# Maps to toric varieties and toric degenerations

Takuya Murata and Lara Bossinger  School of Mathematics, Institute for
Research in Fundamental Sciences (IPM), Teheran, Iran<EMAIL_ADDRESS>Instituto de Matemáticas Unidad Oaxaca, Universidad Nacional Autónoma de
México, León 2, altos, Centro Histórico, 68000 Oaxaca, Mexico<EMAIL_ADDRESS>
###### Abstract.

We study and construct maps to toric varieties. Our motivation comes from
toric degenerations –flat families whose special fibers are not-necessarily-
normal toric varieties– and the desire to construct a map from the generic to
the special fiber. In the process, we give a negative answer to a question of
I. Dolgachev and K. Kaveh as to whether a projective toric degeneration can be
constructed as a degeneration by projection.

In the classical topology over $\mathbb{C}$, given a proper family (not just
projective one), we construct a deformation-retract from a neighborhood of the
special fiber to the special fiber (using tubular neighborhoods of the
strata). We recover the construction of integral systems in [HK15] and
moreover extend it to the singular locus.

##  0\. Introduction

In this paper, we study maps to a not-necessarily-normal toric variety; both,
those that are algebraic and those that are not. While the topic is of general
interest, our primary motivation comes from applications to toric
degenerations, flat families of varieties with a special fiber isomorphic to a
possibly non-normal toric variety. In this case, having a map from a general
to the special fiber turns out to be very useful in applications: the
existence of such a map allows to _pull-back_ objects on the limit toric
variety to the original variety. For example, in characteristic zero, a torus-
equivariantly quasi-projective possibly non-normal toric variety admits a
moment map (smoothness is not needed). The moment map composed with a map from
general to special fiber yields a map from the original variety to a convex
polytope, known as the Newton–Okounkov polytope.

A normal toric variety can be characterized combinatorially by a fan of convex
polyhedral cones. In a similar way we expect, maps (algebraic or not) to not-
necessarily-normal toric varieties admit a combinatorial characterization– a
consequence of the fact that a not-necessarily-normal toric variety is locally
described by semigroup algebras and a semigroup algebra admits a universal
property similar to that of a polynomial ring (see the end of §1).

The paper is split into two parts: first, we discuss an algebraic construction
of the aforementioned maps and then turn to a local non-algebraic
construction.

###  0.1. Algebraic construction

In literature, there are several characterizations of morphisms (algebraic
maps) from a scheme to a normal toric variety. For example, [AMRT75], one of
the original works on toric varieties, describes the functor represented by a
normal toric variety in terms of a polyhedral cone decomposition defining a
toric variety. On the other hand, Cox gives a characterization of morphisms to
smooth toric varieties based on a geometric-invariant-theory quotient
interpretation of a normal toric variety [Cox93]. The GIT quotient
interpretation is also used in [Kol06] to interpret a normal toric variety as
a moduli of polarizied smooth projetive varieties.

In this paper, we give a characterization of morphisms to a not-necessarily-
normal but projective toric variety. Recall the following fact:

######  Proposition.

Let $W\subset\mathbb{P}^{q}$ be a projective variety, $X$ a scheme and
$\mathcal{L}$ an invertible sheaf on $X$. There is a natural bijection between
the set of all morphisms $X\to W$ along which $\mathcal{O}_{W}(1)$ pulls-back
to $\mathcal{L}$ and the set of all $q+1$ tuples of global sections
$g_{0},\dots,g_{q}$ of $\mathcal{L}$ that generate $\mathcal{L}$ and satisfy
the defining homogeneous equations of $W$.

When $W=\mathbb{P}^{q}$ is a projective space, the defining homogeneous
equations are vacuous and the above follows from the case of projective space
(e.g., [Har77, Ch. II, Theorem 7.1]).

Now, suppose $W$ is a not-necessarily-normal toric variety (i.e., the closure
of a torus orbit when a torus acts linearly on $\mathbb{P}^{q}$). Even without
normality, $W$ is defined by binomial equations of form $y_{0}^{n_{0}}\cdots
y_{q}^{n_{q}}=y_{0}^{m_{0}}\cdots y_{q}^{m_{q}}$. We then have the following
from the above proposition (for the non-projective case, see Remark 0.3):

######  Theorem (Theorem 1.3).

A morphism from a projective variety $X$ to a not-necessarily-normal toric
variety $W$ such that $\mathcal{O}_{W}(1)$ pulls back to a given invertible
sheaf $\mathcal{L}$ on $X$ is characterized by the existence of global
sections that generate $\mathcal{L}$ (i.e., base-point-free) that satisfy the
binomial equations of $W$.

When $X=\mathbb{G}_{m}^{r}$ is a torus and $\mathcal{L}=\mathcal{O}_{X}$,
suitable $g_{i}$’s are polynomials in the coordinates $t_{i}$ of $X$ that
satisfy the binomial relations. In this case the morphism
$\mathbb{G}_{m}^{r}\to W$ is a torus embedding. Hence, the above is also a
generalization of a torus embedding. In fact, the converse is also true for
arbitrary $X$ in the sense that any morphism $\varphi:X\to W$ pulling back
$\mathcal{O}_{W}(1)$ to $\mathcal{L}$ is a unique extension of a monomial map
(Proposition 1.14). In this sense, a morphism $X\to W$ is a _compactification
of a monomial map_.

We proceed to give some existence results. First, when $\dim X\geq\dim W$ and
$\mathcal{L}$ is big (e.g., ample), we show it is always possible to choose
$g_{i}$’s in some power $\mathcal{L}^{\otimes m}$ satisfying the defining
binomial relations of $W$ in such a way that the corresponding morphism from
some open dense subset of $X$ is well-defined and dominant (Proposition 1.14).

In the case of toric degenerations, we give a more precise construction.
Throughout this paper, unless mentioned otherwise a toric degeneration of an
algebraic variety $X$ is a flat surjective morphism of schemes
$\xi:\mathfrak{X}\to\mathbb{A}^{1}$ that is trivial away from the _special
fiber_ $\xi^{-1}(0)$ which is isomorphic to a not-necessarily-normal toric
variety and moreover the _generic fiber_ $\xi^{-1}(1)$ is isomorphic to $X$
(see Definition 2.1).

