##
**Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory.**
*(English)*
Zbl 1270.14022

The main aim of the paper is to generalize the notion of the Newton polytope of a polynomial to several more abstract contexts. The authors begin with proving general results on semigroups of integral points. Let \(S\) be an additive semigroup in the lattice \(\mathbb{Z}^n\subset\mathbb{R}^n\). Consider the group \(G(S)\) consisting of all the linear combinations \(\sum_i k_ia_i\), where \(a_i\in S\) and \(k_i\in\mathbb{Z}\), and the closed convex cone \(\text{Con}(S)\) that is the closure of the set of all linear combinations \(\sum_i \lambda_ia_i\) for \(a_i\in S\) and \(\lambda_i\geq 0\). Let us define the regularization of \(S\) as the semigroup \(\text{Reg}(S)=G(S)\cap\text{Con}\). Theorem 1.6 states that, for every closed strongly convex cone \(\text{Con}\) inside \(\text{Con}(S)\) that intersects the boundary of \(\text{Con}(S)\) only at the origin, there exists a constant \(N>0\) such that each point in the group \(G(S)\) that lies in \(\text{Con}\) and whose distance from the origin is bigger than \(N\) belongs to \(S\).

Assume that the cone \(\text{Con}(S)\) is strognly convex and denote by \(L\) its linear span in \(\mathbb{R}^n\). Let \(\dim L=q+1\). Fix a rational \(q\)-dimensional subspace \(M_0\subset L\) intersecting \(\text{Con}(S)\) only at the origin. Let \(M_k\), \(k\in\mathbb{Z}_{\geq 0}\), be the family of \(q\)-dimensional affine subspaces parallel to \(M_0\) such that each \(M_k\) intersects the cone \(\text{Con}(S)\) as well as the group \(G(S)\). The Hilbert function \(H_S(k)\) of a semigroup \(S\) is defined as the number of points in \(S\cap M_k\). Let us define the Newton-Okounkov body of \(S\) as the intersection \(\Delta(S)=\text{Con}(S)\cap M_1\). The authors prove that the function \(H_S(k)\) grows like \(a_qk^q\), where the coefficient \(a_q\) is equal to the (normalized in the appropriate way) \(q\)-dimensional volume of \(\Delta(S)\). Also the growth of the Hilbert function corresponding to the semigroup generated by \(S\cap M_k\) is characterized.

The second part is devoted to graded algebras and their Hilbert functions. Let \(F\) be a finitely generated extension of an algebraically closed field \(\mathbb{K}\) and \(F[t]\) be the algebra of polynomials over \(F\). Consider a nonzero finite dimensional subspace \(L\) of \(F\) over \(\mathbb{K}\) and let \(A_L:=\bigoplus_{k\geq 0}L^kt^k\). It is a homogeneous \(\mathbb{K}\)-subalgebra of \(F[t]\) generated by finitely many elements of degree \(1\). A homogeneous subalgebra \(A\) in \(F[t]\) is said to be of integral type if it is a finite module over some subalgebra \(A_L\). Moreover, a homogeneous subalgebra \(A\) is of almost integral type if it is contained in an algebra of integral type.

Every \(\mathbb{Z}^n\)-valued valuation of the field \(F\) maps the set of nonzero elements of a homogeneous subalgebra \(A\subseteq F[t]\) to a semigroup of integral points in \(\mathbb{Z}^n\times\mathbb{Z}_{\geq 0}\). This allows to define the Newton-Okounkov body of a graded algebra \(A\) and interpret the results on semigroups stated above in terms of graded algebras. In particular, it is proved that the Hilbert function of an algebra of almost integral type has polynomial growth.

The third part of the paper deals with algebraic geometry. Let \(X\) be an \(n\)-dimensional irreducible variety over \(\mathbb{K}\) with \(F=\mathbb{K}(X)\) being the field of rational functions. With a finite-dimensional subspace \(L\subset F\) one associates the Kodaira rational map \(\Phi_L: X \to \mathbb{P}(L^*)\). Let \(Y_L\) be the closure of the image of this map. The algebra \(A_L\) can be identified with the homogenenous coordinate ring of \(Y_L\subseteq\mathbb{P}(L^*)\). Algebras of integral type are related in these terms to the rings of sections of ample line bundles, while algebras of almost integral type are related to the rings of sections of arbitrary line bundles (see Theorems 3.7 and 3.8).

The Fujita approximation theorem in the theory of divisors states that the so-called volume of a big divisor can be approximated by the self-intersection numbers of ample divisors. The above results on graded algebras can be regarded as an abstract analogue of this result. In fact, it leads to a generalization of Fujita’s result for any divisor or even any graded linear system on any complete variety (Corollary 3.11).

