Projective completions of affine varieties via degree-like functions

Mondal, Pinaki

doi:10.48550/arxiv.1012.0835

Cited by 2 publications

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“…While dealing with such ν, we always work with δ := −ν (which we call a semidegree) instead of ν, since for polynomials f ∈ C[x, y], δ(f ) has a more 'natural' meaning than ν(f ), in the same sense that degree is a 'more natural' function on polynomials than negative degree. More generally, semidegrees are special types of degree-like functions [Mon10] which correspond to compactifications of affine varieties.…”

Section: Effective Answermentioning

confidence: 99%

“…We prove Proposition 4.2 in Section 5 based on some lemmas, whose proofs we defer to Section 6. Our proofs are self-contained modulo some properties of key polynomials which we list in Section 3 and are motivated by the general theory of projective completions via 'degree-like functions' [Mon10].…”

Section: Further Applications and Comments On The Structure Of The Proofmentioning

confidence: 99%

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An effective criterion for algebraic contractibility of rational curves

Mondal

2013

Preprint

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Let π : Ỹ → 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 π. 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.

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Section: Effective Answermentioning

confidence: 99%

Section: Further Applications and Comments On The Structure Of The Proofmentioning

confidence: 99%

An effective criterion for algebraic contractibility of rational curves

Mondal

2013

Preprint

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“…A clearer exposition (with more complete reference) is in order and this submission will be updated in a few days. This article is a sequel to [9]. In it we develop affine Bezout type theorems.…”

Section: Introductionmentioning

confidence: 99%

“…

In this sequel to [9] we develop Bezout type theorems for semidegrees (including an explicit formula for 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 [5] and Lazarsfeld-Mustata [6]) * The idea of looking at the construction of theorem 2.5 is due to Professor A. Khovanskii.

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confidence: 99%