Suppose we are given a Gröbner toric degeneration111A toric degeneration being
Gröbner essentially means that is it torus-equivalent and embedded, see §2.3
and Theorem 2.6. $\mathfrak{X}\subset\mathbb{P}^{n}\times\mathbb{A}^{1}$ of a
projective variety $X$ or equivalently $\mathfrak{X}$ is given by a full-rank
graded valuation on the homogeneous coordinate ring of $X$ with the finitely
generated image, such that $X$ is the general fiber and $W$ the special fiber.
In this case, the homogeneous coordinate rings $R_{X}$, respectively $R_{W}$,
of $X$, respectively $W$, both have a basis indexed by the same set of
standard monomials (i.e. monomials that are not contained in a monomial
initial ideal of simultaneously $I(X)$ and $I(W)$, the defining ideals of $X$,
respectively $W$). Hence, there is an induced bijection. We make use of this
fact to obtain the following:

######  Theorem (Theorem 2.23).

Let $\chi^{a_{0}},\dots,\chi^{a_{q}}$ be the generators of the semigroup
algebra $R_{W}$ of $W$. Then there is an injective ring homomorphism

(1) $R_{W}=k[\chi^{a_{0}},\cdots,\chi^{a_{q}}]\hookrightarrow R_{X}$

that maps the basis of $R_{W}$ (consisting of all monomials in the
$\chi^{a_{0}}s$’s) to the standard monomial basis of $R_{X}$. Moreover, the
homomorphism is graded of some degree $m$ so that a homogeneous element of
degree $l$ goes to a homogeneous element of degree $lm$.

Geometrically, $R_{W}\hookrightarrow R_{X}$ in the theorem induces

$X-B\underset{\sim}{\overset{v}{\longrightarrow}}v(X-B)\overset{p}{\longrightarrow}W$

where $v$ is the $m$-th Veronese embedding from $\mathbb{P}^{q}$ to
$\mathbb{P}^{N}$, $p$ the projection from $\mathbb{P}^{N}$ to $\mathbb{P}^{q}$
and $B=V(R_{W,+}R_{X})$. When $X$ is projective and $\mathcal{L}$ ample, by
Proposition 1.14 any morphism $\varphi:X-B\to W$ along which
$\mathcal{O}_{\mathbb{P}^{q}}(1)$ pulls-back to $\mathcal{L}^{\otimes m}$ has
to have a ring homomorphism of the above form, at least locally.

Note that, except in the trivial case, we cannot require the above
homomorphism (1) to preserve the degree (i.e., $m=1$) since $m=1$ implies
$R^{\prime}=R$ by consideration of the Hilbert function. That we cannot take
$m=1$ has an implication to the following question of I. Dolgachev that the
first author learned from K. Kaveh:

######  0.1 Problem.

Construct a toric degeneration as a degeneration by projection (or a sequence
of the latter).

It is a classical fact that given a projective space $\mathbb{P}^{N}$ and a
linear subspace $L\subset\mathbb{P}^{N}$, the projection of a closed
subvariety $X\subset\mathbb{P}^{N}$ (disjoint from the center) to $L$ induces
a flat degeneration (Theorem 3.1). The above discussion implies that the
solving the naive formulation of Problem 0.1 is not possible. If it were
possible then, algebraically, we would have $R_{W}\hookrightarrow R_{X}$
preserving the grading, a contradiction away from trivial cases.

Next, we may ask whether it is possible to construct a degeneration by
projection after a Veronese embedding of the original variety $X$ to some
other projective space. However, we show that this is not possible neither in
Theorem 3.12. However, we do nonetheless show that there is a degeneration by
projection containing the toric degeneration.

As of this writing, it is unclear how the different characterizations of
morphisms to toric varieties found in literature are related to each other
(and to our approach). Hence, we want to mention:

######  0.2 Problem.

Give a unified moduli interpretaion of a not-necessarily-normal toric variety.

For the non-projective-case, we have the following idea:

######  0.3 Remark (non-projective case).

While the restriction to the projecive case is reasonable, generalizing the
projective case to the non-projective-case is useful in particular for the
problem of constructing toric degenerations embedded in a smooth toric variety
(see Problem 5.12) being considered by the first author. We believe such a
generalization can be obtained by using a Cox ring in place of a graded ring
defining a projective variety. This, however, would require the definition of
the Cox ring of a Noetherian scheme (possible using divisorial sheaves instead
of Weil divisors) and thus is beyond the scope of the present paper.

###  0.2. Local non-algebraic construction

There is a local algebraic construction that is completely analogous to the
global construction described above. However, it turns out not to be very
interesting due to a key issue in the theory of Khovanskii bases that do not
work well in the local situation because of the lack of grading. We therefore
have adapted a construction in differential topology of stratified spaces,
namely, that of Goresky and MacPherson ([Gor81, §7]). Given a proper family
over $\mathbb{C}$ embedded in a complex manifold, we construct a map that
collapses a neighborhood of the special fiber to the special fiber. Roughly
speaking it collapses tubular neighborhoods of the strata of the special fiber
(strata being smooth submanifolds of the ambient smooth manifold). For a toric
degeneration, strata may be taken to be torus orbits and then obtain a
continuous surjective map from a general fiber to the special fiber.
Precisely, we show:

######  Theorem (Theorem 4.1).

Suppose we are given a toric degeneration $\mathfrak{X}$ from $X$ (a generic
fiber) to $X^{\prime}$ (the special fiber) over $\mathbb{C}$ that is embedded
in a complex manifold as a proper family. Then there exists a continuous
surjective map (not a morphism of varieties)

$\varphi:X\to X^{\prime}$

such that, for each torus orbit $O\subset X^{\prime}$, $\varphi^{-1}(O)$
naturally has the structure of a reducible variety so that
$\varphi:\varphi^{-1}(O)\to O$ is a finite étale covering. If
$\mathfrak{X}\subset\mathbb{P}^{n}\times\mathbb{A}^{1}$ and $\mathfrak{X}$ is
induced by a valuation, then for the moment map $\mu$ of $X^{\prime}$, the
composition $\mu\circ\varphi$ is a surjection from $X$ to the Newton–Okounkov
polytope of $X$.

