Algebraicity of normal analytic compactifications of with one irreducible curve at infinity
We present an effective criterion to determine if a normal analytic compactification of with one irreducible curve at infinity is algebraic or not. As a by product we establish a correspondence between normal algebraic compactifications of with one irreducible curve at infinity and algebraic curves contained in with one place at infinity. Using our criterion we construct pairs of homeomorphic normal analytic surfaces with minimally elliptic singularities such that one of the surfaces is algebraic and the other is not. Our main technical tool is the sequence of key forms - a ‘global’ variant of the sequence of key polynomials introduced by MacLane [maclane-key] to study valuations in the ‘local’ setting - which also extends the notion of approximate roots of polynomials considered by Abhyankar-Moh [abhya-moh-tschirnhausen].
2010 Mathematics Subject Classification:Primary 32J05, 14J26; Secondary 14E05, 14E15
Algebraic compactifications of (i.e. compact algebraic surfaces containing ) are in a sense the simplest compact algebraic surfaces. The simplest among these are the primitive compactifications, i.e. those for which the complement of (a.k.a. the curve at infinity) is irreducible. It follows from a famous result of Remmert and Van de Ven that up to isomorphism, is the only nonsingular primitive compactification of . In some sense a more natural category than nonsingular algebraic surfaces is the category of normal algebraic surfaces111This is true for example from the perspective of valuation theory: the irreducible components of the curve at infinity of a normal compactification of correspond precisely to the discrete valuations on which are centered at infinity with positive dimensional center on . Therefore is primitive and normal iff corresponds to precisely one discrete valuation centered at infinity on . In this article we tackle the problem of understanding the simplest normal algebraic compactifications of :
What are the normal primitive algebraic compactifications of ?
We give a complete answer to this question; in particular, we characterize both algebraic and non-algebraic primitive compactifications of . Our answer is equivalent to an explicit criterion for determining algebraicity of (analytic) contractions of a class of curves: indeed, it follows from well known results of Kodaira, and independently of Morrow, that any normal analytic compactification of is the result of contraction of a (possibly reducible) curve from a non-singular surface constructed from by a sequence of blow-ups. On the other hand, a well known result of Grauert completely and effectively characterizes all curves on a nonsingular analytic surface which can be analytically contracted: namely it is necessary and sufficient that the matrix of intersection numbers of the irreducible components of is negative definite. It follows that the question of understanding algebraicity of analytic compactifications of is equivalent to the following question:
Let be a birational morphism of nonsingular complex algebraic surfaces and be a line. Assume restricts to an isomorphism on . Let be the exceptional divisor of (i.e. is the union of curves on which map to points in ) and be irreducible curves contained in . Let be the union of the strict transform (on ) of and all components of excluding . Assume is analytically contractible; let be the contraction of . When is algebraic?
Question 1 is equivalent to the case of Question 2. We give a complete solution to this case of Question 2 (Theorem 1). Our answer is in particular effective, i.e. given a description of (e.g. if we know a sequence of blow ups which construct from , or if we know precisely the discrete valuation on associated to the unique curve on which does not get contracted), our algorithm determines in finite time if the contraction is algebraic. In fact the algorithm is a one-liner: “Compute the key forms of . is algebraic iff the last key form is a polynomial.” The only previously known effective criteria for determining the algebraicity of contraction of curves on surfaces was the well-known criteria of Artin [artractability] which states that a normal surface is algebraic if all its singularities are rational. We refer the reader to [morrow-rossi], [brenton-algebraicity], [franco-lascu], [schroe-traction], [badescu-contractibility], [palka-Q1] for other criteria - some of these are more general, but none is effective in the above sense. Moreover, as opposed to Artin’s criterion, ours is not numerical, i.e. it is not determined by numerical invariants of the associated singularities. We give an example (in Section 2) which shows that in fact there is no topological, let alone numerical, answer to Question 2 even for .