In the last part, a far-reaching generalization of the Kushnirenko theorem is obtained and a new version of the Hodge inequality is found. Also the authors give elementary proofs of the Alexandrov-Fenchel inequality and its analogue in algebraic geometry.

Assume that the cone \(\text{Con}(S)\) is strognly convex and denote by \(L\) its linear span in \(\mathbb{R}^n\). Let \(\dim L=q+1\). Fix a rational \(q\)-dimensional subspace \(M_0\subset L\) intersecting \(\text{Con}(S)\) only at the origin. Let \(M_k\), \(k\in\mathbb{Z}_{\geq 0}\), be the family of \(q\)-dimensional affine subspaces parallel to \(M_0\) such that each \(M_k\) intersects the cone \(\text{Con}(S)\) as well as the group \(G(S)\). The Hilbert function \(H_S(k)\) of a semigroup \(S\) is defined as the number of points in \(S\cap M_k\). Let us define the Newton-Okounkov body of \(S\) as the intersection \(\Delta(S)=\text{Con}(S)\cap M_1\). The authors prove that the function \(H_S(k)\) grows like \(a_qk^q\), where the coefficient \(a_q\) is equal to the (normalized in the appropriate way) \(q\)-dimensional volume of \(\Delta(S)\). Also the growth of the Hilbert function corresponding to the semigroup generated by \(S\cap M_k\) is characterized.

The second part is devoted to graded algebras and their Hilbert functions. Let \(F\) be a finitely generated extension of an algebraically closed field \(\mathbb{K}\) and \(F[t]\) be the algebra of polynomials over \(F\). Consider a nonzero finite dimensional subspace \(L\) of \(F\) over \(\mathbb{K}\) and let \(A_L:=\bigoplus_{k\geq 0}L^kt^k\). It is a homogeneous \(\mathbb{K}\)-subalgebra of \(F[t]\) generated by finitely many elements of degree \(1\). A homogeneous subalgebra \(A\) in \(F[t]\) is said to be of integral type if it is a finite module over some subalgebra \(A_L\). Moreover, a homogeneous subalgebra \(A\) is of almost integral type if it is contained in an algebra of integral type.

Every \(\mathbb{Z}^n\)-valued valuation of the field \(F\) maps the set of nonzero elements of a homogeneous subalgebra \(A\subseteq F[t]\) to a semigroup of integral points in \(\mathbb{Z}^n\times\mathbb{Z}_{\geq 0}\). This allows to define the Newton-Okounkov body of a graded algebra \(A\) and interpret the results on semigroups stated above in terms of graded algebras. In particular, it is proved that the Hilbert function of an algebra of almost integral type has polynomial growth.

The third part of the paper deals with algebraic geometry. Let \(X\) be an \(n\)-dimensional irreducible variety over \(\mathbb{K}\) with \(F=\mathbb{K}(X)\) being the field of rational functions. With a finite-dimensional subspace \(L\subset F\) one associates the Kodaira rational map \(\Phi_L: X \to \mathbb{P}(L^*)\). Let \(Y_L\) be the closure of the image of this map. The algebra \(A_L\) can be identified with the homogenenous coordinate ring of \(Y_L\subseteq\mathbb{P}(L^*)\). Algebras of integral type are related in these terms to the rings of sections of ample line bundles, while algebras of almost integral type are related to the rings of sections of arbitrary line bundles (see Theorems 3.7 and 3.8).

The Fujita approximation theorem in the theory of divisors states that the so-called volume of a big divisor can be approximated by the self-intersection numbers of ample divisors. The above results on graded algebras can be regarded as an abstract analogue of this result. In fact, it leads to a generalization of Fujita’s result for any divisor or even any graded linear system on any complete variety (Corollary 3.11).

In the last part, a far-reaching generalization of the Kushnirenko theorem is obtained and a new version of the Hodge inequality is found. Also the authors give elementary proofs of the Alexandrov-Fenchel inequality and its analogue in algebraic geometry.

Reviewer: Ivan V. Arzhantsev (Moscow)

### MSC:

14M17 | Homogeneous spaces and generalizations |

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |

### Keywords:

semigroup of integral points; convex body; graded algebra; Hilbert function; Hodge index theory; mixed volume; Alexandrov-Fenchel inequality; Bernstein-Kushnirenko theorem; Cartier divisor; linear series
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\textit{K. Kaveh} and \textit{A. G. Khovanskii}, Ann. Math. (2) 176, No. 2, 925--978 (2012; Zbl 1270.14022)

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