Titlebook: Extended Abstracts February 2016; Positivity and Valua Maria Alberich-Carramiñana,Carlos Galindo,Joaquim Conference proceedings 2018 Sprin

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,Notes on Local Positivity and Newton–Okounkov Bodies,We explore the notion of local numerical equivalence in higher dimension and its relationship with Newton–Okounkov bodies with respect to flags centered at a given point.

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,Newton–Okounkov Bodies and Reified Valuations of Higher Rank,We study the shape change of the Newton–Okounkov body of a fixed divisor . with respect to a valuation . moving in a suitable space of (higher-rank) valuations.

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,Semigroup and Poincaré Series for Divisorial Valuations,the Poincaré series associated to ., assuming that . has a finite generating sequence. First, if . is infinite, this semigroup is finitely generated. Secondly, for any ., the Poincaré series associated to . is a rational function whose denominator can be expressed in terms of the valuation vectors of the elements in the generating sequence.

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On the Containment Hierarchy for Simplicial Ideals,etry. Our considerations are mainly inspired by results of Bocci–Harbourne in (Bocci and Harbourne. Proc. Am. Math. Soc. 138, 1175–1190 (2010) [.]). The results presented on a poster are from (Lampa-Baczyńska and Malara. J. Pure Appl. Algeb. 219, 5402–5412 (2015) [.]), the common paper of both authors.

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