The advantage of our construction over Goresky–MacPherson’s is that our map is
the restriction of a deformation-retract. We use this fact to construct
integrable systems much the same way as is done by Harada and Kaveh in [HK15].
In loc.cit. a map from the general fiber to the special fiber is obtained
using a Hamiltonian gradient flow. This approach requires the extension of the
map from the regular locus to the singular locus by a limit argument. In
contrast, our construction is applied to each stratum (torus orbits, in this
case) and the global map is obtained from gluing the local maps using a
partition-of-unity-type argument. In particular, the integral system we
construct is defined on the whole variety. More precisely:

######  Theorem (improved Harada–Kaveh; Theorem 5.9).

Let $\mathfrak{X}\subset M$ be a toric degeneration of a variety $X$ embedded
in some complex smooth variety $M$ in such a way that
$\mathfrak{X}\to\mathbb{A}^{1}$ is the restriction of a proper morphism
$M\to\mathbb{A}^{1}$. Then there exists a continuous surjective map

$\varphi:X\to X^{\prime}$

satisfying the étale property of Theorem 4.1 above. Moreover, given the moment
map $\mu$ of $X^{\prime}$ we find that

$(f_{1},\dots,f_{r}):X\overset{\varphi}{\longrightarrow}X^{\prime}\overset{\mu}{\longrightarrow}\mathbb{R}^{r}$

is an integral system in the stratified sense (Definition 5.5), i.e. linear
combinations of $f_{1},\dots,f_{r}$ form an integral system on each stratum of
$X$ (with respect to an essentially canonical stratification222A complex
algebraic variety admits a stratification by complex-algebraic smooth
subvarieties and it is known that such a stratification is essentially
canonical; see the discussion following [Mat73, Theroem 4.9,]).

The key improvement here over [HK15] is that the integrable system is
constructed also on the boundary, in addition to the fact that $\varphi$ is
defined explicitly not as a continuous extension.

###  0.3. Toric degenerations as limits

We comment on the following idea of constructing a map from a projective
variety to the limit of a toric degeneration of $X$ when such a toric
degeneration exists333We learned about this construction from Nathan Ilten in
the course of preparing the current paper.. Suppose we are given some linear
$\mathbb{G}_{m}$-action on a projective space $\mathbb{P}^{N}$. We propose to
define the map $\psi:\mathbb{P}^{N}\to\mathbb{P}^{N}$ by the formula

$\psi(x)=\lim_{t\to 0}t\cdot x.$

It is not obvious that this map is _well-defined_ , (i.e. a limit exists and
is unique) or that it is a morphism of varieties. Ignoring these issues, let
us suppose $\psi$ is in fact a morphism. Suppose we are further given a
subvariety $X\subset\mathbb{P}^{N}$ and a toric degeneration
$\mathfrak{X}\subset\mathbb{P}^{N}\times\mathbb{A}^{1}$ of $X$ obtained as the
closure $\overline{\mathbb{G}_{m}\cdot X}$. For example, Gröbner degenerations
are this form (see Remark 2.8). Then the image $\psi(X)$ lies in the limit
$X^{\prime}$ of $\mathfrak{X}$ but, in general, it need not coincide with the
latter or not even be dense there. Consider, for example, a toric degeneration
of an elliptic curve. In that case, a toric degeneration of an elliptic curve
is given for instance by the action
$t\cdot[c_{0}:c_{1}:c_{2}]=[t^{-1}c_{0}:c_{1}:t^{-3}c_{2}]=[t^{2}c_{0}:t^{3}c_{1}:c_{2}]$
(see Example 2.12). So, a general point maps to $[0:0:1]$. So $\psi(X)$ is
finite and irreducible, thus is a point. In this $\psi$ is somehow trivial. In
fact, $\psi$ is an instance of a specialization map which is usually not
surjective.

Some other related work is [EHS13], where the authors consider a monomial map.
The image of a standard cube $[0,1]^{n}$ under such a map called a toric cube.
One of the main results in loc.cit. is the (explicit) decomposition of a toric
cube into a CW-complex. It seems that such a decomposition should be related
to decompositions of monomial maps; i.e., closely related to the algebraic
maps considered in this paper.444While in [EHS13, Example 4] the structure of
a CW-complex is obtained by decomposing a monomial map the general case is
different.

###  0.4. Existence of toric degenerations

The present paper is part of the investigation of the existence of toric
degenerations, a broader research project to which many people have
contributed in the past decade; for example, Kaveh and Manon building a bridge
between toric degenerations by valuations and the tropicalization of an ideal
[KM19a]. Their work was used by the first author to formulate a criterion for
the finite generation of a value semigroup in [Bos21]); i.e. a criterion for
the existence of a toric degeneration given by a valuation. Moreover, [BLMM17]
gave a procedure whose aim is to construct a prime cone in the tropicalization
of an ideal (i.e. a toric Gröbner degeneration). The motivation of [IW20] was
to verify if this procedure terminates, which it does not in general.

For the authors of this paper, on the one hand an essential question is how to
embed a toric degeneration into a projective space (see the beginning of §2)
or some other ambient space. On the one hand, it is generally understood that
there is a polarized projective variety that does not admit a toric
degeneration preserving a power of the polarization; for example, a polarized
smooth projective curve of genus at least two over an uncountable base field
([KMM22, Proposition 3.2.] or a comment just after Corollary 2.14). However,
it is expected, as we understand, that an abstract toric degeneration of a
variety (i.e., a degeneration not taking place in any ambient space) is
possible unconditionally (Problem 2.16). Hence, the interesting question is
regarding an ambient space.

We now venture into some speculation. Although a toric variety is typically
defined by a _finite_ fan, the original construction (see [KKMSD73]) allows
for an infinite fan (so the resulting toric variety is not necessarily a
variety). Similarly, I. Satake [Sat73] constructs a torus embedding into an
infinite projective space. Since a map to a toric variety can be thought of as
a generalization of a torus embedding (Example 1.7), it is presumably possible
to extend the results of this paper to these more general infinite-dimensional
setup. For example, Theorem 1.3 should extend so to generalize Satake’s torus
embedding. An interesting example to study this question is given by [BFMN20]
where a toric degeneration of a cluster variety to a toric scheme defined by
an infinite simplicial fan is constructed.

The Gröbner approach in §2.5 in particular suggests a tight connection between
a map to a toric variety and a toric degeneration. That is, a toric
degeneration may be thought of as a particular kind of a map to a toric
variety (we have not justified this statement yet; a necessary first step
would be solving the problem in Remark 2.26). In classical algebraic geometry,
a linear system plays a central role in classifications and understandings of
varieties; e.g., whether a given variety admits a certain linear system or
not. So, in an analogous fashion, asking whether a variety admits a toric
degeneration may be phrased as asking whether a variety admits a certain map
to a toric variety.

###  0.5. Open problems

We comment on some of the several problems related to or arising from the
current paper that have not yet been fully investigated.

######  0.4 Problem.

Give the coordinate-free construction of a map to a toric variety.