As a corollary of our criterion, we establish a new correspondence between normal primitive algebraic compactifications of and algebraic curves in with one place at infinity222Let be an algebraic curve, and let be the closure of in and be the desingularization of . has one place at infinity iff . (Theorem 3). Curves with one place at infinity have been extensively studied in affine algebraic geometry (see e.g. [abhya-moh-tschirnhausen], [abhya-moh-line], [ganong], [russburger], [naka-oka], [suzuki], [wightwick]), and we believe the connection we found between these and compactifications of will be useful for the study of both333E.g. we use this connection in [cpl] to solve completely the main problem studied in [campillo-piltant-lopez-cones-surfaces]..
Our main technical tool is the sequence of key forms, which is a direct analogue of the sequence key polynomials introduced by MacLane [maclane-key]. The key polynomials were introduced (and have been extensively used - see e.g. [moyls], [favsson-tree], [vaquie], [herrera-olalla-spivakovsky]) to study valuations in a local setting. However, our criterion shows how they retain information about the global geometry when computed in ‘global coordinates.’
The example in Section 2 shows that algebraicity of from Question 2 can not be determined only from the (weighted) dual graph (Definition 25) of . However, at least when , it is possible to completely characterize the weighted dual graphs (more precisely, augmented and marked weighted dual graphs - see Definition 26) which correspond to only algebraic contractions, those which correspond to only non-algebraic contractions, and those which correspond to both types of contractions (Theorem 4). The characterization involves two sets of semigroup conditions (S1-k) and (S2-k). We note that the first set of semigroup conditions (S1-k) are equivalent to the semigroup conditions that appear in the theory of plane curves with one place at infinity developed in [abhya-moh-tschirnhausen], [abhyankar-expansion], [abhyankar-semigroup], [sathaye-stenerson].
Finally we would like to point that Question 1 is equivalent to a two dimensional Cousin-type problem at infinity: let be points at infinity. Let be coordinates near , be a Puiseux series (Definition 2) in , and be a positive rational number, .
Determine if there exists a polynomial such that for each analytic branch of the curve at infinity, there exists , , such that
intersects at ,
has a Puiseux expansion at such that .
We use Puiseux series in an essential way in this article. However, instead of the usual Puiseux series, from Section 3 onward, we almost exclusively work with descending Puiseux series (a descending Puiseux series in is simply a meromorphic Puiseux series in - see Definition 4). The choice was enforced on us ‘naturally’ from the context - while key polynomials and Puiseux series are natural tools in the study of valuations in the local setting, when we need to study the relation of valuations corresponding to curves at infinity (on a compactification of ) to global properties of the surface, key forms and descending Puiseux series are sometimes more convenient.
We start with an example in Section 2 to illustrate that the answer to Question 2 can not be numerical or topological. The construction also serves as an example of non-algebraic normal Moishezon surfaces444Moishezon surfaces are analytic surfaces for which the fields of meromorphic functions have transcendence degree 2 over with the ‘simplest possible’ singularities (see Remark 1). In Section 3 we recall some background material and in Section 4 we state our results. The rest of the article is devoted to the proof of the results of Section 4. In Section 5 we recall some more background material needed for the proof; in particular in Section 5.1 we state the algorithm to compute key forms of a valuation from the associated descending Puiseux series, and illustrate the algorithm via an elaborate example (we note that this algorithm is essentially the same as the algorithm used in [lenny] for a different purpose). In Section 6 we build some tools for dealing with descending Puiseux series and in Section 7 we use these tools to prove the results from Section 4. The appendices contain proof of two lemmas from Section 6 - the proofs were relegated to the appendix essentially because of their length.
This project started during my Ph.D. under Professor Pierre Milman to answer some of his questions, and profited enormously from his valuable suggestions and speaking in his seminars. I would like to thank Professor Peter Russell for inviting me to present this result and for his helpful remarks. The idea for presenting the ‘effective answer’ in terms of key polynomials came from a remark of Tommaso de Fernex, and Mikhail Zaidenberg asked the question of characterization of (non-)algebraic dual graphs during a poster presentation. I thank Mattias Jonsson and Mark Spivakovsky for bearing with my bugging at different stages of the write up, and Leonid Makar-Limanov for crucial encouragement during the (long) rewriting stage and pointing out the equivalence of the algorithm of [lenny] and our Algorithm 1. Finally I have to thank Dmitry Kerner for forcing me to think geometrically by his patient (and relentless) questions. Theorem 4, and the example from Section 2 appeared in [trento].
