Intersections of Hypersurfaces and Ring of Conditions of a Spherical
Homogeneous Space

Intersections of Hypersurfaces and Ring
of Conditions of a Spherical Homogeneous Space††This paper is a contribution
to the Special Issue on Algebra, Topology, and Dynamics in Interaction in
honor of Dmitry Fuchs. The full collection is available at
https://www.emis.de/journals/SIGMA/Fuchs.html

Kiumars KAVEH † and Askold G. KHOVANSKII ‡§

K. Kaveh and A.G. Khovanskii

† Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA
<EMAIL_ADDRESS>
‡ Department of Mathematics, University of Toronto, Toronto, Canada
<EMAIL_ADDRESS>
§ Moscow Independent University, Moscow, Russia

Received November 04, 2019, in final form March 14, 2020; Published online
March 20, 2020

We prove a version of the BKK theorem for the ring of conditions of a
spherical homogeneous space $G/H$. We also introduce the notion of ring of
complete intersections, firstly for a spherical homogeneous space and secondly
for an arbitrary variety. Similarly to the ring of conditions of the torus,
the ring of complete intersections of $G/H$ admits a description in terms of
volumes of polytopes.

BKK theorem; spherical variety; Newton–Okounkov polytope; ring of conditions

14M27; 14M25; 14M10

##  1 Introduction

Let $\Gamma_{1},\ldots,\Gamma_{n}$ be a set of $n$ hypersurfaces in the
$n$-dimensional complex torus $(\mathbb{C}^{*})^{n}$ defined by equations
$P_{1}=0,\ldots,P_{n}=0$ where $P_{1},\ldots,P_{n}$ are generic Laurent
polynomials with given Newton polyhedra $\Delta_{1},\ldots,\Delta_{n}$. The
Bernstein–Koushnirenko–Khovanskii theorem (or BKK theorem, see [2, 16, 18])
claims that the intersection number of the hypersurfaces
$\Gamma_{1},\ldots,\Gamma_{n}$ is equal to the mixed volume of
$\Delta_{1},\ldots,\Delta_{n}$ multiplied by $n!$. In particular when
$\Delta_{1}=\cdots=\Delta_{n}=\Delta$ this theorem can be regarded as giving a
formula for the degree of an ample divisor class in a projective toric
variety.

Spherical varieties are a generalization of toric varieties to actions of
reductive groups. Let $G$ be a connected reductive algebraic group. A normal
variety $X$ with an action of $G$ is called spherical if a Borel subgroup of
$G$ has an open orbit. When $G$ is a torus we get the definition of a toric
variety. On the other hand, spherical varieties are generalizations of flag
varieties $G/P$ as well. Similarly to toric varieties, the geometry of
spherical varieties and their orbit structure can be read off from
combinatorial and convex geometric data of fans and convex polytopes.

The BKK theorem has been generalized to spherical varieties by B.
Kazarnovskii, M. Brion and the authors of the present paper. In particular,
this generalization provides a formula for the degree of an ample
$G$-linearized line bundle $L$ on a projective spherical variety $X$ (see [4,
14, 10]). The formula is given as the integral of a certain explicit function
over the moment polytope (also called the Kirwan polytope) of $(X,L)$. This
formula can be equivalently expressed as volume of a larger polytope
associated to $(X,L)$. It is usually referred to as a Newton–Okounkov polytope
of $(X,L)$ (see Section 6).

In [11, 19], the BKK theorem has been generalized to intersection numbers of
generic members of linear systems on any irreducible algebraic variety. The
Newton–Okounkov bodies play a key role in this generalization. All
generalizations mentioned above deal with the intersection numbers of
hypersurfaces in an algebraic variety.

Following the original ideas of Schubert, in the early 1980s De Concini and
Procesi developed an intersection theory for algebraic cycles (which are not
necessarily hypersurfaces) in a symmetric homogeneous space $X=G/H$ (see [5]).
Their intersection theory, named the ring of conditions of $X$, can be
automatically generalized to a spherical homogeneous space $X$.

For an algebraic torus $(\mathbb{C}^{*})^{n}$ and more generally a
horospherical homogeneous space $X$, nice descriptions of the ring of
conditions are known. Moreover, combinatorial descriptions of cohomology rings
as well as rings of conditions of interesting subclasses of spherical
varieties have been obtained via equviariant cohomology methods by several
authors (see [3, 20, 24]). In our opinion, it is unlikely that for an
arbitrary spherical homogeneous space, one can find such nice and transparent
descriptions for the ring of conditions in terms of combinatorial data.

The contributions of the present paper are the following:

  * (1)

We give a version of the BKK theorem in the ring of conditions of a spherical
homogeneous space $X$, namely a formula for the intersection numbers of
hypersurfaces in the ring of conditions in terms of volumes of polytopes
(Theorem 7.3). This modified version is as transparent as the original BKK
theorem.

  * (2)

Along the way, we introduce the notion of ring of complete intersections of a
spherical homogeneous space. Unlike the ring of conditions, this ring admits a
nice description in terms of volumes of polytopes (see Theorem 9.2, also cf.
[8]). In the case when $X$ is a torus $(\mathbb{C}^{*})^{n}$, the ring of
complete intersections is isomorphic to the ring of conditions.

  * (3)

We also introduce the ring of complete intersections for any irreducible
algebraic variety (not necessarily equipped with a group action) and give a
description of this ring in terms of volumes of convex bodies (Section 10).

We would like to point out that there is a big difference between the
construction/definition of the ring of complete intersections for a spherical
homogeneous space (as in (2) above) and the construction/definition of the
ring of complete intersections for an arbitrary variety (as in (3) above). The
first ring is defined similarly to the ring of conditions except that in the
definition of a cycle one should be more careful and consider (strongly)
transversal hypersurfaces, The second ring is much larger and has a very
different nature. It is defined by a general algebra construction that
associates a (Poincaré duality) algebra to a vector space equipped with an
$n$-linear form (Section 8). Nevertheless, the ring of complete intersections
of a spherical homogeneous space can be described using the same general
algebra construction.

##  2 Preliminaries on spherical varieties

In the rest of the paper we will use the following notation about reductive
groups:

  * –

$G$ denotes a connected complex reductive algebraic group.

