dual graph of the curve at infinity on ˜ X has the same shape as Γ of figure 3. In particular,contracting all curves at infinity on ¯ X other than E and E results in a compactification ¯ X ∗ withprecisely two irreducible curves E ∗ and E ∗ at infinity, • the bimeromorphic correspondence ¯ X ( x,y ) ¯ X ∗ maps L ∞ dominantly on to E ∗ . • the bimeromorphic correspondence ¯ X ¯ X ∗ maps C dominantly on to E ∗ .This implies that(1) ¯ X ∗ is precisely the compactification guaranteed by assertion 1 of Theorem 1.5 in the casethat k = 2 and ν is the divisorial valuation associated to C .(2) ˜ X is precisely the minimal resolution of singularities of ¯ X ∗ .Moreover, let P ∗ be the point of intersection of E ∗ and E ∗ . We claim that E ∗ \ { P ∗ } ∼ = C . Indeed,this is clear if Γ is as in figure 3(b). On the other hand, if Γ is as in figure 3(a), then it suffices toshow that E ∗ is non-singular at the point Q ∗ to which the curves corresponding to the right-most NALYTIC COMPACTIFICATIONS OF C PART I - CURVETTES AT INFINITY 13 vertical string of Γ contracts. But the singularity at Q ∗ is a cyclic quotient (or Hirzebruch-Jung )singularity, and E is transversal to the string of exceptional divisor of its resolution. It then followsfrom the well known properties of cyclic-quotient singularities (see e.g. [BHPVdV04, Section III.5])that E ∗ does not acquire any singularity at Q ∗ .As mentioned in Remark 4.3, the resolution of singularities of ¯ X constructed in Proposition 4.5may not be minimal. Understanding the minimal resolution of ¯ X requires a more detailed analysisof the change of the initial exponent of a Puiseux series under blow up. This is the content of thenext theorem. Theorem 4.5.
Let the assumptions and notations be as in Proposition 4.2; in particular the initialexponent of ψ is in the normal form, and Γ is as in figure 3(a) if p l +1 > and as in figure 3(b)if p l +1 = 1 .(1) ¯ X is non-singular iff ψ ( x, ξ ) = ξx .(2) Otherwise if q /p > / , then u > and Γ is the augmented dual graph of the minimalresolution of ¯ X .(3) Otherwise let ˜ q := p − q . Then we must have p = ˜ q + r and ˜ q = m r + r for somepositive integers r , m , r with r < r < ˜ q . Moreover, if t is as in Γ (see the ‘leftmost’string of figure 2), then t ≥ m and u m ≥ and u j = 2 for all j , ≤ j < m . Theaugmented dual graph of the minimal resolution of ¯ X is gotten from Γ by deleting all thevertices to the left of e m and changing the weight of e m to − u m + 1 .Proof. The ( ⇐ ) implication of the first assertion follows from Proposition 4.2. Now we assumethat ψ ( x, ξ ) = ξx and show that either 2nd or the 3rd assertion of the theorem is true. Note thatthis will also prove the ( ⇒ ) implication of assertion (1) (since a surface is non-singular iff the dualgraph of the minimal resolution of singularity is non-empty) and complete the proof of the theorem.Since the initial exponent of ψ is in the normal form, it follows that deg x ( ψ ) <
1. We nowdivide our proof based on different possibilities for deg x ( ψ ). For each case we construct the mini-mal resolution ˜ X min of singularities of ¯ X and show that the exceptional divisor of the morphism˜ X min → ¯ X is of the required form. Case 1: deg x ( ψ ) = 1 /n , n ≥ . In this case ψ = ξx /n . Consequently (5) implies that a genericcurvette γ associated to C has Puiseux expansion near P of the form u = ξ ′ v n/ ( n − + h.o.t.for a generic ξ ′ ∈ C . Let ¯ X = ¯ X ( x,y ) , ¯ X , . . . be the sequence of surfaces constructed in theresolution Algorithm 2.14. Then it follows that the strict transform of γ on ¯ X i has a Puiseuxexpansion of the form v i = ξ ′ u n − ii + h.o.t.where ( u i , v i ) := ( u/v, v i /u i − ). In particular, the bimeromorphic correspondence ¯ X ¯ X i maps C to the point ( u i , v i ) = 0 for i < n , and dominantly on to the line u n = 0 (which is preciselythe exceptional divisor of the last blow up) for i = n . It follows that ˜ X = ¯ X n is precisely theresolution of singularity of ¯ X achieved via algorithm 2.14 with the augmented dual graph as infigure 4. − E − E − E n − − E n − nE Figure 4.