2. Algebraic and non-algebraic compactifications with homeomorphic singularities
Let be a system of ‘affine’ coordinates near a point (‘affine’ means that both and are lines on ) and be the line . Let and be curve-germs at defined respectively by and . For each , let be the surface constructed by resolving the singularity of at and then blowing up more times the point of intersection of the (successive) strict transform of with the exceptional divisor. Let be the last exceptional curve, and be the union of the strict transform (on ) of and (the strict transforms of) all exceptional curves except .
Note that the pairs of germs and are isomorphic via the map . It follows that ‘weighted dual graphs’ (Definition 25) of ’s are identical; they are depicted in Figure 0(a) (we labeled the vertices according to the order of appearance of the corresponding curves in the sequence of blow-ups). It is straightforward to compute that the matrices of intersection numbers of the components of ’s are negative definite, so that there is a bimeromorphic analytic map contracting . Note that each is a normal analytic surface with one singular point . It follows from the construction that the weighted dual graphs of the minimal resolution of singularities of are identical (see Figure 0(b)), so that the numerical invariants of the singularities of ’s are also identical.
It is straightforward to verify that the weighted dual graph of Figure 0(b) is precisely the graph labeled in [laufer-elliptic]. It then follows from [laufer-elliptic] that the singularities at are Goerenstein hypersurface singularities of multiplicity and geometric genus , which are also minimally elliptic (in the sense of [laufer-elliptic]). Minimally elliptic Gorenstein singularities have been extensively studied (see e.g. [yau], [ohyanagi], [nemethi-weakly-elliptic]), and in a sense they form the simplest class of non-rational singularities555Indeed, every connected proper subvariety of the exceptional divisor of the minimal resolution of a minimally elliptic singularity is the exceptional divisor of the minimal resolution of a rational singularity [laufer-elliptic].. Since having only rational singularities imply algebraicity of the surface [artractability], it follows that the surface we constructed above is a normal non-algebraic Moishezon surface with the ‘simplest possible’ singularity.
It follows from [laufer-elliptic, Table 2] that the singularity at the origin of (Figure 2) is of the same type as the singularity of each , .
3. Background I
Here we compile the background material needed to state the results. In Section 5 we compile further background material that we use for the proof.
Throughout the rest of the article we use to denote with coordinate ring and to denote copy of such that is embedded into via the map . We also denote by the line at infinity , and by the point of intersection of and (closure of) the -axis. Finally, if are positive integers, we denote by the complex -dimensional weighted projective space corresponding to weights .
3.1. Meromorphic and descending Puiseux series
Definition 2 (Meromorphic Puiseux series).
A meromorphic Puiseux series in a variable is a fractional power series of the form for some , and for all . If all exponents of appearing in a meromorphic Puiseux series are positive, then it is simply called a Puiseux series (in ). Given a meromorphic Puiseux series in , write it in the following form:
where is the smallest non-integer exponent, and for each , , we have that , , , and the exponents of all terms with order between and (or, if , then all terms of order ) belong to . Then the pairs , are called the Puiseux pairs of and the exponents , , are called characteristic exponents of . The polydromy order [casas-alvero, Chapter 1] of is , i.e. the polydromy order of is the smallest such that . Let be a primitive -th root of unity. Then the conjugates of are
for (i.e. is constructed by multiplying the coefficients of terms of with order by ).
We recall the standard fact that the field of meromorphic Puiseux series in is the algebraic closure of the field of Laurent polynomials in :
Let be an irreducible monic polynomial in of degree . Then there exists a meromorphic Puiseux series in of polydromy order such that
where ’s are conjugates of .
Definition 4 (Descending Puiseux Series).
A descending Puiseux series in is a meromorphic Puiseux series in . The notions regarding meromorphic Puiseux series defined in Definition 2 extend naturally to the setting of descending Puiseux series. In particular, if is a descending Puiseux series and the Puiseux pairs of are , then has Puiseux pairs , polydromy order , and characteristic exponents for .
