Newton-Okounkov bodies of exceptional curve valuations

2017/05/10 by Galindo, Carlos, Monserrat, Francisco, Moyano-Fernández, Julio José +1
#13A18 #14C20 #14E15 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1705.03948

Abstract

We prove that the Newton-Okounkov body of the flag E_\bullet:= \ X=X_r ⊃ E_r ⊃ \q\ \, defined by the surface X and the exceptional divisor E_r given by any divisorial valuation of the complex projective plane ℙ², with respect to the pull-back of the line-bundle O_ℙ² (1) is either a triangle or a quadrilateral, characterizing when it is a triangle or a quadrilateral. We also describe the vertices of that figure. Finally, we introduce a large family of flags for which we determine explicitly their Newton-Okounkov bodies which turn out to be triangular.

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