Augmented dual graph for resolution when deg x ( ψ ν ) = 1 /n , n ≥ E n is precisely the pre-image of C , it follows that the exceptional divisor of the resolution˜ σ : ˜ X → ¯ X is ˜ E := E ∪ · · · ∪ E n − . Note that ˜ E has two connected components: E and ˜ E := E ∪ E ∪ · · · ∪ E n − . By Castelnuovo’s criterion ˜ X min is formed from ˜ X by contracting ˜ E to a non-singular point. In particular, the exceptional divisor of the minimal resolution ˜ X min → ¯ X is precisely (the isomorphic image of) E . It is straightforward to check that this is precisely theform of the exceptional divisor prescribed by assertion 3 of the theorem. Case 2: > deg x ( ψ ) > / . In this case a generic curvette γ associated to C has Puiseuxexpansion near P of the form u = av α + h.o.t.where a = 0 ∈ C and α >
2. It follows that Algorithm 2.14 requires at least 3 blow ups, andstrict transforms of L ∞ contain the centers of at least the first three blow ups. In particular, theintersection number of the strict transform of L ∞ on ˜ X (which is precisely negative of the labelof the vertex e in figure 3) is ≤ −
2. This implies that all the irreducible curves with support inthe exceptional divisor of ˜ σ : ˜ X → ¯ X has self intersection ≤ −
2. Consequently, ˜ σ is precisely theminimal resolution of singularities of ¯ X and assertion 2 of the theorem holds. Case 3: < deg x ( ψ ) < / , deg x ( ψ ) = 1 /n for all n ∈ Z . The hypothesis of this case impliesthat a generic curvette γ associated to C has Puiseux expansion near P of the form u = av α + h.o.t.where a = 0 ∈ C and 1 < α < α = ( n + 1) /n for all n ≥
1. Note that α = p / ˜ q where˜ q is as in (5). In particular p , ˜ q are integers with no common factors. Let us follow the steps ofthe computation of gcd( p , ˜ q ) = 1 via Euclidean algorithm. The assumptions on α translate tothe following observations: p = ˜ q + r for some r ∈ Z , < r < q , and˜ q = m r + r for some m , r ∈ Z , ≤ m , ≤ r < r . Let the next step of the computation of gcd( p , q ) be r = m r + r for some m > , and 0 ≤ r < r . Then straightforward arguments as in Case 1 shows that after m + m + 1 blow-ups the dualgraph of the union of strict transforms of E i ’s for 1 ≤ i ≤ m + m + 1 on ¯ X m + m +1 is as infigure 5, and the Puiseux expansion for the strict transform γ m + m +1 of γ on ¯ X m + m +1 is givenby: u m + m +1 = a ′ ( v m + m +1 ) r /r + h.o.t. for some a ′ = 0 ∈ C . where ( u m + m +1 , v m + m +1 ) = ( u m m /v m + m m , v m +1 /u m ). Moreover, u m + m +1 = 0and v m + m +1 = 0 are respectively the local equations of the strict transform of E m +1 and E m + m +1 near γ m + m +1 . We divide the rest of the proof for this case into the following twosubcases: − E − E − E m − ( m + 1) E m +1 − E m + m +1 − E m + m − E m +2 − ( m + 2) E Figure 5.
Dual graph of E ∪ · · · ∪ E m + m +1 after m + m + 1 blow-ups NALYTIC COMPACTIFICATIONS OF C PART I - CURVETTES AT INFINITY 15
Subcase 3.1: r = 0 . Since a ′ = 0, this implies that γ m + m +1 does not belong to the stricttransform of E m +1 on ¯ X m + m +1 . It follows from Algorithm 2.14 all the remaining blow-ups forthe construction of ˜ X keep (the strict transforms of) E , . . . , E m +1 unchanged and the dual graphof the exceptional divisor of the morphism ˜ X → ¯ X is of the form as in figure 6(a). Moreover, r = 0 implies that r = gcd( p , q ) = 1, so that m = r ≥
2. The same arguments as in Case 1then show that the dual graph of the exceptional divisor of the minimal resolution ˜ X min → ¯ X isof the form as in figure 6(b). This is precisely the form of the dual graph prescribed by assertion3 of the theorem. − E − E − E m − ( m + 1) E m +1 rest ofthe graph − E m + m − E m +2 − ( m + 2) E (a) Non-minimal resolution − m E m +1 rest ofthe graph − E m + m − E m +2 − ( m + 2) E (b) Minimal resolution Figure 6.
Dual graphs of the exceptional divisor of the resolution of singularitiesof ¯ X for the case r = 0 Subcase 3.2. r > : In this case γ m + m +1 intersects the point P m + m +1 of intersection of E m + m +1 and the strict transform of E m +1 on ¯ X m + m +1 . It follows that the bimeromorphiccorrespondence ¯ X ¯ X m + m +1 maps C to P m + m +1 and therefore Algorithm 2.14 requires atleast one more blow up to construct ˜ X . The dual graph of the union of strict transforms of E i ’sfor 1 ≤ i ≤ m + m + 2 on ¯ X m + m +2 is as in figure 7(a). Also, since r < r , it follows thatthe strict transform of γ on ¯ X m + m +2 does not intersect the strict transform of E m +1 , and thesame reasoning as in Subcase 3.1 then implies that the dual graph of the exceptional divisor ofthe minimal resolution ˜ X min → ¯ X is of the form as in figure 7(b). It is straightforward to checkthat this agrees with assertion 3, which completes the proof of the theorem. (cid:3) − E − E − E m − ( m + 2) E m +1 − E m + m +2 − E m + m +1 − E m + m − E m +2 − ( m + 2) E (a) E ∪ · · · ∪ E m + m +2 after m + m + 2 blow-ups − ( m + 1) E m +1 rest ofthe graph − E m + m − E m +2 − ( m + 2) E (b) Exceptional divisor of the mini-mal resolution Figure 7.
Dual graphs for the case r > Corollary 4.6 ([RvdV60]) . Up to an (analytic) isomorphism P is the only smooth primitivecompactification of C .Proof. This follows from combining the first assertions of Theorem 4.5 and Proposition 4.2. (cid:3)
Remark 4.7.