As perhaps clear by now, an act of making a choice of sections determining a
map is a critical component in the algebraic construction presented; this
poses the question if it is even possible to do the construction in a
coordinate-free manner. This problem is related to the following problem in
the theory of toric varieties: while a smooth toric variety generalizes a
projective space, it is not clear how to generalize the projectivization of a
vector space (a choice of a fan fixes unordered coordinates on a projective
space). Or said differently, what is a fan without a choice of cones?

Moreover, it is worthwhile finding a generalization of a linear system (or
rather results involving a linear system). The key difference is that divisor
theory for toric varieties is not quite as simple as it is on a projective
space where every divisor is a hypersurface.

######  0.5 Problem (Embedding into toric varieties).

Generalize the facts about ample line bundles and closed embedding into a
projective space to maps to toric varieties.

For example, what are necessary and sufficient conditions for a map to a toric
variety to be a closed embedding? Is it possible to embed a projective variety
into an arbitrary toric variety, if the dimension is not an obstruction?

######  0.6 Problem (Relative version of the map).

Construct and describe a map to a toric variety over a base other than the
spectrum of a field.

In fact, our construction of the map goes through in such a relative setup,
just as the construction of a map to a projective space is valid in the
relative setup. However finiteness and properness in a relative setting are
more involved as the Noether normalziation famously fails in this case. A
toric degeneration, like a toric scheme, can be thought of as a relative
generalization of a toric variety; thus, one may contemplate a map to a toric
degeneration.

######  0.7 Problem.

Classify maps to toric varieties. Also, classify topological coverings of
complex toric varieties that are not normal.

A special case of a map to a toric variety is a torus-equivariant vector
bundle over a toric variety, the topic studied by, for example, C. Manon and
K. Kaveh [KM19b]. Hence, the above problem is quite natural. For the
importance of the case of topological coverings, see a discussion around
Theorem 4.17.

######  0.8 Problem.

Show the local invariant cycle theorem for a non-semistable degeneration.

Clemens proves the local invariant cycle theorem for a semistable degeneration
(“semistable” means the special fiber is a divisor with normal-crossing) in
[Cle77] by constructing a map from a general fiber to the special fiber. It
seems reasonable that this result extends to a non-semistable case by using
the map in §4. Even more generally, nowadays the local invariant cycle theorem
can be obtained as an immediate consequence of the BBG decomposition –the
decomposition theorem of Beilinson, Bernstein and Deligne. As the main
application of the construction in §6 of [GM83] is to the specialization of
perverse sheaves, it is possible to do a version of the BBG decomposition in
the setup of §4.

###  0.6. Outline of the paper

In §1 we discuss the algebraic construction of a map to a toric variety, the
main results are Theorem 1.3 and Proposition 1.14 discussed above. In §2 we
review constructions of toric degenerations and apply the previous results to
this case. This is followed by a discussion on degenrartions by projection and
Problem 0.1 in §3. Next, in §4 we turn to the local non-algebraic constrcution
using Whitney startifications and conical structures. A key result is the
relative Poincaré Lemma (Corollary 4.16) necessary to prove the extension of
Weinstein’s Theorem (Theorem 5.8) which in turn allows to strengthen
Harada–Kaveh’s result on integrable systems (Theorem 5.9, discussed above) in
the last section §5.

###  0.7. Acknowledgements

We thank Nathan Ilten for his comment on the early work as well as informing
us of the paper [Cox93]. We also thank Kiumars Kaveh for helping us clarify
the exposition of the paper. Further we are greatful to Eduardo González for a
detailed reading of an early version of the paper and his valuable feedback.
L.B. acknowledges support of the PAPIIT project IA100122, Dirección General de
Asuntos del Personal Académico, Universidad Nacional Autónoma de México 2022.

###  0.8. Conventions and notations

The following terminology, notations and conventions are used throughout the
paper.

  * •

Following Bourbaki, $\mathbb{N}=\\{0,1,2,\dots\\}.$

  * •

Following [Har77], a “graded ring” means an $\mathbb{N}$-graded ring.

  * •

Generally, $k$ denotes the base field. We assume $k$ is algebraically closed
but make no blanket assumption on its characteristic.

  * •

A variety is assumed to be reduced and irreducible. A reducible variety means
a not-necessarily-irreducible variety.

  * •

$\mathbb{P}(V)$ denotes the projectivization of a finite-dimensional vector
space $V$ (not the dual of $V$). That is, as a set, $\mathbb{P}(V)$ is the set
of lines passing through the origin in $V$.

##  1\. Maps to toric varieties

We first recall the construction of maps to a projective space. For the sake
of completeness and future work we prefer to give a coordinate-free version
here. Our discussion of this follows §4.4. of [Ful98]. The non-coordinate-free
version is well-known; e.g., [Har77, Ch. II, Theorem 7.1].

Let $X$ be a variety and $\mathcal{L}$ an invertible sheaf on it. Given a
finite-dimensional vector subspace $V\subset\Gamma(X,\mathcal{L})$ generating
$\mathcal{L}$, we have that
$\mathcal{O}_{X}\otimes_{k}V\to\mathcal{L},\,(f,s)\mapsto fs$ is surjective
and thus we get the closed embedding:

$X\hookrightarrow\mathbb{P}(V^{*})\times X$

such that the embedding followed by the projection:

$\varphi:X\to\mathbb{P}(V^{*})$

has the property
$\mathcal{L}=\varphi^{*}(\mathcal{O}_{\mathbb{P}(V^{*})}(1))$.555This can be
seen as follows. Let $V_{X}=V\otimes_{k}\mathcal{O}_{X}$. Since
$\operatorname{Sym}(V_{X}\otimes\mathcal{L}^{*})\to\bigoplus_{0}^{\infty}\mathcal{O}_{X}$
is surjective and
$X=\operatorname{Proj}_{X}(\bigoplus_{0}^{\infty}\mathcal{O}_{X})$, we get
$X\hookrightarrow\mathbb{P}(V_{X}^{*}\otimes\mathcal{L})\simeq\mathbb{P}(V_{X}^{*})=\mathbb{P}(V^{*})\times
X$ as a projetive bundle is stable under a twist by an invertible sheaf
([Har77, Ch. II, Lemma 7.9.]).  The vector space $V$ and its projectivization
$\mathbb{P}(V)$ are called a (base-point-free) linear system on $X$ and
$\varphi$ the map determined by the linear system $\mathbb{P}(V)$. The reason
for the terminology is as follows: each hyperplane $H$ in $\mathbb{P}(V^{*})$
is represented by a nonzero linear functional on $V^{*}$; i.e., a nonzero
vector $s$ in $V$ and $\varphi^{-1}(H)=X\times_{\mathbb{P}(V^{*})}H$ is an
effective (Cartier) divisor on $X$ that is the zero-locus of $s$ in $X$. Thus,
$\varphi^{-1}(H)$ for various $H$ form a family of effective divisors on $X$
that are linearly equivalent to one another; whence, the terminology.

