On projective completions of affine varieties determined by 'degree-like' functions

We study projective completions of affine algebraic varieties which are given by filtrations, or equivalently, 'degree like functions' on their rings of regular functions. For a quasifinite polynomial map P (i.e. with all fibers finite) of affine varieties, we prove that there are completions of the source that do not add points at infinity for P (i.e. in the intersection of completions of the hypersurfaces corresponding to a generic fiber and determined by the component functions of P). Moreover we show that there are 'finite type' completions with the latter property, determined by the maximum of a finite number of 'semidegrees', i.e. maps of the ring of regular functions excluding zero, into integers, which send products into sums and sums into maximas (with a possible exception when the summands have the same semidegree). We characterize the latter type completions as the ones for which the ideal of the 'hypersurface at infinity' is radical. Moreover, we establish a one-to-one correspondence between the collection of minimal associated primes of the latter ideal and the unique minimal collection of semidegrees needed to define the corresponding degree like function. We also prove an 'affine Bezout type' theorem for quasifinite polynomial maps P which admit semidegrees such that corresponding completions do not add points at infinity for P.Comment: 32 page

Mori dream surfaces associated with curves with one place at infinity

We study a class of rational surfaces (considered in [Campillo, Piltant and Reguera, 2005]) associated to curves with one place at infinity and explicitly describe generators of the Cox ring and global sections of line bundles on these surfaces. In particular, we show that their Cox rings are finitely generated, i.e. they are Mori dream spaces. We also compute their "global Zariski semigroups at infinity" (consisting of line bundles which have no base points `at infinity') and "global Enriques semigroups" (generated by closures of curves in C^2). In particular, we show that the global Zariski semigroups at infinity and Enriques semigroups of surfaces corresponding to pencils which are equisingular at infinity are isomorphic, which answers a question of [Campillo, Piltant and Reguera-Lopez, 2002]. We also give an effective algorithm to determine if a (rational) surface `admits systems of numerical curvettes' (these surfaces were also considered in [Campillo, Piltant and Reguera, 2005])

An effective criterion for algebraic contractibility of rational curves

Let f: Y -> CP^2 be a birational morphism of non-singular (rational) surfaces. We give an effective (necessary and sufficient) criterion for algebraicity of the surfaces resulting from contraction of the union of the strict transform of a line on CP^2 and all but one of the exceptional divisors of f. As a by-product we construct normal non-algebraic Moishezon surfaces with the `simplest possible' singularities, which in particular completes the answer to a remark of Grauert. Our criterion involves `global variants' of `key polynomials' introduced by MacLane. The geometric formulation of the criterion yields a correspondence between normal algebraic compactifications of C^2 with one irreducible curve at infinity and algebraic curves in C^2 with one place at infinity.Comment: 5 figures, 12 + 30 pages (the first part introduction and statements of results, the second part proofs). Any comments would be greatly appreciated. arXiv admin note: text overlap with arXiv:1211.433

How to determine the sign of a valuation on C[x,y]?

Given a divisorial discrete valuation 'centered at infinity' on C[x,y], we show that its sign on C[x,y] (i.e. whether it is negative or non-positive on non-constant polynomials) is completely determined by the sign of its value on the 'last key form' (key forms being the avatar of 'key polynomials' of valuations (introduced by [Maclane, 1936]) in 'global coordinates'). The proof involves computations related to the cone of curves on certain compactifications of C^2 and gives a characterization of the divisorial valuations centered at infinity whose 'skewness' can be interpreted in terms of the 'slope' of an extremal ray of these cones, yielding a generalization of a result of [Favre-Jonsson, 2007]. A by-product of these arguments is a characterization of valuations which 'determine' normal compactifications of C^2 with one irreducible curve at infinity in terms of an associated 'semigroup of values'.Comment: 12 page

An effective criterion for algebraicity of rational normal surfaces

We give a novel and effective criterion for algebraicity of rational normal analytic surfaces constructed from resolving the singularity of an irreducible curve-germ on $CP^2$ and contracting the strict transform of a given line and all but the `last' of the exceptional divisors. As a by-product we construct a new class of analytic non-algebraic rational normal surfaces which are `very close' to being algebraic. These results are local reformulations of some results in (Mondal, 2011) which sets up a correspondence between normal algebraic compactifications of $C^2$ with one irreducible curve at infinity and algebraic curves in $C^2$ with one place at infinity. This article is meant partly to be an exposition to (Mondal, 2011) and we give a proof of the correspondence theorem of (Mondal, 2011) in the `first non-trivial case'.Comment: 20 pages, 6 figure