In [RvdV60] Remmert and Van de Ven essentially proved that compactificationsof C are algebraic spaces, i.e. Theorem 2.15 (which is essentially the point of departure of thisarticle), and then used it to prove the result of Corollary 4.6 by arguments different from ours.Our proof of Corollary 4.6 therefore is in fact a new proof of the implication “Theorem 2.15 ⇒ Corollary 4.6.”
Corollary 4.8 ([Jun42]) . The group of C -algebra automorphisms of the ring of complex polyno-mials in two variables is generated by linear automorphisms and triangular automorphisms (i.e.Type II automorphisms of Lemma 3.11).Proof. Let ∆ be the group of C -algebra automorphisms of C [ x ′ , y ′ ] and Σ be the subgroup of∆ generated by linear and triangular automorphisms. Pick F = ( F , F ) ∈ ∆. Set ( u, v ) :=( F ( x ′ , y ′ ) , F ( x ′ , y ′ )). Let ¯ X ∼ = P be the compactification of X := Spec C [ x ′ , y ′ ] ∼ = C via theembedding ( x ′ , y ′ ) [1 : F ( x ′ , y ′ ) : F ( x ′ , y ′ )]. Lemma 3.11 implies that there exists G =( G , G ) ∈ ∆ such that initial exponent of the generic descending Puiseux series ψ ( x, ξ ) of C with respect to ( x, y ) := ( G ( x ′ , y ′ ) , G ( x ′ , y ′ )) coordinates is in the normal form. Since ¯ X isnon-singular, first assertions of Theorem 4.5 and Proposition 4.2 imply that ψ ( x, ξ ) = ξx and G ◦ F − : ¯ X → ¯ X ( x,y ) is an isomorphism. It follows that G ◦ F − is a non-invertible linear mapin ( x, y )-coordinates; in particular, G ◦ F − ∈ Σ. Therefore F ∈ Σ, as required. (cid:3)
The following result is immediate from the arguments of the proof of Theorem 4.5. We will useit in the proof of Theorem 1.1.
Corollary 4.9.
Let ¯ X be a primitive compactification of X . Choose coordinates ( x, y ) on X suchthat the initial exponent of the generic descending Puiseux series ψ ( x, ξ ) of the curve C at infinityis in the normal form. Then one of the following must hold:(1) ¯ X ∼ = P , C ∼ = P .(2) σ − x,y ) : ¯ X ( x,y ) ¯ X contracts L ∞ to a point P ∈ C . In this case(a) either P is a singular point of ¯ X ,(b) or ¯ X is isomorphic to the weighted projective space P (1 , , n ) for some n ≥ , and P is a non-singular point of both ¯ X and C . (cid:3) Singularities and curves at infinity
Definition 5.1.
Let ¯ X be a compactification of C . We say that ¯ X is a minimal compactification (of C ) if none of the curves at infinity can be (analytically) contracted. Example 5.2.
Note that a minimal compactification of C may not be a minimal surface. Indeed,let ¯ X = P . Pick a line L on ¯ X and a point Q ∈ L . Fix k ≥
1. Choose points P , . . . , P k ∈ ¯ X \ L such that the lines L i joining P i and Q , 1 ≤ i ≤ k , are pairwise distinct. Let ¯ X k bethe blow up of ¯ X at P , . . . , P k . Let C i ⊆ ¯ X k be the strict transform of L i , 0 ≤ i ≤ k , and X k := ¯ X k \ S ki =0 C i . Note that X ∼ = C . It then follows by induction that X k ∼ = C (indeed, weneed only the following observation: if Y is the blow up of C at a point P and L is a line through P on C , then the complement in Y of the strict transform of L is also isomorphic to C ). Weclaim that ¯ X k is a minimal compactification of X k . Indeed, the matrix of intersection numbers( C i , C j ) is: I = · · · · · · · · · · · · ... ...1 1 1 1 · · · · · · It then follows from Theorem 2.12 that no C i can be contracted, i.e. ¯ X k is a minimal compactifi-cation of X k . Also note that the configuration of the curves at infinity is as in figure 1. NALYTIC COMPACTIFICATIONS OF C PART I - CURVETTES AT INFINITY 17
Proof of Theorem 1.1.
Let σ : ¯ X ¯ X ∼ = P be the bimeromorphic correspondence induced byidentification of X with C . Algorithm 2.14 shows that a resolution of singularities ˜ σ : ˜ X → ¯ X can be constructed from ¯ X via a sequence of blow-ups ¯ X i +1 → ¯ X i , with ˜ X = ¯ X s for some s ≥ E be the line at infinity on ¯ X , and for each i ≥
1, let E i be the exceptional divisor of the i -th blow-up. Finally, for each i ≥
0, let Γ i be the augmented dual graph at the i -th step, i.e. Γ i isthe dual graph for the union of strict transforms on ¯ X i of E , . . . , E i . Then it is straightforwardto see (see e.g. [Spi90b, Remark 5.5]) that for each i , there are only two possibilities for thetransformation from Γ i to Γ i +1 which are described in figure 8. In particular, it follows that forall i ≥ i is a tree (i.e. every pair of vertices is connected by a unique minimal path).(ii) E i is connected to at most two distinct E j ’s in Γ i ; denote them as E t i and E t ′ i ( t i = t ′ i inthe case of figure 8(a)). (Here we used i > i (resp. ˜Γ ′ i ) be the connected component of Γ s \ { E i } which contains E t i (resp. E t ′ i ).Then the vertex corresponding to E is in ˜Γ i ∪ ˜Γ ′ i (from now we will abuse the notationand simply write E ∈ ˜Γ i ∪ ˜Γ ′ i ). W.l.o.g. we assume that E ∈ ˜Γ i .(iv) Let ˜Γ be a connected component of Γ s \ { E i } which does not contain E t i or E t ′ i and let˜ E := S { E j : E j ∈ ˜Γ } . Then there is a point Q ∈ E i such that ˜ E is precisely the union ofexceptional curves arising from blow-up of Q and points infinitely close to Q . In particular,˜ E can be (analytically) contracted to a non-singular point and the image of E i under thiscontraction is also non-singular. E t i Γ i E t i E i +1 Γ i +1 (a) Possibility 1 E t i Γ i E t ′ i E t i E i +1 Γ i +1 E t ′ i (b) Possibility 2 Figure 8.