Conversely, given a morphism $\varphi:X\to\mathbb{P}(V^{*})$ for some finite-
dimensional vector space $V$, the pullback
$\mathcal{L}=\varphi^{*}(\mathcal{O}_{\mathbb{P}(V^{*})}(1))$ is an invertible
sheaf on $X$ and, moreover, if $\varphi(X)$ is not contained in any
hyperplane, then
$\varphi^{*}:V=\Gamma(\mathbb{P}(V^{*}),\mathcal{O}_{\mathbb{P}(V^{*})}(1))\hookrightarrow\Gamma(X,\mathcal{L})$
is injective. Thus, in that case, $V$ can be viewed as a subspace of
$\Gamma(X,\mathcal{L})$ and the $\varphi$ here coincides with the $\varphi$
constructed earlier from $V$. Because of this, it is common to view a linear
system and a map to a projective space interchangeably.

It is not straightforward to repeat the above discussion with a projective
space replaced by a proejctive toric variety. It is, however, fairly
straightforward to generalize the coordinate version of the above discussion.

We begin by introducing a projective (possibly non-normal) toric variety
playing a role analogous to that of a projective space. Let the torus
$\mathbb{G}_{m}^{r}$ act linearly on $\mathbb{P}^{q}$ in such a way that the
homogeneous coordinates $y_{0},\dots,y_{q}$ have weights $a_{0},\dots,a_{q}$
and a generic stabilizer is finite (so necessarily $r\leq q$). Then let
$W\subset\mathbb{P}^{q}$ be the closure of a $\mathbb{G}_{m}^{r}$-orbit where
a stabilizer is finite. If $W$ is normal, then it is a toric variety. For this
reason $W$ is usually referred to as a possibly non-normal toric variety. By
passing from $W$ to a Veronese embedding (if necessary) we shall also assume
the section ring $R^{\prime}=R(W,\mathcal{O}_{W}(1))$ coincides with the
homogeneous coordinate ring; thus, the section ring is generated by degree-one
elements. Since $R^{\prime}$ is a $\mathbb{G}_{m}^{r}$-module and the base
field $k$ is algebraically closed, each degree piece of $R^{\prime}$ has a
$\mathbb{G}_{m}^{r}$-weight decomposition. If $f,g$ are $\mathbb{G}_{m}^{r}$
weight vectors in $R^{\prime}$ and homogeneous of the same degree, then $f/g$
is a rational function on $W$ of weight zero; i.e., it is constant on a
$\mathbb{G}_{m}^{r}$-orbit. Thus, $f/g$ is constant. This is to say
$R^{\prime}$ is the semigroup algebra

$R^{\prime}=\bigoplus_{(n,a)\in S(R^{\prime})}k\cdot\chi^{(n,a)}$

of some subsemigroup $S(R^{\prime})\subset\mathbb{N}\times\mathbb{Z}^{r}.$
Here, $\chi^{(1,a_{i})}$ are the restrictions of homogeneous coordinates
$y_{i}$ on $\mathbb{P}^{q}$.

######  1.1 Definition.

The convex polytope $\Delta(1)$ associated to $W\subset\mathbb{P}^{q}$ is
defined to be the convex hull of $\\{a_{0},\dots,a_{q}\\}$ in
$\mathbb{R}^{r}$.

Since $R^{\prime}$ is generated by $R^{\prime}_{1}$, the degree-one elements
$\chi^{(1,a_{i})}$ generate $R^{\prime}$ as a $k$-algebra. Thus, $(1,a_{i})$
generate the semigroup $S(R^{\prime})$ and are also integral points of
$\Delta(1)$. Moreover,

######  1.2 Proposition.

We have:

  1. (i)

$\Delta(1)\cap\mathbb{Z}^{r}=\\{a_{0},\dots,a_{q}\\}$ if and only if $W$ is
normal (i.e., projectively normal666as we assumed the section ring of
$\mathcal{O}_{W}(1)$ coincides with the homogeneous coordinate ring.)

  2. (ii)

Each vertex of $\Delta(1)$ is $a_{i}$ for some $i$.

###### Proof.

(i) It is known and is not difficult to show

  1. (1)

The saturation $\widetilde{S(R^{\prime})}$ of the semigroup $S(R^{\prime})$ is
the semigroup $(1\times\Delta(1))\cap(\mathbb{N}\times\mathbb{Z}^{r})$

  2. (2)

The integral closure of $R^{\prime}$ is the semigroup algebra of
$\widetilde{S(R^{\prime})}$.

The assertion follows from the above.

(ii) Clear. ∎

With the above setup, we can now state:

######  1.3 Theorem.

Given an invertible sheaf $\mathcal{L}$ on a variety $X$ and some nonzero
generators $g_{0},\dots,g_{q}$ of $\mathcal{L}$, the following are equivalent.

  1. (i)

There exists a unique morphism

$\varphi:X\to W$

such that $\varphi^{*}(\mathcal{O}_{W}(1))=\mathcal{L}$ and
$\varphi^{*}(\chi^{(1,a_{i})})=g_{i}$.

  2. (ii)

Inside the section ring
$R(X,\mathcal{L}):=\bigoplus_{n=0}^{\infty}\Gamma(X,\mathcal{L}^{\otimes n})$,
the sections $g_{0},\dots,g_{q}$ satisfy the following binomial relations:

$g^{\alpha}=g^{\beta}$

where $\alpha,\beta$ are arbitrary points in $\mathbb{N}^{q+1}$ such that
$\sum\alpha_{i}(1,a_{i})=\sum\beta_{i}(1,a_{i})$.

Conversely, if $\varphi:X\to W$ is a morphism, then
$\varphi^{*}(\mathcal{O}_{W}(1))$ is an invertible sheaf on $X$ and
$g_{i}=\varphi^{*}(\chi^{(1,a_{i})})$’s satisfy the relations in (ii).

###### Proof.

(i) $\Rightarrow$ (ii): We note that $\varphi^{*}:R(W,\mathcal{O}_{W}(1))\to
R(X,\mathcal{L})$ is a ring homomorphism. Then the relations in (ii) hold
since $\chi^{(1,a_{i})}$ satisfy the required relations.