Analytic Compactifications of C^2 part I - curvettes at infinity

We study normal analytic compactifications of C^2 and describe their singularities and configuration of curves at infinity, in particular improving and generalizing results of (Brenton, Math. Ann. 206:303--310, 1973). As a by product we give new proofs of Jung's theorem on polynomial automorphisms of C^2 and Remmert and Van de Ven's result that CP^2 is the only smooth analytic compactification of C^2 for which the curve at infinity is irreducible. We also give a complete answer to the question of existence of compactifications of C^2 with prescribed divisorial valuations at infinity. In particular, we show that a valuation on C(x,y) centered at infinity determines a compactification of C^2 iff it is "positively skewed" in the sense of (Favre and Jonsson, Ann. Sci. Ecole Norm. Sup. 40(2):309--349, 2007).Comment: Final and substantially improved version, accepted in the Mathematical Reports of the Academy of Science, Royal Society of Canad

When is the intersection of two finitely generated subalgebras of a polynomial ring also finitely generated?

We study two variants of the following question: "Given two finitely generated subalgebras R_1, R_2 of C[x_1, \ldots, x_n], is their intersection also finitely generated?" We show that the smallest value of $n$ for which there is a counterexample is 2 in the general case, and 3 in the case that R_1 and R_2 are integrally closed. We also explain the relation of this question to the problem of constructing algebraic compactifications of C^n and to the moment problem on semialgebraic subsets of R^n. The counterexample for the general case is a simple modification of a construction of Neena Gupta, whereas the counterexample for the case of integrally closed subalgebras uses the theory of normal analytic compactifications of C^2 via "key forms" of valuations centered at infinity.Comment: Includes a solution to the general question thanks to a construction of Neena Gupta communicated by Wilberd van der Kallen. Explanation has been added regarding the connection to the problem of constructing algebraic compactifications of C^n and to the moment problem on semialgebraic subsets of R^n. Accepted at Arnold Mathematical Journa

Algebraicity of normal analytic compactifications of C^2 with one irreducible curve at infinity

We present an effective criterion to determine if a normal analytic compactification of C^2 with one irreducible curve at infinity is algebraic or not. As a by product we establish a correspondence between normal algebraic compactifications of C^2 with one irreducible curve at infinity and algebraic curves contained in C^2 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 to study valuations in the 'local' setting - which also extends the notion of "approximate roots" of polynomials considered by Abhyankar and Moh.Comment: Proof of Theorem 4.4 has been corrected - may thanks to the anonymous referee who pointed out the error. To appear in Algebra & Number Theory. arXiv admin note: text overlap with arXiv:1301.012

Projective completions of affine varieties via degree-like functions

We study projective completions of affine algebraic varieties induced by filtrations on their coordinate rings. In particular, we study the effect of the 'multiplicative' property of filtrations on the corresponding completions and introduce a class of projective completions (of arbitrary affine varieties) which generalizes the construction of toric varieties from convex rational polytopes. As an application we recover (and generalize to varieties over algebraically closed fields of arbitrary characteristics) a 'finiteness' property of divisorial valuations over complex affine varieties proved in the article "Divisorial valuations via arcs" by de Fernex, Ein and Ishii (Publ. Res. Inst. Math. Sci., 2008). We also find a formula for the pull-back of the 'divisor at infinity' and apply it to compute the matrix of intersection numbers of the curves at infinity on a class of compactifications of certain affine surfaces.Comment: Improved exposition, added discussion of weighted projective spaces with possibly non-positive weights. Accepted in the Asian Journal of Mathematic

General Bezout-type theorems

In this sequel to arxiv:arXiv:1012.0835 we develop Bezout type theorems for semidegrees (including an explicit formula for {\em iterated semidegrees}) and an inequality for subdegrees. In addition we prove (in case of surfaces) a Bernstein type theorem for the number of solutions of two polynomials in terms of the mixed volume of planar convex polygons associated to them (via the theory of Kaveh-Khovanskii and Lazarsfeld-Mustata