  * –

$B$ a Borel subgroup of $G$ and $U$, $T$ the maximal unipotent subgroup and a
maximal torus contained in $B$ respectively.

  * –

$G/H$ denotes a spherical homogeneous space, we let $\dim(G/H)=n$.

We recall some basic background material about spherical varieties. A normal
$G$-variety $X$ is called _spherical_ if a Borel subgroup (and hence any Borel
subgroup) has a dense orbit. If $X$ is spherical, it has a finite number of
$G$-orbits as well as a finite number of $B$-orbits. Spherical varieties are a
generalization of toric varieties for actions of reductive groups. Analogous
to toric varieties, the geometry of spherical varieties can be read off from
associated convex polytopes and convex cones. For an overview of the theory of
spherical varieties, we refer the reader to [22].

It is a well-known fact that if ${\mathcal{L}}$ is a $G$-linearized line
bundle on a spherical variety, then the space of sections
$H^{0}(X,{\mathcal{L}})$ is a multiplicity free $G$-module. For a quasi-
projective $G$-variety $X$, this is equivalent to $X$ being spherical.

Some important examples of spherical varieites and spherical homogeneous
spaces are the following:

  * (1)

When $G$ is a torus, the spherical $G$-varieties are exactly toric varieties.

  * (2)

By the Bruhat decomposition the flag variety $G/B$ and the partial flag
varieties $G/P$ are spherical $G$-varieties.

  * (3)

Let $G\times G$ act on $G$ from left and right. Then the stabilizer of the
identity is $G_{{\rm diag}}=\\{(g,g)\,|\,g\in G\\}$. Thus $G$ can be
identified with the homogeneous space $(G\times G)/G_{{\rm diag}}$. Also by
the Bruhat decomposition, this is a spherical $(G\times G)$-homogeneous space.

  * (4)

Let $\mathcal{Q}$ be the set of all smooth quadrics in $\mathbb{P}^{n}$. The
group $G={\rm PGL}(n+1,\mathbb{C})$ acts transitively on $\mathcal{Q}$. The
stabilizer of the quadric $x_{0}^{2}+\cdots+x_{n}^{2}=0$ (in the homogeneous
coordinates) is $H={\rm PO}(n+1,\mathbb{C})$ and hence $\mathcal{Q}$ can be
identified with the homogeneous space ${\rm PGL}(n+1,\mathbb{C})/{\rm
PO}(n+1,\mathbb{C})$. The subgroup ${\rm PO}(n+1,\mathbb{C})$ is the fixed
point set of the involution $g\mapsto(g^{t})^{-1}$ of $G$ and hence
$\mathcal{Q}$ is a symmetric homogeneous space. In particular, $\mathcal{Q}$
is spherical. The homogeneous space $\mathcal{Q}$ plays an important role in
classical enumerative geometry (see [5]).

Throughout the rest of the paper we will fix a spherical homogeneous space
$G/H$.

##  3 Good compactification theorem

The ring of conditions of $X=G/H$ is a version of the Chow ring for a (usually
not complete) spherical homogenous space $X$. The existence of a good
compactification plays a crucial role in this intersection theory. One can
define the ring of conditions more geometrically considering algebraic cycles
as stratified analytic varieties and using cohomology rings instead of Chow
rings. In this section we recall preliminaries on transversal intersections of
stratified varieties and will state the theorem on existence of a good
compactification.

Let $Y$ be an algebraic variety. A stratification of $Y$ is a decomposition of
$Y$ into a disjoint union of smooth algebraic subvarieties (possibly of
different dimensions). Each algebraic variety $Y$ admits the following
canonical stratification: Let $Y=Y_{0}\supset
Y_{1}\supset\dots\supset\varnothing$ be the decreasing set of subvarieties
where $Y_{i+1}$ is the set of singular points in $Y_{i}$. The canonical
stratification of $Y$ is the partition $Y=\bigcup_{i}Y_{i}^{0}$ where
$Y_{i}^{0}=Y_{i}\setminus Y_{i+1}.$

Two subvarieties $Y,Z$ of an ambient smooth variety $X$ are transversal if
there are stratifications $Y=\bigcup_{i}Y_{i}^{0}$, $Z=\bigcup Z_{j}^{0}$ such
that any pair $Y_{i}^{0}$, $Z_{j}^{0}$ of smooth subvarieties are transversal
in $X$. Similarly one can define transversality of several subvarieties.

Let $X=G/H$ be a homogeneous space and $Y$, $Z$ smooth subvarieties. The
Kleiman transversality theorem (which is a version of the famous Thom
transversality theorem) states that: _for almost all $g\in G$ the varieties
$Y$ and $g\cdot Z$ are transversal in $X$, i.e., the subset $G^{0}\subset G$
such that the subvarieties $Y$ and $gZ$ are transversal in $X$ contains a
nonempty Zariski open in $G$_.

The next important result is due to De Concini and Procesi [5]. They
considered the case when $G/H$ is a symmetric homogeneous space but their
proof, more or less without change, works for a spherical homogeneous space as
well.

######  Theorem 3.1 (existence of good compactification).

Let $X=G/H$ be a spherical homogeneous space. Let $Y\subset X$ be a
subvariety. Then there exists a complete $G$-variety $M$ that contains $X$ as
its open $G$-orbit and for each $G$-orbit $O\subset M$ intersecting
$\overline{Y}\subset M$ we have $\operatorname{codim}(\overline{Y}\cap
O)=\operatorname{codim}(O)+\operatorname{codim}(Y)$.

Let us say that two subvarieties $Y$, $Z$ of $X$ are strongly transversal in
$X$ if there is a complete $G$-variety $M\supset X$ such that: (1) $M$ is a
good compactification for $Y$ and $Z$ and (2) for each orbit $O\subset M$ the
intersections $\overline{Y}\cap O$ and $\overline{Z}\cap O$ of their closures
$\overline{Y}$ and $\overline{Z}$ with $O$ are transversal in $O$. Similarly,
one defines strong transversality of several subvarieties.

Using Kleiman’s theorem and the good compactification theorem one can show for
any subvarieties $Y,Z\subset X$ there is a Zariski open subset $G_{0}\subset
G$ such that, for any $g\in G_{0}$, the varieties $Y$ and $g\cdot Z$ are
strongly transversal in $X$.