We use to denote the field of descending Puiseux series in . For and , we denote by the descending Puiseux polynomial (i.e. descending Puiseux series with finitely many terms) consisting of all terms of of degree . If is also in , then we write iff iff .
The following is an immediate Corollary of Theorem 3:
Let . Then there are (up to conjugacy) unique descending Puiseux series in , a unique non-negative integer and such that
3.2. Divisorial discrete valuation and semidegree
Let be a birational correspondence of normal complex algebraic surfaces and be an irreducible analytic curve on . Then the local ring of on is a discrete valuation ring. Let be the associated valuation on the field of rational functions on ; in other words is the order of vanishing along . We say that is a divisorial discrete valuation on ; the center of on is , where is the set of points of indeterminacy of (the normality of ensures that is a discrete set, so that ). Moreover, if is an open subset of , we say that is centered at infinity with respect to iff .
Definition 7 (Semidegree).
Let be an affine variety and be a divisorial discrete valuation on the ring of regular functions on which is centered at infinity with respect to . Then we say that is a semidegree on .
The following result, which connects semidegrees on with descending Puiseux series in is a reformulation of [favsson-tree, Proposition 4.1].
Let be a semidegree on . Assume that . Then there is a descending Puiseux polynomial (i.e. a descending Puiseux series with finitely many terms) (unique up to conjugacy) in and a (unique) rational number such that for every ,
where is an indeterminate.
In the situation of Theorem 8, we say that is the generic descending Puiseux series associated to . Moreover, if is an analytic compactification of and is a curve at infinity such that is the order of pole along , then we also say that is the generic descending Puiseux series associated to .
If is a weighted degree in -coordinates corresponding to weights for and for with positive integers, then the generic descending Puiseux series corresponding to is . Note that if we embed into the weighted projective space via , then is precisely the order of the pole along the curve at infinity.
Recall the set up of the example from Section 2. Then ’s have Puiseux expansions at , where
Now note that are coordinates on , and with respect to coordinates the has a descending Puiseux expansion of the form . Similarly, has a descending Puiseux expansion of the form . Let be the order of pole along , . Then the generic descending Puiseux series corresponding to and are respectively of the form
Definition 12 (Formal Puiseux pairs of generic descending Puiseux series).
Let and be as in Definition 9. Let the Puiseux pairs of be . Express as where and . Then the formal Puiseux pairs of are , with being the generic formal Puiseux pair. Note that
it is possible that (as opposed to other ’s, which are always ).
3.3. Geometric interpretation of generic descending Puiseux series
In this subsection we recall from [sub2-1] the geometric interpretation of generic descending Puiseux series. We use the notations introduced in Notation 1.
An irreducible analytic curve germ at infinity on is the image of an analytic map from a punctured neighborhood of the origin in to such that as (in other words, is analytic on and has a pole at the origin). If is an analytic compactification of , then there is a unique point such that as . We call the center of on , and write .
Let be a primitive normal analytic compactification of with an irreducible curve at infinity. Let be the natural bimeromorphic map, and let be a resolution of indeterminacies of , i.e. is a non-singular rational surface equipped with analytic maps and such that . Let be the strict transform of on and be (the unique point) such that . Let .
Proposition 14 ([sub2-1, Proposition 3.5]).
Let be the order of pole along , be the generic descending Puiseux series associated to and be an irreducible analytic curve germ on . Then iff has a parametrization of the form
for some , where l.d.t. means ‘lower degree terms’ (in ).
We call a center of -infinity on . is in fact unique in the case of ‘generic’ primitive normal compactifications of (we do not use this uniqueness in this article, so we state it without a proof):
If for some , then every point of is a center of -infinity on .
If for some , then has two singular points, and these are precisely the centers of -infinity on .
In all other cases, there is a unique center of -infinity on - it is precisely the unique point on which has a non-quotient singularity.
3.4. Key forms of a semidegree
Let be a semidegree on such that . Pick such that . Set . Then is a discrete valuation on which is centered at the origin. It follows that can be completely described in terms of a finite sequence of key polynomials in [maclane-key]. The key forms of that we introduce in this section are precisely the analogue of key polynomials of . We refer to [favsson-tree, Chapter 2] for the properties of key polynomials that we used as a model for our definition of key forms:
Definition 16 (Key forms).