Change of the augmented dual graph in ( i + 1)-th step Proof of assertion (1) : ¯ X is constructed from ˜ X by contracting some of the E i ’s. Let E i , . . . , E i k be the non-contracted curves; w.l.o.g. we may assume C j = ˜ σ ( E i j ), 1 ≤ j ≤ k . Observation (iv)can be reformulated as:(v) Fix j , 1 ≤ j ≤ k . Let P ∈ Sing( ¯ X ) ∩ C j . Then one of the following folds:(a) P ∈ C j ∩ C j ′ for some j ′ = j ,(b) i j ≥
1, ˜ σ contracts ˜Γ i j ∋ E , and P = ˜ σ ( E ) = ˜ σ (˜Γ i j ).(c) i j ≥ t i j = t ′ i j , ˜ σ contracts ˜Γ ′ i j , and P = ˜ σ (˜Γ ′ i j ).Define Σ := { i j : ˜ σ contracts ˜Γ i j ∪ ˜Γ ′ i j } (6) S = [ ≤ j 1. If Σ = ∅ , then observation (v) implies that for all j ,1 ≤ j ≤ k , | Sing( ¯ X ) ∩ C j \ S | ≤ 1. It follows that | Sing( ¯ X ) | ≤ k + | S | + 1 ≤ k . On the other hand,if Σ = ∅ , then observations (iii) and (i) imply that ˜ σ contracts E to some point P ∈ ¯ X and P isthe unique point of intersection of all C j such that i j ∈ Σ. Observation (v) then implies that forall j , 1 ≤ j ≤ k , | Sing( ¯ X ) ∩ C j \ ( S ∪ { P } ) | ≤ 1. It follows that | Sing( ¯ X ) | ≤ k + | S ∪ { P }| ≤ k .This completes the proof of assertion (1). Proof of assertion (4) : Since ¯ X is minimal, it follows from Theorem 2.12 that either ˜ σ contracts E or ¯ X ∼ = ¯ X . W.l.o.g. we may assume the former. Consider the surface ¯ X ′ obtained from ˜ X by contracting all curves at infinity other than the strict transforms of C , . . . , C k and the line E at infinity on ¯ X (which is possible e.g. by Theorem 2.12). The bimeromorphic correspon-dences π ′ : ¯ X ′ ¯ X and π : ¯ X ′ ¯ X extend to holomorphic maps. In particular, for each j , 1 ≤ j ≤ k , the strict transform C ′ j of C j on ¯ X ′ is contractible, so that ( C ′ j , C ′ j ) < 0. On theother hand, the minimality assumption on ¯ X and Theorem 2.12 imply that ( C j , C j ) ≥ j ,1 ≤ j ≤ k . Since π ′ | ¯ X ′ \ E ′ (where E ′ is the strict transform of E on ¯ X ′ ) is an isomorphism, itfollows that E ′ intersects each C ′ j , 1 ≤ j ≤ k , so that P := π ′ ( E ′ ) ∈ T kj =1 C j . This, together withobservation (i) above, implies that π ′− ( P ) ∩ C ′ j consists of a single point P ′ j . In particular thisproves assertion (4a) and implies that S ∪ ˜ σ ( E ) = { P } , where S is as in (7). Observation (v) thenimplies that (cid:12)(cid:12) Sing( ¯ X ) \ { P } (cid:12)(cid:12) ≤ k , which is precisely assertion (4c). Now fix j , 1 ≤ j ≤ k , and let π ′ j : ¯ X ′ → ¯ X ∗ j be the contraction of all C ′ i , i = j . Then ¯ X ∗ j is precisely the compactification ¯ X ∗ ofRemark 4.4 for C = C j . Since π ′ j is an isomorphism on a neighborhood of C ′ j \ { P ′ j } , assertions(4b) and (4d) follows from Remark 4.4. Proof of assertion (2) : At first note that if ˜ σ does not contract E , then ¯ X dominates ¯ X , andtherefore all singularities of ¯ X are sandwiched. So assume ¯ σ contracts ˜ E to a point P ∈ ¯ X . Let¯ X ′ be as in the preceding paragraph. Then ¯ X ′ dominates ¯ X , and therefore all the singularitiesof ¯ X ′ are sandwiched. Since ¯ X \ { P } ∼ = ¯ X ′ \ E ′ , this implies assertion (2). Proof of assertion (3) : At first note that if C j is the image of E , then C j ∼ = P (since then thebirational map ¯ X → ¯ X maps C j on to L ∞ ). So assume that E does not map on to C j . Let ¯ X ∗ j be as in the proof of assertion (4) and ˜ π ∗ j : ˜ X → ¯ X ∗ j be the corresponding map. Recall that E i j is the strict transform of C j on ˜ X . Let ˜ Q j be the point of intersection of E i j and ˜Γ i j (where ˜Γ i ’sare as in observation (iii)). Then ˜ π ∗ j ( ˜ Q j ) is precisely the point of intersection of the two curvesat infinity on ¯ X ∗ j . Since the bimeromorphic correspondence ¯ X ¯ X ∗ j restricts to a holomorphicmap on a neighborhood of C j \ ˜ σ ( ˜ Q j ), assertion (3a) follows from Remark 4.4.It remains to prove assertion (3b). Let Q be a singular point of C j such that Q ∈ C j \ S i = j C i .Recall that our proof of Theorem 1.1 started with the choice of an arbitrary compactification ¯ X of X which is isomorphic to P . Now we choose coordinates ( x, y ) on X such that the initialexponent of the generic descending Puiseux series associated to C j is in the normal form , and set¯ X = ¯ X ( x,y ) and σ = σ ( x,y ) . The arguments in the preceding paragraph imply that Q = ˜ σ ( ˜ Q j )and ˜ σ contracts E to Q . Since Q ∈ ¯ X \ S i = j C i , this in turn implies that the bimeromorphiccorrespondence ¯ X ¯ X ∗ j restricts to a holomorphic map on a neighborhood of E ∗ := ˜ π ∗ j ( E ). Inparticular, this implies that E ∗ is analytically contractible. Let µ ∗ j : ¯ X ∗ j → Z be the contraction of E ∗ . Then Z is a primitive compactification of X , and µ ∗ j induces a holomorphic map µ j : ¯ X → Z such that Z \ X = µ j ( C j ) and µ j is an isomorphism near Q . Assertion (3b) now follows fromCorollary 4.9. (cid:3) Example 5.3 (Compactifications with maximal number of singular points) . Pick relatively primeintegers p, q > X be the weighted projective surface P (1 , p, q ), so that ¯ X is a com-pactification of C with two singular points at infinity. Pick P ∈ C := ¯ X \ X such that ¯ X isnon-singular at P . Then perform a sequence of 3 blow-ups as follows: at first blow up ¯ X at P ,then blow up the resulting surface at a point on the exceptional divisor E which is not on thestrict transform of C , and then blow up the point of intersection of the new exceptional divisor E and the strict transform of E . This produces a compactification of C with the dual graph ofthe union of the curves at infinity as in Figure 9.It follows that blowing down E and E produces a compactification of C with 2 irreduciblecurves and 4 singular points at infinity. For each k ≥ 1, applying this procedure to k distinct points NALYTIC COMPACTIFICATIONS OF C PART I - CURVETTES AT INFINITY 19 C E − E − E − Figure 9. Construction of ¯ X such that | Sing( ¯ X ) | is maximalon C \ (Sing ¯ X ) produces a compactification of C with k + 1 irreducible curves and 2( k + 1)singular points at infinity.6. Intersection numbers of curves at infinity Proof of Theorem 1.5. Since each ν j is centered at infinity, it follows that there exists a compact-ification ¯ X j of X such that ν j is the order of vanishing along a curve C ′ j at infinity on ¯ X j . Byassumption we can assume ¯ X ∼ = P . Let ˜ X be the simultaneous resolution of singularities of ¯ X j ,1 ≤ j ≤ k . Let ˜ C j be the strict transform of C ′ j on ˜ X . Let ˜ E be the union of the exceptionalcurves of the map ˜ σ : ˜ X → ¯ X and let ˜ E be the union of all curves in ˜ E which are differentfrom ˜ C , . . . , ˜ C k . Since ˜ E is contractible, it follows that ˜ E is also contractible. Let ˜ σ : ˜ X → ¯ X be the contraction of ˜ E . Then ¯ X is precisely the compactification Question 1.3 asks for. Since˜ σ factors through ˜ σ , it follows that every singularity of ¯ X is sandwiched , and therefore rational [Spi90a, Remark 1.15]. A criterion of Artin [Art62, Theorem 2.3] then shows that ¯ X is projective.This completes the proof of assertions (1) and (2) of Theorem 1.5.We now prove assertion (3). Remark 1.6 shows that m ij = − ν i ( g ν j ( x, y, ˜ ξ )) for generic ˜ ξ ∈ C ,where g ν j is the generic key form of ν j . For all ˜ ξ ∈ C , let D , ˜ ξ be the closure in ¯ X of the curve g ν ( x, y, ˜ ξ ) = 0. Recall (from Example 3.8) that D , ˜ ξ is a line with ‘slope’ ˜ ξ . Therefore for generic˜ ξ , D , ˜ ξ intersects C transversally at one point and does not intersect any C j for j ≥ 1. Since theWeil divisor on ¯ X of g ν ( x, y, ˜ ξ ) is D , ˜ ξ + P kl =1 ν l ( g ν ( x, y, ˜ ξ )) C l , it follows that k X l =1 m l ( C l , C j ) = δ j for all j, ≤ j ≤ k, (8)where δ ij is the usual Kronecker delta. Now fix i , 2 ≤ i ≤ k and pick n ≥ x n g ν i ∈ C [ x, y, ξ ]. For all ˜ ξ ∈ C , let D i, ˜ ξ be the closure in ¯ X of the curve { x n g ν i ( x, y, ˜ ξ ) = 0 } ⊆ X . Let Z i, ˜ ξ be the image of D i, ˜ ξ under the natural birational morphism σ : ¯ X → ¯ X . Note that(1) ¯ X = ¯ X ( x,y ) and σ = σ ( x,y ) in the notation of Section 3.(2) Z i, ˜ ξ is precisely the curve Z ˜ ξ from Proposition 3.7 when applied to ν = ν i .