##  4 Ring of conditions of $\boldsymbol{G/H}$

In this section we consider a variant of intersection theory for a spherical
homogeneous space $G/H$ (which is often a non-complete variety) called the
ring of conditions $\mathcal{R}(G/H)$. Similar to the Chow ring, the elements
of $\mathcal{R}(G/H)$ are formal linear combinations of subvarieties in $G/H$
but considered up to a different and stronger equivalence. The definition of
ring of conditions goes back to De Concini and Procesi in their fundamental
paper [5]. They introduced it as a natural ring in which one can study many
classical problems from enumerative geometry (this is related to Hilbert’s
Fifteenth problem). They also showed that $\mathcal{R}(G/H)$ can be realized
as a direct limit of Chow rings of all smooth equivariant compactifications of
$G/H$.

Consider the set $\mathcal{C}$ of algebraic cycles in $G/H$. That is, every
element of $\mathcal{C}$ is a formal linear combination $V=\sum_{i}a_{i}V_{i}$
with $a_{i}\in\mathbb{Z}$ and $V_{i}$ irreducible subvarieties. Clearly, with
the formal addition operation of cycles, $\mathcal{C}$ is an abelian group. If
all the subvarieties $V_{i}$ in $V$ have the same dimension $k$ we say that
$V$ is a $k$-cycle. For $0\leq k\leq n$, the subgroup of $k$-cycles is denoted
by $\mathcal{C}_{k}$. For a cycle $V=\sum_{i}a_{i}V_{i}$ and $g\in G$ we
define $g\cdot V$ to be $\sum_{i}a_{i}(g\cdot V_{i})$. A $0$-cycle is just a
formal linear combination of points. If $P=\sum_{i}a_{i}P_{i}$ is a $0$-cycle
where the $P_{i}$ are points, we let $|P|=\sum_{i}a_{i}$.

Let $Y$ and $Z$ be strongly transversal irreducible subvarieties, and let
$Y\cap Z$ be a union of irreducible components $T$. We then define the
intersection product $Y\cdot Z$ to be the cycle

$Y\cdot Z=\sum_{T}T.$

By linearity the intersection product can be extended to algebraic cycles
$Y=\sum_{i}a_{i}Y_{i}$ and $Z=\sum_{j}b_{j}Z_{j}$ such that for all $i$, $j$
varieties $Y_{i}$ and $Z_{j}$ are strongly transversal to each other.

We now define an equivalence relation $\sim$ on the set of algebraic cycles as
follows. Let $V=\sum_{i}a_{i}V_{i}$,
$V^{\prime}=\sum_{j}b_{j}V^{\prime}_{j}\in\mathcal{C}_{k}$ be algebraic cycles
of dimension $k$. Let $Z$ be an irreducible subvariety of complementary
dimension $n-k$. One knows that for generic $g\in G$, the subvariety $g\cdot
Z$ intersects all the $V_{i}$ and $V^{\prime}_{j}$ in a strongly transversal
fashion. Thus, for generic $g\in G$, the intersection products $V\cdot(g\cdot
Z)$ and $V^{\prime}\cdot(g\cdot Z)$ are defined and are $0$-cycles. We say
that $V\sim V^{\prime}$ if for any $Z$ as above and for generic $g\in G$ we
have:

$\displaystyle|V\cap(g\cdot Z)|=|V^{\prime}\cap(g\cdot Z)|.$ (4.1)

That is, $V\sim V^{\prime}$ if they intersect general translates of any
subvariety of complementary dimension at the same number of points.

One verifies using good compactifications that if $Y_{1}\sim Y_{2}$ and
$Z_{1}\sim Z_{2}$ then for generic $g\in G$ the intersection products
$Y_{1}\cdot gZ_{1}$ and $Y_{2}\cdot gZ_{2}$ are equivalent to each other. Thus
the intersection product of strongly transversal subvarieties induces an
intersection operation on the quotient $\mathcal{C}/{\sim}$.

######  Definition 4.1.

The ring of conditions of $G/H$ is $\mathcal{C}/{\sim}$ with the ring
structure coming from addition and intersection product of cycles.

The following example, due to De Concini and Procesi, shows the assumption
that $G/H$ is spherical is important and the ring of conditions is not well-
defined for all homogeneous spaces.

######  Example 4.2.

Take the $3$-dimensional affine space $\mathbb{C}^{3}$ regarded as an additive
group. We would like to show that the ring of conditions of $\mathbb{C}^{3}$
does not exist. By contradiction suppose that it exists. Consider the surface
(quadric) $S$ in $\mathbb{C}^{3}$ defined by the equation $y=zx$. The
intersection of a horizontal plane $z=a$ and $S$ is the line $y=ax$. Thus for
almost all $a\in\mathbb{C}$ the lines $y=ax$, $z=a$ must be equivalent in the
ring of conditions of $\mathbb{C}^{3}$. On the other hand we claim that two
skew lines $\ell_{1}$ and $\ell_{2}$ cannot be equivalent. This is because one
can find a $2$-dimensional plane $P$ such that any translate of $P$ intersects
$\ell_{1}$ but no translate of $P$ intersects $\ell_{2}$ unless it contains
$\ell_{2}$. The contradiction shows that the ring of conditions of
$\mathbb{C}^{3}$ is not well-defined.

##  5 Ring of complete intersections of $\boldsymbol{G/H}$

In this section we propose an analogue of the ring of conditions constructed
using only (non-degenerate) complete intersections in a spherical homogeneous
space $G/H$.

######  Definition 5.1.

A non-degenerate complete intersection in $G/H$ is an intersection of several
strongly transversal hypersurfaces $\Gamma_{1},\ldots,\Gamma_{k}\subset G/H$.

Consider the collection $\mathcal{C}^{\prime}_{k}$ of all $k$-dimensional
complete intersection cycles, that is, formal linear combinations
$\sum_{i}a_{i}V_{i}$ where each $V_{i}$ is a $k$-dimensional non-degenerate
complete intersection in $G/H$. We let
$\mathcal{C}^{\prime}=\bigoplus_{i=0}^{n}\mathcal{C}^{\prime}_{k}$.