(ii) $\Rightarrow$ (i): First we identify the defining ideal $I$ of
$W\subset\mathbb{P}^{q}$; i.e., the kernel of the surjection
$k[y_{0},\dots,y_{q}]\to k[S(R^{\prime})].$ We say that
$y^{\beta},\beta\in\mathbb{N}^{q+1}$ is a non-standard monomial if there is
some $\alpha\in\mathbb{N}^{q+1}$ such that

  1. (1)

$\displaystyle\sum_{i=0}^{q}(\alpha_{i}-\beta_{i})(1,a_{i})=0,$

  2. (2)

$\beta<\alpha$ in lexicographical ordering.

Then $y^{\beta}$ is called a standard monomial if it is not non-standard.
Clearly, the images of the standard monomials form a basis of
$k[S(R^{\prime})]$ and then the ideal $I$ is generated by all
$y^{\beta}-y^{\alpha}$ for nonstandard monomials $y^{\beta}$ and standard
monomials $y^{\alpha}$ such that $\sum_{i}(\alpha_{i}-\beta_{i})(1,a_{i})=0$
called a straightening law.

Next, by the discussion at the beginning of this section or by [Har77, Ch. II,
Theorem 7.1.], we have the morphism

$\varphi:X\to\mathbb{P}^{q}$

such that $\mathcal{O}_{\mathbb{P}^{q}}(1),y_{0},\dots,y_{q}$ pull-back to
$\mathcal{L},g_{0},\dots,g_{q}$ under $\varphi$. Explicitly (namely, by the
construction in loc. cit.), the restriction $\varphi:X_{g_{i}}=\\{g_{i}\neq
0\\}\to\mathbb{P}^{q}_{y_{i}}$ is given by the ring homomorphism

$k[y_{0}/y_{i},\dots,y_{q}/y_{i}]\to\Gamma(X_{g_{i}},\mathcal{O}_{X}),\,y_{j}/y_{i}\mapsto
g_{j}/g_{i}.$

The relations in (ii) imply tha above factors through
$k[\chi^{a_{j}-a_{i}}\mid 0\leq j\leq r]$. That is, $\varphi(X_{g_{i}})\subset
W\cap\\{y_{i}\neq 0\\}$. Hence, the image of $\varphi$ lies in $W$.

Finally, the morphism in (i) is unique since each $X\to W$ gives $X\to
W\hookrightarrow\mathbb{P}^{q}$ and the latter is uniquely determined by the
condition that $\mathcal{O}_{\mathbb{P}^{q}}(1),y_{i}$ pull-back to
$\mathcal{L},g_{i}$. ∎

Several remarks and examples are in order.

######  1.4 Remark.

When $g_{0},\dots,g_{q}$ do not generate $\mathcal{L}$; i.e., when the common
zero-locus $B=V(g_{0},\dots,g_{q})$ is nonempty, we still have the map

$\varphi:X-B\to W$

that satisfies the conditions in (i), as the theorem applies to the variety
$X-B$ and $\mathcal{L},g_{i}$ restricted to $X-B$.

Moreover, when $X$ is complete and $\mathcal{L}$ is ample (so that
$X=\operatorname{Proj}R(X,\mathcal{L})$), we can use the Proj construction to
construct the above map. Indeed, $k[g_{0},\dots,g_{q}]$ is a graded subring of
$R=R(X,\mathcal{L})$. Also, $k[g_{0},\dots,g_{q}]$ is a quotient of the
homogeneous coordinate ring of $W\subset\mathbb{P}^{q}$ because the generators
$g_{i}$ satisfy the relations in (ii) of the theorem. Thus, the Proj
construction applied to $k[g_{0},\dots,g_{q}]\hookrightarrow R$ gives the map

$X-B\to W,$

the closure of whose image is the Proj of $k[g_{0},\dots,g_{q}]$.

######  1.5 Remark.

In literature, the relations in (ii) are often phrased in terms of a matrix.
Namely, let $M=\begin{bmatrix}1&\cdots&1\\\ a_{0}&\cdots&a_{q}\end{bmatrix}$
where we view $a_{i}$ as column vectors. Then the relations in (ii) says that
$g^{\alpha}=g^{\beta}$ whenver the column vector $\alpha-\beta$ is in the
kernel of $M$. Since $M$ is surjective (by the assumption on stabilizers), the
rank of the kernel of $M$ as an abelian group is $q-r$. That is, the number of
the binomial relations in (ii) is $q-r$.

######  1.6 Example.

Let $\mathbb{G}_{m}$ act linearly on $\mathbb{P}^{2}$ in such a way the
weights $a_{0},a_{1},a_{2}$ of the homogeneous coordinates $y_{0},y_{1},y_{2}$
are $1,0,3$, respectively and then take $W$ to be the closure of the orbit
$\mathbb{G}_{m}\cdot[1:1:1]$.

Let $X=\mathbb{P}^{1}$ be a projecitve line with homogeneous coordinates $u,v$
and take $g_{0}=u^{2}v,g_{1}=u^{3},g_{2}=v^{3}$. Then we have
$g_{1}^{2}g_{2}=g_{0}^{3}$; i.e., $g^{\alpha}=g^{\beta}$ for
$\alpha=(0,2,1),\beta=(3,0,0)$. We have $\alpha-\beta$ is in the kernel of the
matrix $M=\begin{bmatrix}1&1&1\\\ 1&0&3\end{bmatrix}$ (see Remark 1.5). Thus,
$g_{i}$ satisfy the binomial relations in (ii). Also,
$V(g_{0},g_{1},g_{2})=\emptyset$. Thus, we get

$\varphi:\mathbb{P}^{1}\to W$

such that $\mathcal{O}_{W{(1)}}$ pulls-back to
$\mathcal{O}_{\mathbb{P}^{1}}(3)$ and $\chi^{(1,a_{i})}$ to $g_{i}$. Let $u,v$
have weight $0,1$. Then $g_{0},g_{1},g_{2}$ have weights $1,0,3$ and thus
$\varphi$ is $\mathbb{G}_{m}$-equivariant. That is to say $\varphi$ is exactly
the normalization map (since $k(X)=k(W)$).

######  1.7 Example (torus embedding).

Take $X=\mathbb{G}_{m}^{r}$ to be the torus. Then Theorem 1.3 reduces to a
torus embedding (which is not necessarily an embedding).

