# Newton–Okounkov bodies sprouting on the valuative tree

Newton–Okounkov bodies sprouting on the valuative tree Given a smooth projective algebraic surface X, a point $$O\in X$$ 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 near O, in the sense that there is a sequence of blowups $$X' \rightarrow X$$ X ′ → X , mapping the smooth, irreducible rational curve $$E\subset X'$$ 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 $$X=\mathbb {P}^2$$ X = P 2 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Rendiconti del Circolo Matematico di Palermo Springer Journals

# Newton–Okounkov bodies sprouting on the valuative tree

, Volume 66 (2) – Nov 15, 2016
34 pages

/lp/springer_journal/newton-okounkov-bodies-sprouting-on-the-valuative-tree-HLe17eN0iu
Publisher
Springer Milan
Subject
Mathematics; Mathematics, general; Algebra; Geometry; Analysis; Applications of Mathematics
ISSN
0009-725X
eISSN
1973-4409
D.O.I.
10.1007/s12215-016-0285-3
Publisher site
See Article on Publisher Site

### Abstract

Given a smooth projective algebraic surface X, a point $$O\in X$$ 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 near O, in the sense that there is a sequence of blowups $$X' \rightarrow X$$ X ′ → X , mapping the smooth, irreducible rational curve $$E\subset X'$$ 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 $$X=\mathbb {P}^2$$ X = P 2 .

### Journal

Rendiconti del Circolo Matematico di PalermoSpringer Journals

Published: Nov 15, 2016

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