As in the construction of ring of conditions (Section 4) we define an
equivalence relation $\sim$ on $\mathcal{C}^{\prime}$ as follows: let $V$,
$V^{\prime}\in\mathcal{C}^{\prime}_{k}$ be complete intersection $k$-cycles.
We say that $V\sim V^{\prime}$ if for any complete intersection $(n-k)$-cycle
$Z$ and generic $g\in G$ we have $|V\cap(g\cdot Z)|=|V^{\prime}\cap(g\cdot
Z)|$ (recall $n=\dim(G/H)$). Note that in defining $\sim$ we only use $Z$ that
are complete intersections.

######  Definition 5.2.

The ring of complete intersections $\mathcal{R}^{\prime}(G/H)$ is
$\mathcal{C}^{\prime}/{\sim}$ with the ring structure coming from addition and
intersection product of complete intersection cycles.

In the same fashion as for the ring of conditions, it can be verified that the
ring $\mathcal{R}^{\prime}(G/H)$ is well-defined. The inclusion
$\mathcal{C}^{\prime}\subset\mathcal{C}$ induces a ring homomorphism
$\mathcal{R}^{\prime}(G/H)\to\mathcal{R}(G/H)$. In general, this homomorphism
is neither injective nor surjective.

In Section 9 we will give a description of $\mathcal{R}^{\prime}(G/H)$ in
terms of volumes of polytopes.

##  6 Newton–Okounkov polytopes for spherical varieties

In this section we recall the notion of a Newton–Okounkov polytope associated
to a $G$-linear system on a spherical homogeneous space $G/H$. The volume of
this polytope gives the self-intersection number of the linear system.

Let $X$ be an $n$-dimensional projective spherical $G$-variety with a
$G$-linearized very ample line bundle ${\mathcal{L}}$. One associates a convex
polytope $\Delta(X,{\mathcal{L}})$ to $(X,{\mathcal{L}})$. The construction
depends on the combinatorial choice of a reduced word decomposition
${\underline{w}_{0}}$ for the longest element $w_{0}$ in the Weyl group of
$G$.

A main property of the Newton–Okounkov polytope $\Delta(X,{\mathcal{L}})$ is
that its volume gives a formula for the self-intersection number of divisor
class of ${\mathcal{L}}$. Namely,

$\displaystyle
c_{1}({\mathcal{L}})^{n}=n!\operatorname{vol}_{n}(\Delta(X,{\mathcal{L}})).$
(6.1)

The formula (6.1) is equivalent to the Brion–Kazarnovskii formula for the
degree of a projective spherical variety (see [8, Theorem 2.5] as well as [14]
and [4]).

######  Remark 6.1.

The original versions of (6.1) in [14] and [4], do not express the answer as
volume of a Newton–Okounkov polytope but rather as integral of a certain
function over the moment polytope $\mu(X,{\mathcal{L}})$. The construction of
$\Delta(X,{\mathcal{L}})$ appears in [21] and [1] motivated by a question of
the second author.

######  Remark 6.2.

If $X$ is not normal the polytope $\Delta(X,{\mathcal{L}})$ can still be
defined using the integral closure of the ring of sections of ${\mathcal{L}}$
(in its field of fractions), see [10, Section 6.2].

More generally, let $X$ be a not-necessarily-projective spherical variety and
let $E$ be a $G$-linear system on $X$, that is, $E$ is a finite-dimensional
$G$-invariant subspace of $H^{0}(X,{\mathcal{L}})$ for a $G$-linearized line
bundle ${\mathcal{L}}$ on $X$. Extending the notion of intersection number of
divisors on complete varieties, one can talk about intersection number of
linear systems. Let $E_{1},\ldots,E_{n}$ be linear systems on $X$. The
intersection index $[E_{1},\ldots,E_{n}]$ is defined to be the number of
solutions in $X$ of a generic system of equations
$f_{1}(x)=\cdots=f_{n}(x)=0$, where $f_{i}\in E_{i}$. When counting the
solutions, we ignore the solutions $x$ at which all the sections in some
$E_{i}$ vanish. An important property of the intersection index is multi-
additivity with respect to product of linear systems. In [10, Section 6] the
authors define the Newton–Okounkov polytope $\Delta(E)$ and prove the
following.

######  Proposition 6.3.

The intersection index $[E,\ldots,E]$ is equal to
$n!\operatorname{vol}_{n}(\Delta(E))$.

From the polarization formula in linear algebra the following readliy follows.

######  Corollary 6.4.

Let ${\mathcal{L}}_{1},\ldots,{\mathcal{L}}_{n}$ be $G$-linearized very ample
line bundles on a spherical variety $X$ and take $G$-invariant linear systems
$E_{i}\subset H^{0}(X,{\mathcal{L}}_{i})$. For any subset
$I\subset\\{1,\ldots,n\\}$ let $\Delta_{I}=\Delta\big{(}\prod_{i\in
I}E_{i}\big{)}$. We have the following formula for the intersection index of
the $E_{i}$:

$\displaystyle(-1)^{n}[E_{1},\ldots,E_{n}]=-\sum_{i}\operatorname{vol}_{n}(\Delta_{i})+\sum_{i<j}\operatorname{vol}_{n}(\Delta_{i,j})+\cdots+(-1)^{n}\operatorname{vol}_{n}(\Delta_{1,\ldots,n}).$

######  Remark 6.5.

The above notion of the Newton–Okounkov polytope of a $G$-linear system over a
spherical variety is a special case of the more general notion of a
Newton–Okounkov body of a graded linear system on an arbitrary variety (see
[11, 19] and the references therein).

Suppose $E$ is a very ample $G$-linear system on $X$. That is, the Kodaira map
of $E$ gives an embedding of $X$ into the dual projective space
$\mathbb{P}(E^{*})$. Let $Y_{E}$ denote the closure of the image of $X$ in
$\mathbb{P}(E^{*})$ and let ${\mathcal{L}}$ be the $G$-linearized line bundle
on $Y_{E}$ induced by $\mathcal{O}(1)$ on $\mathbb{P}(E^{*})$. One then has
$\Delta(E)=\Delta(Y_{E},{\mathcal{L}})$. We note that in general $Y_{E}$ may
not be normal, nevertheless one can still define the polytope
$\Delta(Y_{E},{\mathcal{L}})$ (see Remark 6.2).