Take $\mathcal{L}=\mathcal{O}_{X}$ to be the structure sheaf and
$g_{0},\dots,g_{q}$ regular functions on $X$ given as $t\mapsto
t^{a_{0}},\dots,t\mapsto t^{a_{q}}$ where we wrote
$t^{a_{i}}=t_{1}^{a_{i,1}}\cdots t_{r}^{a_{i,r}}$. Then $g_{i}$ satisfy the
relations in (ii) of the theorem and so we get the map

$\varphi:\mathbb{G}_{m}^{r}\to W$

which is given as $t\mapsto[t^{a_{0}}:\dots:t^{a_{q}}]$.

Now, let $W^{nor}$ be the normalization of $W$. Then $W^{nor}$ contains
$\mathbb{G}_{m}^{r}$ as an open dense subset. Then the above map extends to
$W^{nor}\to W$, which is the normalization map.

######  1.8 Example (Cox’s theorem).

Let $W^{nor}$ denote the normalization of $W$ and $\Sigma$ the fan defining
$W^{nor}$. Let $\Sigma(1)$ denote the set of all rays in $\Sigma$. The affine
space $\mathbb{A}^{\Sigma(1)}$ has coordinates $x_{\rho}$ parametrized by
$\rho\in\Sigma(1)$. For each cone $\sigma$ in $\Sigma$, let

$x^{\overline{\sigma}}=\prod_{\rho\not\subset\sigma}x_{\rho}.$

If $\Sigma(1)$ spans $\mathbb{R}^{r}$, then we have ([Cox05, Theorem 3.1.]):

$\varphi:\mathbb{A}^{\Sigma(1)}-B\to W^{nor}\to W$

where the first map is a GIT quotient map777Here, the quotient need not be a
geometric quotient unless the fan of $W^{nor}$ is simplicial. See for example,
Theorem 5.1.11. in [CLS11]. and
$B=V(x^{\overline{\sigma}}\mid\sigma\in\Sigma)$ = the unstable locus. Note
that the normalization map is given by extending the torus embedding as in the
previous example (Example 1.7).

Note that since the normalization is finite, in particular, affine and since a
GIT quotient map is affine (by construction), the above $\varphi$ is affine;
in other words, $\varphi$ is the structure map corresponding to the relative
Spec of a quasi-coherent $\mathcal{O}_{W}$-algebra.

For example, if $W=W^{nor}=\mathbb{P}^{1}$, then $\Sigma(1)$ has two rays and
the above is the usual map from the the affine cone over $\mathbb{P}^{1}$ off
the vertex to $\mathbb{P}^{1}$. The case of
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ is similar ([Cox05, Example 3.5.]).

Even though investigating when the map in the above theorem is a closed
embedding is outside the scope of this paper, we mention the following fact
generalizing the corresponding fact on a homogeneous coordinate ring.

######  1.9 Corollary.

For $\varphi$ appearing in the theorem, assume $\varphi$ is a closed embedding
(this implies that $X$ is projective). Then $X$ is the Proj of the image $R$
of the ring homomorphism below

$k[\chi^{(1,a_{0})},\dots,\chi^{(1,a_{q})}]\to R(X,\mathcal{L}).$

Another interesting case is when the map $\varphi$ is finite surjective, which
is our main concern. First we mention:

######  1.10 Remark (connection to the Noether normalization).

If $X$ is complete, $\mathcal{L}$ is ample, $\dim X=r$ and $g_{i}$ are in
general positions, then $\varphi$ is finite; we see this exactly the same way
we see the Noether normalization is finite (precisely by (iii) of the graded
Nakayama lemma below).

######  1.11 Lemma (graded Nakayama Lemma).

Let $R$ be a graded ring and $M$ a $\mathbb{Z}$-graded module over $R$ such
that $M_{l}=0$ for $l\ll 0$.

  1. (i)

If $IM=M$ for some ideal $I\subset R_{+}$, then $M=0$.

  2. (ii)

$M$ is finite over $R$ if and only if $M/IM$ is finite over $R/I$.

  3. (iii)

Let $S$ be a Noetherian graded ring. Given a grade-preserving ring
homomorphism $R\to S$, we have: $R\to S$ is finite if and only if $R_{0}\to
S_{0}$ is finite and $\sqrt{R_{+}S}=S_{+}$.

###### Proof.

(i) is easy to see and (ii) is proved by the same argument used for the usual
Nakayama Lemma.

(iii) We only need to prove ($\Leftarrow$). Let $x_{1},\dots,x_{r}$ be
homogeneous generators of the ideal $S_{+}$. Then $S$ is spanned over $S_{0}$
by monomials $x_{1}^{n_{1}}\dots x_{r}^{n_{r}}$ of degree $\sum
n_{i}\deg(x_{i})$. Given an integer $m>0$, if $\sum n_{i}\deg(x_{i})\geq
rm\max_{i}\deg(x_{i})$, then $n_{i}\geq m$ for some $i$. Thus, $S/S_{+}^{m}$
is a finite direct sum of finite $S_{0}$-modules and so is finite over
$S_{0}$. If $m$ is large enough, then $S_{+}^{m}\subset R_{+}S$ and so
$S/R_{+}S$ is finite over $S_{0}$; thus is finite over $R_{0}=R/R_{+}$ by
assumption. Hence, by (ii), $S$ is finite over $R$. ∎

######  1.12 Remark.

In Theorem 1.3, it is not important that the base ring is a field; in fact
[Har77, Ch. II, Theorem 7.1.] that we used is valid over an arbitrary ring and
the proof of Theorem 1.3 also does not use the fact that the base ring is a
field.

######  1.13 Remark.

It is well-known888Here is a quick proof: it is enough to consider a map
$\varphi:X-B\to\mathbb{P}^{r}$ (since if the image of $\varphi$ is contained
in $W$, the image of any extension is still contained in $W$). Let $Z\subset
B$ be the union of the scheme-theoretic irreducible components of $B$ of
codimension one (called the fixed part of the linear system) and $\pi:Y\to X$
the blow-up along $B$. Then, by [Har77, Ch. II, Example 7.17.3.],
$\varphi:X-B\to\mathbb{P}^{r}$ extends to $Y\to\mathbb{P}^{r}$ when $X-B$ is
identified as an open subset of $Y$. Now, since $X$ is smooth, $Z$ is an
effective Cartier divisor on $X$ and $\pi$ is an isomorphism on $Z$; i.e.,
$\varphi$ can be defined on $Z$. that a morphism from a nonempty open subset
of a smooth variety to a projective variety extends to an open subset whose
complement has codimension $\geq 2$. This fact applies to the above $\varphi$.
Thus, if $X$ is smooth, then $\varphi$ can be extended to

$\varphi:X-B^{\prime}\to W$

where $B^{\prime}$ has codimension $\geq 2$.

