Newton-Okounkov bodies and normal toric degenerations

TL;DR

This paper studies conditions under which Newton-Okounkov bodies induce normal toric degenerations of varieties, providing algorithms and flags for del Pezzo surfaces to ensure normality and finite generation of associated semigroups.

Contribution

It introduces criteria for the normality of the toric degenerations and develops an algorithm for del Pezzo surfaces to find flags ensuring normal, finitely generated value semigroups.

Findings

Identifies conditions for the normality of the toric degeneration.

Provides an algorithm to find flags on del Pezzo surfaces ensuring normal semigroups.

Shows existence of flags inducing normal degenerations for all divisors on certain surfaces.

Abstract

Anderson proved that the finite generation of the value semigroup $\Gamma_{Y_\bullet}(D)$ in the construction of the Newton-Okounkov body $\Delta_{Y_\bullet}(D)$ induces a toric degeneration of the corresponding variety $X$ to some toric variety $X_0$. In this case the normalization of $X_0$ is the normal toric variety corresponding to the rational polytope $\Delta_{Y_\bullet}(D)$. Since $X_0$ is not normal in general this correspondence is rather implicit. In this article we investigate in conditions to assure that $X_0$ is normal, by comparing the Hilbert polynomial with the Ehrhart polynomial. In the case of del Pezzo surfaces this will result in an algorithm which outputs for a given divisor $D$ a flag $Y_\bullet$ such that the value semigroup in question is indeed normal. Furthermore, we will find flags on del Pezzo surfaces and on some particular weak del Pezzo surfaces which induce normal toric degenerations for all possible divisors at once. We will prove that in this case the global value semigroup $\Gamma_{Y_\bullet}(X)$ is finitely generated and normal.