One can show that the map $E\mapsto\Delta(E)$ is piecewise additive in the
following sense. Let $C$ be a rational polyhedral cone generated by a finite
number of $G$-linear systems $E_{1},\ldots,E_{s}$. Then the there is a
rational polyhedral cone $\tilde{C}$ (living in some appropriate Euclidean
space) and a linear projection $\pi\colon\tilde{C}\to C$ such that for each
$G$-linear system $E\in C$ we have $\Delta(E)=\pi^{-1}(E)$. It is then not
hard to see that there is a fan $\Sigma$ supported on $C$ such that the map
$E\mapsto\Delta(E)$ is additive when restricted to each cone in $\Sigma$ (see
[13, Proposition 1.4]).

When the line bundles belong to a cone of the above fan in which the
Newton–Okounkov polytope is additive, the formula for their intersection
number in Corollary 6.4 can be simplified to the mixed volume of their
corresponding Newton–Okounkov polytopes.

######  Corollary 6.6.

Suppose $E_{1},\ldots,E_{n}$ are $G$-linear systems which lie in a cone on
which the map $E\mapsto\Delta(E)$ is additive. We then have

$\displaystyle E_{1}\cdots E_{n}=n!V(\Delta(E_{1}),\ldots,\Delta(E_{n})),$

where $V$ denotes the mixed volume of polytopes in $n$-dimensional Euclidean
space.

###### Proof.

The corollary immediately follows from Proposition 6.3, additivity of the
Newton–Okounkov polytope and the multi-linearity of mixed volume. ∎

##  7 A version of BKK theorem for ring of conditions

Proposition 6.3 and Corollary 6.6 express the intersection numbers of generic
elements of $G$-linear systems on $X=G/H$ in terms of volumes/mixed volumes of
certain (virtual) polytopes. In this section we will modify this so that it
computes the intersection numbers of hypersurfaces in the ring of conditions
of $X$.

We have to deal with the following two problems: (1) In general an algebraic
hypersurface in $X$ is not a section of a $G$-linearized line bundle on $X$.
(Although in some cases, for example for $X=G=(\mathbb{C}^{*})^{n}$, any
hypersurface in $X$ is indeed a section of some $G$-linearized line bundle on
$X$.) (2) One has to show that it is possible to make any given finite
collection of hypersurfaces “generic enough” by moving its members via generic
elements of the group $G$.

Fortunately, it is known how to overcome the first problem: any hypersurface
after multiplication by a natural number $m$ becomes a section of a
$G$-linearized line bundle. The second problem is also not complicated to
solve (see below).

Let $G/H$ be a spherical homogeneous space of dimension $n$ and let $D$ be an
effective divisor (i.e., a linear combination of prime divisors with
nonnegative coefficients) in $G/H$. Let ${\mathcal{L}}=\mathcal{O}(D)$ be the
corresponding line bundle on $G/H$. We would like to associate a
Newton–Okounkov polytope to the divisor $D$ such that the volume of this
polytope gives the self-intersection number of $D$ in the ring of conditions
of $G/H$. To this end, we need to equip ${\mathcal{L}}$ with a
$G$-linearization. The following results are well-known (see [17, 23]).

######  Theorem 7.1.

Let $X$ be a normal $G$-variety and let $\mathcal{L}$ be a line bundle on $X$.
Then there is $m>0$ such that $\mathcal{L}^{\otimes m}$ has a
$G$-linearization.

Alternatively, there exists a finite covering $\tilde{G}\to G$ where
$\tilde{G}$ is a connected reductive group with trivial Picard group.
Moreover, after replacing $G$ with $\tilde{G}$, every line bundle on $X$
admits a linearization (see remark after Proposition 2.4 and Proposition 4.6
in [17]). Thus, every hypersurface is a section of a linearizable line bundle.

######  Theorem 7.2.

The assignment $\chi\mapsto\mathcal{L}_{\chi}$ gives a one-to-one
correspondence between characters of $H$ and $G$-linearized line bundles
$\mathcal{L}$ on $G/H$.

Let $m>0$ be such that ${\mathcal{L}}^{\otimes m}$ has a $G$-linearization and
fix a $G$-linearization for ${\mathcal{L}}^{\otimes m}$. Let $s_{m}\in
H^{0}\big{(}G/H,{\mathcal{L}}^{\otimes m}\big{)}$ be the section defining
$mD$, i.e., $mD=\operatorname{div}(s_{m})$. Let $E_{m}\subset
H^{0}\big{(}G/H,{\mathcal{L}}^{\otimes m}\big{)}$ be the $G$-invariant
subspace generated by $s_{m}$. Let us denote the Newton–Okounkov polytope
associated to $E_{m}$ by $\Delta(mD)$ (see Section 6). We note that the linear
system $E_{m}$ and the corresponding polytope $\Delta(mD)$ depend on the
choice of linearization of $G/H$. We put

$\displaystyle\Delta(D)=(1/m)\Delta(mD).$

######  Theorem 7.3.

The self-intersection number of $D$ in the ring of conditions of $G/H$ is
$n!\operatorname{vol}_{n}(\Delta(D))$.

###### Proof.

The theorem can be reduced to Proposition 6.3 via Kleiman’s transversality
theorem and existence of a good compactification for $D$ (Theorem 3.1).
Without loss of generality let $m=1$ and put $E=E_{m}$. Let $\overline{X}$ be
a spherical embedding of $G/H$ which provides a good compactification for $D$,
i.e., for each $G$-orbit $O\subset\overline{X}$ such that $\overline{D}\cap
O\neq\varnothing$ we have $\overline{D}\cap O$ is a hypersurface in $O$. By
Kleiman’s transversality theorem if $g_{1},\ldots,g_{n}\in G$ are in general
position then for any $G$-orbit $W$ in $\overline{X}$ all the
$\overline{g_{i}\cdot D}\cap W$ intersect strongly transversally in $W$. In
particular if $W\neq G/H$ the intersections are empty. Thus the self-
intersection number of $D$ in the ring of conditions is equal to the self-
intersection number of the divisors $\overline{g_{i}\cdot D}$ in
$\overline{X}$.