Let be a semidegree on such that . A sequence of elements is called the sequence of key forms for if the following properties are satisfied with , :
for , where
’s are integers such that for (in particular, only ’s are allowed to be negative).
For , there exists such that
Let be indeterminates and be the weighted degree on corresponding to weights for and for , (i.e. the value of on a polynomial is the maximum ‘weight’ of its monomials). Then for every polynomial ,
The properties of key forms of semidegrees compiled in the following theorem are straightforward analogues of corresponding (standard) properties of key polynomials of valuations.
Every semidegree on such that has a unique and finite sequence of key forms.
Recall Notation 1. Assume is a composition of point blow-ups and is an exceptional curve of . Let be the order of pole along . Assume . Then the following data are equivalent (i.e. given any one of them, there is an explicit algorithm to construct the others in finite time):
a minimal sequence of points on successive blow-ups of such that factors through the composition of these blow-ups and is the strict transform of the exceptional curve of the last blow-up.
a generic descending Puiseux series of .
the sequence of key forms of .
Let be a semidegree on such that . Let be the generic descending Puiseux series and be the last key form of . Then the descending Puiseux factorization of is of the form
for some such that (see Notation 5).
Let be the weighted degree from Example 10. The key forms of are and .
Definition 20 (Essential key forms).
We say that are the essential key forms of . The following properties of essential key forms follow in a straightforward manner from the defining properties of key forms:
Let the notations be as in Definition 20. Let be the generic descending Puiseux series of and be the formal Puiseux paris of . Then
, i.e. the number of essential key forms of is precisely .
Set , . Then the sequence depend only on the formal Puiseux pairs of . More precisely, with , we have
Let be as in Property (P0) of key forms. Then
Pick , . Assume for some , . Then is in the group generated by .
We call of Proposition 21 the sequence of essential key values of .
3.5. Resolution of singularities of primitive normal compactifications
Given two birational algebraic surfaces , we say that dominates if the birational map is in fact a morphism. Let be a primitive normal analytic compactification of and be a resolution of singularities of . We say that or is -dominating if dominates . is a minimal -dominating resolution of singularities of if up to isomorpshism (of algebraic varieties) is the only -dominating resolution of singularities of which is dominated by .
Every primitive normal analytic compactification of has a unique minimal -dominating resolution of singularities.
We have not found any proof of Theorem 24 in the literature. We give a proof in [sub2-2] (using Theorem 1 of this article). In this section we recall from [sub2-1] a description of the dual graphs of minimal -dominating resolutions of singularities of primitive normal analytic compactifications of .
Let be non-singular curves on a (non-singular) surface such that for each , either , or and intersect transversally at a single point. Then is called a simple normal crossing curve. The (weighted) dual graph of is a weighted graph with vertices such that
there is an edge between and iff ,
the weight of is the self intersection number of .
Usually we will abuse the notation, and label ’s also by .
Let be a primitive normal analytic compactification of and be a resolution of singularities of such that is a simple normal crossing curve. The augmented dual graph of is the dual graph (Definition 25) of . If is -dominating, we define the augmented and marked dual graph of to be its augmented dual graph with the strict transforms of the curves at infinity on and marked (e.g. by different colors or labels).
Given a sequence of pairs of relatively prime integers, and positive integers such that , we denote by the weighted graph in figure 3,
where the right-most vertex in the top row has weight , and the other weights satisfy: and for , and are uniquely determined from the continued fractions:
is the weighted dual graph of the exceptional divisor of the minimal resolution of an irreducible plane curve singularity with Puiseux pairs (see, e.g. [mendris-nemethi, Section 2.2]).
Theorem 28 ([sub2-1, Proposition 4.2, Corollary 6.3]).
Let be a primitive normal compactification of .
If is nonsingular, then .
After a (polynomial) change of coordinates of if necessary, we may assume that and either or .
Conversely, let , and be pairs of integers such that