Okounkov bodies and restricted volumes along very general curves
Let \(X\) be a projective variety of dimension \(d\) and \(D\) a big divisor on \(X\). The \textit{Okounkov body} is a convex body \(\Delta(D)\) contained in a real vector space of dimension \(d\) which encodes some asymptotic invariants of the linear series \(|mD|\) as \(m\) goes to infinity. For example the volume of \(D\) (defined as the limit when \(m\) goes to infinity of the quotient \(d!\;h^0(X,mD)/m^d\)) is equal to \(d!\) times the Euclidean volume of \(\Delta (D)\). The construction of the Okounkov bodies depends on the choice of an admissible flag of irreducible subvarieties \[ Y_{\bullet}:=\{X=Y_0 \supset Y_1 \supset \dots \supset Y_d=\{pt\}\} \] where codim\(_X(Y_i)=i\) and \(Y_i\) is nosingular at \(Y_d\) for any \(i\). This flag allows to define for any \(s \in H^0(X,mD)\) a \(d\)-tuple of nonnegative integers \((v_1(s), \dots, v_d(s))\) in the following way: work in an open set of nonsingular points of \(X\) in such a way that the corresponding \(Y_i\)'s are smooth; define \(v_1(s)=\)ord\(_{Y_i}(s)\); then choose a local equation for \(Y_1\) in \(X\) and consider the corresponding \(\tilde{s_1} \in H^0(X,mD-v_1(s)Y_1)\); it does not vanish identically on \(Y_1\) so that \(v_2(s)=\)ord\(_{Y_2}(s_1)\), being \(s_1\) the restriction to \(Y_1\) of \(\tilde{s_1}\); and so on. In this way we get \(\Gamma(D)_m\) the set of \(d\)-tuples constructed as described above. Then \(\Delta_{Y_{\bullet}}(D)\) is the closed convex hull of the set \(\bigcup_{m \geq 1}{1 \over m}\Gamma(D)_m\). It was known that if \(D_1\) and \(D_2\) are numerically equivalent then their Okounkov bodies associated to any admissible flag are the same \(\Delta_{Y_{\bullet}}(D_1)=\Delta_{Y_{\bullet}}(D_2)\). The paper under review shows that in the complex case and when \(X\) is normal also the inverse of this statement is true: if \(\Delta_{Y_{\bullet}}(D_1)=\Delta_{Y_{\bullet}}(D_2)\) for any admissible flag \(Y_{\bullet}\) then \(D_1\) and \(D_2\) are numerically equivalent (Theorem A). This result is a consequence of Theorem B in which it is shown that the restricted volume of \(D\) to a very general complete-intersection curve can be read off from its Okounkov body.
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