On the other hand, we know the following: take $s_{1},\ldots,s_{n}\in E$ and
let $H_{i}=\\{x\in G/H\,|\,s_{i}(x)\allowbreak=0\\}$ with closure
$\overline{H}_{i}\subset\overline{X}$. Suppose all the $\overline{H}_{i}$
intersect transversally. In particular,
$\overline{H}_{1}\cap\cdots\cap\overline{H}_{n}$ lies in $G/H$, that is, there
are no intersection points at infinity. Then $[E,\ldots,E]$ is equal to
$|H_{1}\cap\cdots\cap H_{n}|$, the number of solutions $x\in G/H$ of the
system $s_{1}(x)=\cdots=s_{n}(x)=0$ (we note that since $E$ is $G$-invariant
the set of common zeros of all the sections in $E$ is empty). It follows that
if the $g_{i}\in G$ are in general position we have

$\displaystyle[E,\ldots,E]=|\overline{g_{1}\cdot
D}\cap\cdots\cap\overline{g_{n}\cdot D}|.$

Now since $G$ is connected, the action of $G$ on the Picard group of
$\overline{X}$ is trivial. Hence for any $g\in G$, the divisor
$\overline{g\cdot D}$ is linearly equivalent to $\overline{D}$. Putting
everything together we conclude that

$\displaystyle[E,\ldots,E]=\overline{D}^{n}.$

The theorem now follows from Proposition 6.3. ∎

Let $D_{1},\ldots,D_{n}$ be $n$ irreducible hypersurfaces in $G/H$. For each
$i$, fix a $G$-linearization for the line bundle $\mathcal{O}(D_{i})$. For
each collection $I=\\{i_{1},\ldots,i_{s}\\}\subset\\{1,\ldots,n\\}$ equip the
line bundle $\mathcal{O}(D_{i_{1}}+\cdots+D_{i_{s}})$ with the
$G$-linearization induced from those of the $D_{i}$. Let $\Delta_{I}$ be the
Newton–Okounkov body $\Delta(D_{i_{1}}+\cdots+D_{i_{s}})$. The following is an
immediate corollary of Theorem 7.3.

######  Corollary 7.4.

The intersection number of the $D_{i}$ in the ring of conditions is given by
the following formula

$\displaystyle(-1)^{n}D_{1}\cdots
D_{n}=-\sum_{i}\operatorname{vol}_{n}(\Delta_{i})+\sum_{i<j}\operatorname{vol}_{n}(\Delta_{i,j})+\cdots+(-1)^{n}\operatorname{vol}_{n}(\Delta_{1,\ldots,n}).$

##  8 Two algebra constructions

In this section we discuss two similar constructions of graded commutative
algebras with Poincaré duality associated with a $\mathbb{Q}$-vector space
$V$. The first construction uses a symmetric $n$-linear function $F$ on $V$.
The second construction uses a homogeneous degree $n$ polynomial $P$ on $V$.
The constructions produce isomorphic graded algebras with Poincaré duality if
$F$ and $P$ satisfy $n!F(x,\dots,x)=P(x)$ for all $x\in V$. We will give the
results without proofs as all the proofs are straightforward.

###  8.1 Algebra constructed from a symmetric multi-linear function

Let $V$ be a (possibly infinite-dimensional) vector space over $\mathbb{Q}$
equipped with a nonzero symmetric $n$-linear function $F\colon
V\times\cdots\times V\to\mathbb{Q}$. To $(V,F)$ one can associate a graded
$\mathbb{Q}$-algebra $A_{F}=A=\bigoplus_{i=0}^{n}A_{i}$ by the following
construction. Let $\operatorname{Sym}(V)=\bigoplus_{i\geq
0}\operatorname{Sym}^{i}(V)$ be the symmetric algebra of the vector space $V$.
With $F$ one can associate the function $F_{s}$ on $\operatorname{Sym}(V)$ as
follows: (1) if $a\in\operatorname{Sym}^{n}(V)$ and
$a=\sum\lambda^{i}v_{1}^{i}\cdots v^{i}_{n}$ where $\lambda_{i}\in\mathbb{Q}$,
$v^{i}_{j}\in V$ then
$F_{s}(a)=\sum\lambda_{i}F\big{(}v_{1}^{i},\ldots,v^{i}_{n}\big{)}$; (2) if
$a\in\operatorname{Sym}^{k}(V)$ where $k\neq n$ then $F_{s}(a)=0$. Let
$I_{F_{s}}\subset\operatorname{Sym}(V)$ be the subset defined by $a\in
I_{F_{s}}$ if and only if for any $b\in\operatorname{Sym}(V)$ the identity
$F_{s}(ab)=0$ holds. It is easy to see that $I_{F_{s}}$ is a homogeneous ideal
in $\operatorname{Sym}(V)$. We define $A$ to be the graded algebra
$\operatorname{Sym}(V)/I_{F_{s}}$. It follows from the construction that $A$
has the following properties:

  * •

$A_{0}=\mathbb{Q}$.

  * •

$A_{k}=0$ for $k>n$.

  * •

$\dim_{\mathbb{Q}}A_{n}=1$, moreover the function $F_{s}$ induces a linear
isomorphism $f_{s}\colon A_{n}\rightarrow\mathbb{Q}$.

  * •

$A_{1}$ coincides with the image of $V=\operatorname{Sym}^{1}(V)$ in
$A=\operatorname{Sym}(V)/I_{F_{s}}$.

  * •

$A$ is generated as an algebra by $A_{0}$ and $A_{1}$.

  * •

The pairing $B_{k}\colon A_{k}\times A_{n-k}\rightarrow\mathbb{Q}$ by formula
$B_{k}(a_{k},a_{n-k})=f_{s}(a_{k}a_{n-k})$ is non-degenerate. It provides a
“Poincaré duality” on $A$.

###  8.2 Algebra constructed from a homogeneous polynomial

Let $V$ be a (possibly infinite-dimensional) $\mathbb{Q}$-vector space
equipped with a homogeneous degree $n$ polynomial $P\colon V\to\mathbb{Q}$
(recall that a function $P\colon V\to\mathbb{Q}$ is a polynomial or a
homogeneous degree $n$ polynomial if its restriction to any finite-dimensional
subspace $V_{1}\subset V$ is correspondingly a polynomial or a homogeneous
degree $n$ polynomial). To $(V,P)$ one can associate a graded
$\mathbb{Q}$-algebra $A_{P}=A=\bigoplus_{i=0}^{n}A_{i}$ as we explain below.