(3) σ − ( Q y ) ∈ C \ (cid:16)S kj =2 C k (cid:17) .Proposition 3.7 then implies that for generic ˜ ξ ∈ C ,( D i, ˜ ξ , C j ) = − n k X l =1 ν l ( x )( C l , C j ) + k X l =1 m li ( C l , C j ) = ( n if j = 1 δ ij if 1 < j ≤ k. = nδ j + δ ij (9)Now recall that by our assumption ν l ( x ) ≤ ν l ( y ) for all l , 1 ≤ l ≤ k . It follows that m l = − ν l ( y − ˜ ξx ) (where ˜ ξ ∈ C is generic) = − ν l ( x ) , which, together with identities (8) and (9) imply that k X l =1 m li ( C l , C j ) = δ ij for all j, ≤ j ≤ k. (10)The theorem now follows from identities (8) and (10). (cid:3) Example 6.1 (Minimal compactifications with maximal number of singular points) . We applyTheorem 1.5 to construct, for each k ≥ 1, minimal compactifications ¯ X k of X with k irreduciblecurves at infinity and | Sing( ¯ X k ) | = k + 1. Choose relatively prime positive integers p, q . For k = 1,the weighted projective space P (1 , p, q ) satisfies the requirement, provided both p and q are ≥ k ≥ 2. Pick distinct complex numbers α , . . . , α k +1 and for each j , 2 ≤ j ≤ k + 1,let ν j be the divisorial valuation on C ( x, y ) corresponding to generic descending Puiseux series˜ ψ j ( x, ξ ) := α j x + ξx − q/p ; in other words, ν j is the negative of the weighted degree on C ( x, y ) withrespect to coordinates ( x, y − α j x ) such that the weight of x is p and the weight of y − α j x is − q .The key forms of ν j are x, y, y − α j x , and the generic key form of ν j is g ν j = ( y − α j x ) p − ξx − q , ≤ j ≤ k + 1 . Let ν = − deg and ¯ X be the surface obtained by applying Theorem 1.5 to ν , . . . , ν k +1 . Since g ν = y − ξx , it follows that M = p p · · · p pp − pq p · · · p p p p − pq · · · p p ... ... ... · · · ... ... p p p · · · p − pq I = M − = − kpp + q p + q p + q · · · p + q p + q p + q − p ( p + q ) · · · · · · ... ... p + q · · · − p ( p + q ) Now assume ( k − p > q . Then ( C , C ) < C is analytically contractible (Theorem2.12); let ¯ X p,q be the surface formed from ¯ X via contracting C . We claim that for a suitablechoice of parameters p and q , ¯ X p,q is a minimal compactification of X and | Sing( ¯ X p,q ) | = k + 1.Let C ′ j be the image of C j on ¯ X p,q via the morphism π ′ : ¯ X → ¯ X p,q . For the minimality of ¯ X p,q itsuffices to show that ( C ′ j , C ′ j ) ≥ j , 2 ≤ j ≤ k + 1. But ( C ′ j , C ′ j ) = ( π ′∗ ( C ′ j ) , π ′∗ ( C ′ j )) =( C j + c j C , C j + c j C ), where c j = − ( C , C j ) / ( C , C ). Consequently,( C ′ j , C ′ j ) = ( C j + c j C , C j ) = ( C , C )( C j , C j ) − ( C , C j ) ( C , C ) = 1 p ( p + q ) q − ( k − p ( k − p − q Since ( k − p > q , it follows that ( C ′ j , C ′ j ) ≥ k − p ≤ q , i.e. ¯ X p,q is indeed a minimalcompactification of X if k ≥ k − p ≤ q < ( k − p .Now we compute | Sing( ¯ X p,q ) | . First note that for 2 ≤ i < j ≤ k + 1,( C ′ i , C ′ j ) = ( C i + c i C , C j + c j C ) = ( C i + c i C , C j ) = c i ( C , C j ) = 1( p + q )(( k − p − q ) ;in particular, ( C ′ i , C ′ j ) is not an integer, which implies that the (unique) point P ′ of intersectionof C ′ i and C ′ j (which is also the point of intersection of all C ′ l , 2 ≤ l ≤ k + 1, due to assertion4a of Theorem 1.1) is singular. To see other singular points of ¯ X p,q , note that for each j , 1 ≤ j ≤ k , there is a morphism π j : ¯ X → ¯ X p,q,j , where ¯ X p,q,j is the surface obtained from ¯ X bycontracting all curves at infinity other than C and C j . Since − ν and − ν j are weighted degreesin ( x, y − α j x )-coordinates, it follows that ¯ X p,q,j is the toric surface corresponding to the polygonof figure 10. It follows from basic toric geometry that if p ≥ 2, then ¯ X p,q,j has a singular point Q j on π j ( C j ) \ π j ( C ). Since π j is invertible near P j , it then follows that P j := π − j ( Q j ) is a singularpoint on C j \ C and consequently the image P ′ j of P j on ¯ X p,q is a singular point on C ′ j \ S i = j C i . Proof of Theorem 1.4. W.l.o.g. we may (and will) assume that no two ν j ’s are mutually propor-tional. We divide the proof in two cases: NALYTIC COMPACTIFICATIONS OF C PART I - CURVETTES AT INFINITY 21 slope p/q slope − Figure 10. Polygon corresponding to ¯ X p,q,j Case 1: there exists j , ≤ j ≤ k , such that ν j = − deg . In this case w.l.o.g. we mayassume j = 1 and Theorem 1.5 shows that the answer is affirmative. So we only have to show thatdet( − M ) < 0. Indeed, let I be the intersection matrix of the curves at infinity on ¯ X and ˜ I be the( k − × ( k − 1) submatrix of I with ( i, j )-th entry being ( C i , C j ), 2 ≤ j ≤ k . Since C ∪ · · · ∪ C k is contractible, Grauert’s theorem (Theorem 2.12) implies that ˜ I is negative definite. Similarly,since C ∪ · · · ∪ C k is not contractible, it follows that I is not negative definite. Since det( I ) = 0,it then follows from the standard test of negative-definiteness via the sign of principal minors that( − k det I < 0. Consequently, ( − k det M = det( −M ) < 0, as required. Case 2: there is no j , ≤ j ≤ k , such that ν j = − deg . In this case, let ν = − degand apply Theorem 1.5 to the collection ν , . . . , ν k . Let ¯ X ′ be the resulting compactification of C and I ′ be the matrix of intersection numbers of curves ( C ′ i , C ′ j ), where C ′ i is the curve atinfinity on ¯ X ′ corresponding to ν i , 0 ≤ i ≤ k . Theorem 1.5 implies that det M is precisely the(1 , M ′ := I ′− . Cramer’s rule then implies that ( C ′ , C ′ ) = det M / det M ′ . Onthe other hand, applying Case 1 to ν , . . . , ν k yields that sign(det M ′ ) = ( − k . Consequently,sign(( C ′ , C ′ )) = sign(( − k det M ) = sign(det( −M )). Now the result follows from Grauert’stheorem. (cid:3) As an application of Theorem 1.4, we give an interpretation of skewness of valuations - aninvariant of valuations defined by Favre and Jonsson in order to study the valuative tree (see[FJ04] for details). Definition 6.2 (see [FJ07, Appendix A]) . Let ν be a divisorial discrete valuation on C [ x, y ]centered at infinity such that ν ( x ) < ν ( x ) ≤ ν ( y ). Assume that ν = − deg, where deg is thedegree in ( x, y )-coordinates. Let P be the center of ν on ¯ X ( x,y ) ∼ = P . For every f ∈ O ¯ X ( x,y ) ,P ,let ˜ m ( f ) be the intersection multiplicity at P of the curve { f = 0 } with the line at infinity. Notethat u := 1 /x is a regular function at P and u = 0 is precisely the equation of the line at infinitynear P . Let ˜ ν := ν/ν ( u ) be the normalized version of ν (in the sense that ˜ ν ( u ) = 1). Thenthe relative skewness of ν is ˜ α ( ν ) := sup { ˜ ν ( f ) / ˜ m ( f ) : f ∈ O ¯ X ( x,y ) ,P } and the skewness of ν is α ( ν ) := 1 − ˜ α ( ν ). Corollary 6.3. Let ν be a divisorial discrete valuation on C ( x, y ) centered at infinity such that ν ( x ) < and ν ( x ) ≤ ν ( y ) . Let g ν be the generic key form of ν with respect to ( x, y ) -coordinates.Then the following are equivalent:(1) ν determines a compactification of X (i.e. there is a (unique) compactification ¯ X of X such that the curve C at infinity on ¯ X is irreducible and ν is the order of vanishing along C ).(2) ν ( g ν ( x, y, ˜ ξ )) < for some (and hence every!) ˜ ξ ∈ C .(3) α ( ν ) > .Proof. Let p := deg y ( g ν ). Recall (from Definition 3.6) that g ν = ˜ g ν /u p = u p (˜ g n l l − ξ Q l − j =0 ˜ g n j j ),where ˜ g j ’s are key polynomials of ν with respect to ( u, v ) := (1 /x, y/x )-coordinates. The defining In [FJ07, Appendix A] skewness was defined only for normalized valuations centered at infinity. We simplydefined the skewness of a valuation centered at infinity to be the skewness of its normalized version. properties of key polynomials then imply that p = n l deg v (˜ g l ) = ν ( u ) , and for all ˜ ξ ∈ C , (11) ν ( g ν ( x, y, ˜ ξ )) = n l ν (˜ g l ) − ν ( u p ) = n l ν (˜ g l ) − pν ( u ) = n l ( ν (˜ g l ) − ν ( u ) deg v (˜ g l )) . (12)In particular, ν ( g ν ( x, y, ˜ ξ )) does not depend on ˜ ξ . The equivalence of assertions 1 and 2 thenimmediately follows from the k = 1 case of Theorem 1.4. On the other hand, [FJ04, Lemma 3.32]implies that ˜ α ( ν ) = ˜ ν (˜ g l )˜ m (˜ g l ) = ν (˜ g l ) ν ( u ) deg v (˜ g l )It follows that α ( ν ) = 1 − ˜ α ( ν ) = ν ( u ) deg v (˜ g l ) − ν (˜ g l ) ν ( u ) deg v (˜ g l ) = − ν ( g ν ( x, y, ˜ ξ )) pν ( u ) = − ν ( g ν ( x, y, ˜ ξ )) p which shows the equivalence of assertions 2 and 3, and completes the proof of the corollary. (cid:3) Remark 6.4. The term ν ( g ν ( x, y, ˜ ξ )) from assertion (2) of Corollary 6.3, or equivalently theskewness α ( ν ) can be calculated in a straightforward way in terms of formal Puiseux pairs of thegeneric descending Puiseux series ψ ν ( x, ξ ) of ν . We present the formula for the sake of completion:let ( q , p ) , . . . , ( q l +1 , p l +1 ) be the formal Puiseux pairs of ψ ν . Set p := p · · · p l +1 . Then for every˜ ξ ∈ C , ν ( g ν ( x, y, ˜ ξ )) = − p (cid:18) ( p · · · p l +1 − p · · · p l +1 ) q p + ( p · · · p l +1 − p · · · p l +1 ) q p p + · · · + ( p l p l +1 − p l +1 ) q l p · · · p l + p l +1 q l +1 p · · · p l +1 (cid:19) References [Art62] Michael Artin. Some numerical criteria for contractability of curves on algebraic surfaces. Amer. J.Math. , 84:485–496, 1962.