Newton–Okounkov bodies sprouting on the valuative tree

Ciro Ciliberto; Michal Farnik; Alex Küronya; Victor Lozovanu; Joaquim Roé; Constantin Shramov

doi:https://doi.org/10.1007/s12215-016-0285-3

Newton–Okounkov bodies sprouting on the valuative tree

Ciro Ciliberto, Michal Farnik, Alex Küronya, Victor Lozovanu, Joaquim Roé, Constantin Shramov

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

6 Citations (Scopus)

Abstract

© 2016, Springer-Verlag Italia. Given a smooth projective algebraic surface X, a point O∈ X and a big divisor D on X, we consider the set of all Newton–Okounkov bodies of D with respect to valuations of the field of rational functions of X centred at O, or, equivalently, with respect to a flag (E, p) which is infinitely nearO, in the sense that there is a sequence of blowups X′→ X, mapping the smooth, irreducible rational curve E⊂ X′ to O. The main objective of this paper is to start a systematic study of the variation of these infinitesimal Newton–Okounkov bodies as (E, p) varies, focusing on the case X= P2.

Original language	English
Pages (from-to)	161-194
Journal	Rendiconti del Circolo Matematico di Palermo
Volume	66
DOIs	https://doi.org/10.1007/s12215-016-0285-3
Publication status	Published - 1 Aug 2017

Keywords

Algebraic geometry
Linear system
Newton-Okounkov body
Valuation
Valuative tree

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https://doi.org/10.1007/s12215-016-0285-3

https://ddd.uab.cat/record/175899

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title = "Newton–Okounkov bodies sprouting on the valuative tree",

abstract = "{\textcopyright} 2016, Springer-Verlag Italia. Given a smooth projective algebraic surface X, a point O∈ X and a big divisor D on X, we consider the set of all Newton–Okounkov bodies of D with respect to valuations of the field of rational functions of X centred at O, or, equivalently, with respect to a flag (E, p) which is infinitely nearO, in the sense that there is a sequence of blowups X′→ X, mapping the smooth, irreducible rational curve E⊂ X′ to O. The main objective of this paper is to start a systematic study of the variation of these infinitesimal Newton–Okounkov bodies as (E, p) varies, focusing on the case X= P2.",

keywords = "Algebraic geometry, Linear system, Newton-Okounkov body, Valuation, Valuative tree",

author = "Ciro Ciliberto and Michal Farnik and Alex K{\"u}ronya and Victor Lozovanu and Joaquim Ro{\'e} and Constantin Shramov",

year = "2017",

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T1 - Newton–Okounkov bodies sprouting on the valuative tree

AU - Ciliberto, Ciro

AU - Farnik, Michal

AU - Küronya, Alex

AU - Lozovanu, Victor

AU - Roé, Joaquim

AU - Shramov, Constantin

PY - 2017/8/1

Y1 - 2017/8/1

N2 - © 2016, Springer-Verlag Italia. Given a smooth projective algebraic surface X, a point O∈ X and a big divisor D on X, we consider the set of all Newton–Okounkov bodies of D with respect to valuations of the field of rational functions of X centred at O, or, equivalently, with respect to a flag (E, p) which is infinitely nearO, in the sense that there is a sequence of blowups X′→ X, mapping the smooth, irreducible rational curve E⊂ X′ to O. The main objective of this paper is to start a systematic study of the variation of these infinitesimal Newton–Okounkov bodies as (E, p) varies, focusing on the case X= P2.

AB - © 2016, Springer-Verlag Italia. Given a smooth projective algebraic surface X, a point O∈ X and a big divisor D on X, we consider the set of all Newton–Okounkov bodies of D with respect to valuations of the field of rational functions of X centred at O, or, equivalently, with respect to a flag (E, p) which is infinitely nearO, in the sense that there is a sequence of blowups X′→ X, mapping the smooth, irreducible rational curve E⊂ X′ to O. The main objective of this paper is to start a systematic study of the variation of these infinitesimal Newton–Okounkov bodies as (E, p) varies, focusing on the case X= P2.

KW - Algebraic geometry

KW - Linear system

KW - Newton-Okounkov body

KW - Valuation

KW - Valuative tree

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DO - https://doi.org/10.1007/s12215-016-0285-3

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VL - 66

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