First we recall the algebra $\mathcal{D}=\mathcal{D}_{V}$ of constant
coefficient differential operators on $V$. For a vector $v\in V$, let $L_{v}$
be the differentiation operator (Lie derivative) on the space of polynomial
functions on $V$ defined as follows. Let $f$ be a polynomial function on $V$,
then

$\displaystyle L_{v}(f)(x)=\lim_{t\to 0}\frac{f(x+tv)-f(x)}{t}.$

The algebra $\mathcal{D}$ is defined to be the subalgebra of the
$\mathbb{Q}$-algebra of linear operators on the space of polynomials on $V$
generated by the Lie derivatives $L_{v}$ for all $v\in V$ and by
multiplication by scalars $c\in\mathbb{Q}$. The algebra $\mathcal{D}$ is
commutative since Lie derivatives against constant vector fields commute. The
algebra $\mathcal{D}$ can be naturally identified with the symmetric algebra
$\operatorname{Sym}(V)$. When $V\cong\mathbb{Q}^{n}$ is finite-dimensional,
$\mathcal{D}$ can be realized as follows: Fix a basis for $V$ and let
$(x_{1},\ldots,x_{n})$ denote the coordinate functions with respect to this
basis. Each element of $\mathcal{D}$ is then a polynomial expression, with
constant coefficients, in the differential operators
$\xi_{1}=\partial/\partial x_{1},\ldots,\xi_{n}=\partial/\partial x_{n}$. That
is,

$\displaystyle\mathcal{D}=\bigg{\\{}f(\partial/\partial
x_{1},\ldots,\partial/\partial
x_{n})\,|\,f=\sum_{\alpha=(a_{1},\ldots,a_{n})}c_{\alpha}\xi_{1}^{a_{1}}\cdots\xi_{n}^{a_{n}}\in\mathbb{Q}[\xi_{1},\ldots,\xi_{n}]\bigg{\\}}.$

Now let $I_{P}$ be the ideal of all differential operators $D\in\mathcal{D}$
such that $D(P)=0$, i.e., those differential operators that annihilate $P$. We
define $A_{P}$ to be the quotient algebra $\mathcal{D}/I_{P}$. The algebra
$A=A_{P}$ has a natural grading $A=\bigoplus_{i=0}^{n}A_{i}$ (this is because
$I_{P}$ is a homogenous ideal). It follows from the construction that
$A=A_{P}$ has the following properties:

  * •

$A_{0}=\mathbb{Q}$.

  * •

$A_{k}=0$ for $k>n$.

  * •

$\dim_{\mathbb{Q}}A_{n}=1$. Moreover $P$ defines a linear function on
homogeneous order $n$ operators $D_{n}$ sending $D_{n}$ to the constant
$D_{n}(P)$. It induces a linear isomorphism $A_{n}\to\mathbb{Q}$.

  * •

$A_{1}$ coincides with the image of $V$ in $A=\mathcal{D}/I_{P}$.

  * •

$A$ is generated as an algebra by $A_{0}$ and $A_{1}$.

  * •

One can define the pairing $B_{k}\colon A_{k}\times
A_{n-k}\rightarrow\mathbb{Q}$ by $B_{k}(a_{k},a_{n-k})=D_{k}\circ D_{n-k}(P)$
where $D_{k}$ and $D_{n-k}$ are any preimages of the elements
$a_{k},a_{n-k}\in\mathcal{D}/I_{P}$ in $D$. This pairing is well-defined since
the preimages $D_{k}$ and $D_{n-k}$ are defined up to addition of elements
from the ideal $I_{P}$ and elements from $I_{P}$ annihilate $P$. It is easy to
show that the pairing is non-degenerate and gives a “Poincaré duality” on $A$.

One has the following.

######  Theorem 8.1.

Let $A=\bigoplus_{i=0}^{n}A_{i}$ be a graded algebra over $\mathbb{Q}$ with
the following properties:

  * $(1)$

$A_{0}=\mathbb{Q}$.

  * $(2)$

There is a linear isomorphism $f\colon A_{n}\to\mathbb{Q}$.

  * $(3)$

$A$ is generated as an algebra by $A_{0}$ and $A_{1}$.

  * $(4)$

There is a linear projection $\pi\colon V\to A_{1}$ of a given
$\mathbb{Q}$-vector space $V$ onto $A_{1}$.

  * $(5)$

For any $0\leq k\leq n$ the pairing $B_{k}\colon A_{k}\times
A_{n-k}\to\mathbb{Q}$ defined by $B_{k}(a_{k},a_{n-k})=f(a_{k}a_{n-k})$ is
non-degenerate.

Then $A$ can be described as follows:

  * $(a)$

$A$ is isomorphic to the algebra associated to $(V,P)$ where $P$ is the
homogeneous degree $n$ polynomial on $V$ defined by

$\displaystyle P(x)=(1/n!)f\big{(}y^{n}\big{)},$

where $x\in V$ and $y=\pi(x)$.

  * $(b)$

$A$ is isomorphic to the algebra associated to $(V,F)$ where $F$ is the
symmetric $n$-linear function on $V$ equal to the polarization of the
polynomial $P$ from $(a)$ multiplied by $n!$.

A generalization of Theorem 8.1 for commutative algebras $A$ with Poincaré
duality that are not necessarily generated by $A_{0}$ and $A_{1}$ can be found
in [7]. Also some related material can be found in [6, Exercise 21.7] and [8].

##  9 A description of ring of complete intersections of $\boldsymbol{G/H}$

This section contains the main result of the paper which gives descriptions of
the ring of complete intersections of a spherical homogeneous space in terms
of volumes of polytopes. It is an analogue of the description of ring of
conditions of torus $(\mathbb{C}^{*})^{n}$ in [15].

Recall $\mathcal{R}^{\prime}(G/H)$ denotes the ring of complete intersections
where $G/H$ is a spherical homogeneous space. We use Theorem 8.1 to give two
descriptions of the ring
$\mathcal{R}^{\prime}_{\mathbb{Q}}(G/H)=\mathcal{R}^{\prime}(G/H)\otimes_{\mathbb{Z}}\mathbb{Q}$
in terms of volumes of polytopes. Let $\operatorname{Div}(G/H)$ be the group
of divisors on $G/H$ with
$\operatorname{Div}_{\mathbb{Q}}(G/H)=\operatorname{Div}(G/H)\otimes_{\mathbb{Z}}\mathbb{Q}$
the corresponding $\mathbb{Q}$-vector space. For an effective divisor $D$ in
$G/H$ consider the associated Newton–Okounkov polytope $\Delta(D)$ introduced
in Section 7.

