Okounkov bodies of finitely generated divisors (Q2929679)

Let \(X\) be an \(n\)-dimensional complex algebraic variety, \(D\) a Cartier divisor on \(X\), and \(v\) a valuation on \(\mathbb C(X)\) with values in \(\mathbb Z^n\). With this data, one can associate the \textit{Newton-Okounkov convex body} \(\Delta_v(X,D)\subset\mathbb R^n\) that generalizes Newton (or moment) polytopes from toric geometry and captures geometric properties of \(X\) and \(D\). In particular, when \(X\) is a projective toric variety, \(D\) is the divisor of a hyperplane section, and \(v=v_0\) assigns to a Laurent polynomial its lowest degree term, then \(\Delta_{v_0}(X,D)\) is the Newton polytope of \(X\). In general, \(\Delta_v(X,D)\) is not necessarily a polytope, and even when it is a polytope it might be non rational.NEWLINENEWLINEFrom a geometric viewpoint, it is natural to define valuations on \(\mathbb C(X)\) via the orders of vanishing of rational functions along hypersurfaces. Such geometric valuations come from a flag of subvarieties \(Y_\bullet:=(X=Y_0\supset Y_1\supset\ldots\supset Y_n=\{\mathrm{pt}\})\) where \(Y_{i+1}\) has codimension one in \(Y_i\). For instance, the valuation \(v_0\) in the toric case can be obtained this way if one takes a torus invariant flag \(Y_\bullet\). The authors show that if \(D\) is a big divisor and the section ring of \(D\) is finitely generated, then there exists a flag \(Y_\bullet\) such that the Newton-Okounkov polytope \(\Delta_{Y_\bullet}(X,D)\) is a rational simplex.NEWLINENEWLINEThis might be viewed as a generalization of the following statement: the Newton polytope of a polynomial is a simplex after a generic change of variables. However, such a change of variables in the toric case is not compatible with the structure of a toric variety unless the Newton polytope is a simplex. Thus geometric information about \(X\) captured by the Newton polytope might get lost when the Newton polytope is replaced by a simplex. In general, the flag \(Y_\bullet\) constructed by the authors does not seem to produce the Newton-Okounkov polytope that captures the most of geometry of \(X\).NEWLINENEWLINEThe authors also show that if \(X\) is a surface, and \(D\) is any big divisor then \(\Delta_{Y_\bullet}(X,D)\) is a rational simplex for an appropriate choice of a flag \(Y_\bullet\).