[BDP81] Lawrence Brenton, Daniel Drucker, and Geert C. E. Prins. Graph theoretic techniques in algebraicgeometry. I. The extended Dynkin diagram ¯ E and minimal singular compactifications of C . volume100 of Ann. of Math. Stud. , pages 47–63. Princeton Univ. Press, Princeton, N.J., 1981.[BHPVdV04] Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven. Compact complex surfaces ,volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveysin Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveysin Mathematics] . Springer-Verlag, Berlin, second edition, 2004.[BK86] Egbert Brieskorn and Horst Kn¨orrer. Plane algebraic curves . Birkh¨auser Verlag, Basel, 1986. Trans-lated from the German by John Stillwell.[Bre73] Lawrence Brenton. A note on compactifications of C . Math. Ann. , 206:303–310, 1973.[Bre80] Lawrence Brenton. On singular complex surfaces with negative canonical bundle, with applications tosingular compactifications of C and to 3-dimensional rational singularities. Math. Ann. , 248(2):117–124, 1980.[CA00] Eduardo Casas-Alvero. Singularities of plane curves , volume 276 of London Mathematical SocietyLecture Note Series . Cambridge University Press, Cambridge, 2000.[FJ04] Charles Favre and Mattias Jonsson. The valuative tree , volume 1853 of Lecture Notes in Mathematics .Springer-Verlag, Berlin, 2004.[FJ07] Charles Favre and Mattias Jonsson. Eigenvaluations. Ann. Sci. ´Ecole Norm. Sup. (4) , 40(2):309–349,2007.[FJ11] Charles Favre and Mattias Jonsson. Dynamical compactifications of C . Ann. of Math. (2) ,173(1):211–248, 2011.[Fur97] Mikio Furushima. On minimal compactifications of C . Math. Nachr. , 186:115–129, 1997.[Gra62] Hans Grauert. ¨Uber Modifikationen und exzeptionelle analytische Mengen. Math. Ann. , 146:331–368,1962.[Jun42] Heinrich W. E. Jung. ¨Uber ganze birationale Transformationen der Ebene. J. Reine Angew. Math. ,184:161–174, 1942.[Koj01] Hideo Kojima. Minimal singular compactifications of the affine plane. Nihonkai Math. J. , 12(2):165–195, 2001.[KT09] Hideo Kojima and Takeshi Takahashi. Notes on minimal compactifications of the affine plane. Ann.Mat. Pura Appl. (4) , 188(1):153–169, 2009. NALYTIC COMPACTIFICATIONS OF C PART I - CURVETTES AT INFINITY 23 [Lip69] Joseph Lipman. Rational singularities, with applications to algebraic surfaces and unique factoriza-tion. Inst. Hautes ´Etudes Sci. Publ. Math. , (36):195–279, 1969.[Mac36] Saunders MacLane. A construction for absolute values in polynomial rings. Trans. Amer. Math. Soc. ,40(3):363–395, 1936.[MN05] Robert Mendris and Andr´as N´emethi. The link of { f ( x, y )+ z n = 0 } and Zariski’s conjecture. Compos.Math. , 141(2):502–524, 2005.[Mon12] Pinaki Mondal. Compactifications of C via pencils of jets of curves. C. R. Math. Acad. Sci. Soc. R.Can. , 34(3):79–96, 2012.[Mon13] Pinaki Mondal. Analytic compactifications of C II: one irreducible curve at infinity. http://arxiv.org/abs/1307.5577 , 2013.[Mor72] James A. Morrow. Compactifications of C . Bull. Amer. Math. Soc. , 78:813–816, 1972.[MZ88] Masayoshi Miyanishi and De-Qi Zhang. Gorenstein log del Pezzo surfaces of rank one. J. Algebra ,118(1):63–84, 1988.[Oht01] Tomoaki Ohta. Normal hypersurfaces as a compactification of C . Kyushu J. Math. , 55(1):165–181,2001.[RvdV60] Reinhold Remmert and Ton van de Ven. Zwei S¨atze ¨uber die komplex-projektive Ebene. Nieuw. Arch.Wisk. (3) , 8:147–157, 1960.[sga73] Groupes de monodromie en g´eom´etrie alg´ebrique. II . Lecture Notes in Mathematics, Vol. 340.Springer-Verlag, Berlin-New York, 1973. S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie 1967–1969 (SGA 7 II), Dirig´e par P. Deligne et N. Katz.[Spi90a] Mark Spivakovsky. Sandwiched singularities and desingularization of surfaces by normalized Nashtransformations. Ann. of Math. (2) , 131(3):411–491, 1990.[Spi90b] Mark Spivakovsky. Valuations in function fields of surfaces.