######  Theorem 9.1.

The ring $\mathcal{R}^{\prime}_{\mathbb{Q}}(G/H)$ is isomorphic to the algebra
associated to the vector space $V=\operatorname{Div}_{\mathbb{Q}}(G/H)$ and
the $n$-linear function $F\colon V\times\cdots\times V\to\mathbb{Q}$ whose
value on an $n$-tuple $D_{1},\ldots,D_{n}$ of effective divisors in $G/H$ is
given by

$(-1)^{n}F(D_{1},\ldots,D_{n})=-\sum_{i}\operatorname{vol}_{n}(\Delta_{i})+\sum_{i<j}\operatorname{vol}_{n}(\Delta_{i,j})+\cdots+(-1)^{n}\operatorname{vol}_{n}(\Delta_{1,\ldots,n}).$

Here for $I\subset\\{1,\ldots,n\\}$ we set $\Delta_{I}=\Delta\big{(}\sum_{i\in
I}D_{i}\big{)}$. Moreover, if $D_{1},\ldots,D_{n}$ lie in a cone in
$\operatorname{Div}_{\mathbb{Q}}(G/H)$ such that $D\mapsto\Delta(D)$ is linear
then $F(D_{1},\ldots,D_{n})=n!V(\Delta(D_{1}),\ldots,\Delta(D_{n}))$ where $V$
is the mixed volume $($see Section 6).

###### Proof.

The theorem follows from Theorem 7.3, Corollaries 7.4 and 6.6, and Theorem
8.1. ∎

Similarly, we have the following description of
$\mathcal{R}^{\prime}_{\mathbb{Q}}(G/H)$ as a quotient of the ring of
differential operators.

######  Theorem 9.2.

The ring $\mathcal{R}^{\prime}_{\mathbb{Q}}(G/H)$ is isomorphic to the algebra
associated to the vector space $V=\operatorname{Div}_{\mathbb{Q}}(G/H)$ and
the polynomial $P$ whose value on an effective divisor $D$ is equal to
$\operatorname{vol}_{n}(\Delta(D))$.

###### Proof.

The theorem follows from Theorem 7.3, Corollary 7.4 and Theorem 8.1. ∎

Finally, the ring of complete intersections $\mathcal{R}^{\prime}(G/H)$ itself
can be realized as the subring of $\mathcal{R}^{\prime}_{\mathbb{Q}}(G/H)$
generated by $\mathbb{Z}$ and the image of $\operatorname{Div}(G/H)$.

##  10 Ring of complete intersections of an arbitrary variety

Let $X$ be a variety (not necessarily complete). In [9, 12] the authors
consider the collection ${\bf K}(X)$ of all finite-dimensional vector
subspaces of the field of rational functions $\mathbb{C}(X)$. This collection
is a semigroup with respect to the multiplication of subspaces. Namely, for
$L,M\in{\bf K}(X)$ we define

$\displaystyle LM=\text{span}\\{fg\,|\,f\in L,g\in M\\}.$

We then consider the Grothendieck group ${\bf G}(X)$ associated to the
semigroup of subspaces ${\bf K}(X)$.

Given $L_{1},\ldots,L_{n}\in{\bf K}(X)$, one introduces intersection index
$[L_{1},\ldots,L_{n}]\in\mathbb{Z}$. The intersection index
$[L_{1},\ldots,L_{n}]$ is the number of solutions $x\in X$ of a generic system
of equations $f_{1}(x)=\cdots=f_{n}=0$, where $f_{i}\in L_{i}$. In counting
the solutions, we neglect the solutions $x$ at which all the functions in some
space $L_{i}$ vanish as well as the solutions at which at least one function
from some space $L_{i}$ has a pole. In [9, 12] it is shown that the
intersection index is well-defined and is multi-additive with respect to
multiplication of subspaces. Thus it naturally extends to the Grothendieck
group ${\bf G}(X)$ and to ${\bf G}_{\mathbb{Q}}(X)={\bf
G}(X)\otimes_{\mathbb{Z}}\mathbb{Q}$ the corresponding $\mathbb{Q}$-vector
space. In [11, Section 4.3] we associate to each element $L\in{\bf K}(X)$ its
Newton–Okounkov body $\Delta(L)\subset\mathbb{R}^{n}$ in such a way that the
following conditions hold:

  1. 1)

$\Delta(L_{1})+\Delta(L_{2})\subset\Delta(L_{1}L_{2})$,

  2. 2)

$[L,\dots,L]=n!\operatorname{vol}_{n}(\Delta(L)$.

We view the intersection index of subspaces of rational functions as a
birational version of the intersection theory of (Cartier) divisors (more
generally linear systems). The main result in [12] is that the Grothendieck
group ${\bf G}(X)$ is naturally isomorphic to the group of b-divisors
(introduced by Shokurov). Roughly speaking, a b-divisor is an equivalence
class of Cartier divisors on any birational model of $X$. The isomorphism
sends the intersection index to the usual intersection number of Cartier
divisors.

######  Definition 10.1 (ring of complete intersections).

We call the ring associated to the $\mathbb{Q}$-vector space ${\bf
G}_{\mathbb{Q}}(X)$ and the intersection index, the ring of complete
intersections over $\mathbb{Q}$ of $X$ and denote it by
$\mathcal{A}_{\mathbb{Q}}(X)$. We call the subring $\mathcal{A}(X)$ of
$\mathcal{A}_{\mathbb{Q}}(X)$ generated by $\mathbb{Z}$ and the image of ${\bf
G}(X)$, the ring of complete intersections of $X$.

As above Theorem 8.1 suggests two descriptions of the ring $\mathcal{A}(X)$ of
complete intersections of $X$ in terms of volumes of Newton–Okounkov bodies
associated to elements of ${\bf G}(X)$.

### Acknowledgements

We are grateful to the anonymous referee for careful reading of the paper and
several invaluable suggestions and corrections. The first author is partially
supported by a National Science Foundation grant (grant ID: 1601303). The
second author is partially supported by the Canadian grant No. 156833-12.

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