2014/11/30 by Thomas Bauer, Piotr Pokora, David Schmitz · 11 citations
Computer Science · Mathematics · #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic number #Arithmetic #Bounded function #Commutative Algebra and Its Applications #Conjecture #Decomposition #Divisor (algebraic geometry) #Geometry #Mathematical analysis #Mathematics #Multiple #Negativity effect #Order (exchange) #Polynomial and algebraic computation #Pure mathematics #Surface (topology) #math.AG #msc:14A25 #msc:14C20
paper · pdf · doi:10.1515/crelle-2015-0058
published in Journal für die reine und angewandte Mathematik (Crelles Journal) 2017(733), 251-259 (De Gruyter) · Example 3.4 expanded; and minor edits in the text
arxiv created 2015/10/05 · openalex publication_date 2015/10/07 · openalex created_date 2016/06/24 · arxiv updated 2017/12/18 · openalex updated_date 2026/08/05
Abstract Zariski decompositions play an important role in the theory of algebraic surfaces. For making geometric use of the decomposition of a given divisor, one needs to pass to a multiple of the divisor in order to clear denominators. It is therefore an intriguing question whether the surface has a “universal denominator” that can be used to simultaneously clear denominators in all Zariski decompositions on the surface. We prove in this paper that, somewhat surprisingly, this condition of bounded Zariski denominators is equivalent to the bounded negativity of curves that is addressed in the Bounded Negativity Conjecture . Furthermore, we provide explicit bounds for Zariski denominators and negativity of curves in terms of each other.