_A priori_ , it is not obvious whether there can even be a nontrivial map from
a given variety to a given toric variety, up to a base locus. The next
proposition settles this question positively.

######  1.14 Proposition.

Let $W$ be a $\mathbb{G}_{m}^{r}$-orbit closure of dimension $r$ in
$\mathbb{P}^{q}$ as in Theorem 1.3. Let $X$ be a variety and $\mathcal{L}$ an
invertible sheaf on $X$. Then there exist sections $g_{0},\dots,g_{q}$ of some
power $\mathcal{L}^{\otimes m}$ that determine a morphism

$\varphi:X-B\to W,\,\,B=V(g_{0},\dots,g_{q})$

along which $\mathcal{O}_{W}(1)$ pulls-back to $\mathcal{L}^{\otimes
m}|_{X-B}$ and whose image has dimension equal to the transcendence degree of
the section ring of $\mathcal{L}$ minus one. In fact, these $g_{i}$’s here are
monomials of some sections $u_{0},\dots,u_{r}$ of some power
$\mathcal{L}^{\otimes l}$.

Conversely (though not the exact converse), if $\varphi:X\to W$ is a morphism
such that $\mathcal{O}_{W}(1)$ pulls-back to $\mathcal{L}$, then $\varphi$ is
a unique extension of the map given by monomials; i.e., there are regular
functions $f_{i},1\leq i\leq r$ on some nonempty open subset $U$ of $X$ such
that $\mathcal{L}|_{U}\simeq\mathcal{O}_{U}$, $\varphi|_{U}$ is given as

$x\mapsto[f^{a_{0}}(x):\cdots:f^{a_{r}}(x)]$

and $f^{a_{i}}$ are the restrictions of sections of some power
$\mathcal{L}^{\otimes m}$ under $\mathcal{O}_{U}\simeq\mathcal{L}^{\otimes
m}|_{U}$.

###### Proof.

Let $d^{\prime}$ be the transcendence degree of the section ring $R$ of
$\mathcal{L}$. First assume $d^{\prime}\geq r+1$. Then we can find some
elements $x_{0},\dots,x_{r}$ in $Q(R)$ that are algebraically independent over
the base field $k$. We can assume the $x_{i}$’s all have the same denominator;
i.e., we can write $x_{i}=u_{i}/z$ where $u_{i},z$ are all in $R$. It is easy
to see that $u_{0},\dots,u_{r}$ are then algebraically independent. Next,
suppose for each homogeneous component $u_{0,i}$ of $u_{0}$ that
$\\{u_{0,i},u_{1},\dots,u_{r}\\}$ is algebraically dependent. Then each
$u_{0,i}$ is algebraic over $k(u_{1},\dots,u_{r})$, which implies $u_{0}$ is
algebraic over $k(u_{1},\dots,u_{r})$, a contradiction. Hence, we can replace
$u_{0}$ by some homogeneous component while we still have that
$\\{u_{0},\dots,u_{r}\\}$ is algebraically independent. Repeating this
procedure to the rest of $u_{i}$’s, we can have that $u_{i}$ are all
homogeneous. Finally, note $\\{u_{0}^{n_{0}},\dots,u_{r}^{n_{r}}\\}$ is
algebraically independent for positive integers $n_{0},\dots,n_{r}$. Thus,
replacing each $u_{i}$ by some power, we can assume the $u_{i}$’s all have the
same degree.

Then the sections $u_{0},\dots,u_{r}$ determine
$\pi:X-B_{0}\to\mathbb{P}^{r},\,B_{0}=V(u_{0},\dots,u_{r})$ such that
$\mathcal{O}_{\mathbb{P}^{r}}(1)$ pulls-back to $\mathcal{L}^{\otimes
l},\,l=\deg(u_{i})$. Since
$\pi^{*}:k[y_{1}/y_{0},\dots,y_{r}/y_{0}]\to\Gamma(X_{u_{0}},\mathcal{O}_{X}),\,y_{i}/y_{0}\mapsto
u_{i}/u_{0}$ is injective, we have that $\pi$ is dominant. On the other hand,
we have the torus embedding (Example 1.7) $\mathbb{G}^{r}\to W$, which extends
to $\iota:\mathbb{P}^{r}-B_{1}\to W$. Namely, the map $\iota$ is given as
$[s:t_{1}:\cdots:t_{r}]\mapsto[s^{m^{\prime}-|a_{0}|}t^{a_{0}}:\cdots:s^{m^{\prime}-|a_{q}|}t^{a_{q}}]$,
where $m^{\prime}=\max\\{|a_{i}|\mid 0\leq i\leq q\\}$. Hence, the composition
$\varphi:X-B\overset{\pi}{\to}\mathbb{P}^{r}-B_{1}\overset{\iota}{\to}W$ is
given by the sections $g_{i}=u_{0}^{m^{\prime}-|a_{i}|}u^{a_{i}}$ of
$\mathcal{L}^{\otimes m^{\prime}l}$. If $d^{\prime}<r+1$, then choose
$u_{0},\dots,u_{d^{\prime}}$ that are algebraically independent over $k$. Then
the same argument as before gives $\varphi$ whose image has dimension
$d^{\prime}-1$.

Next we show the converse. Given $\varphi:X\to W$, let $O$ be an open dense
orbit in $W$ and $U$ an open subset of $\varphi^{-1}(O)$. We can factor
$\varphi:U\to O$ as

$\varphi:U\to\mathbb{G}_{m}^{r}\overset{\iota}{\underset{\sim}{\to}}O\subset
W$

where $\iota$ is a torus embedding. Here, the first map
$U\to\mathbb{G}_{m}^{r}$ is then given by some $f_{i}:U\to\mathbb{G}_{m},1\leq
i\leq r$. Shrinking $U$, we can assume $U=X_{s}$ for some nonzero section $s$
of $\mathcal{L}^{\otimes l}$. Then $s^{m^{\prime}}f_{i}$ are global sections
of $\mathcal{L}^{\otimes lm^{\prime}}$ for large enough $m^{\prime}>0$
([Har77, Ch. II, Lemma 5.14. (b)]), which are $f_{i}$ on $U$ under
$\mathcal{O}_{U}\simeq\mathcal{L}^{\otimes lm^{\prime}}|_{U}$. ∎

In general, it is not possible to extend $\varphi$ in the proposition, even as
a continuous map, to the whole $X$, as the next example shows:

######  1.15 Example.

Take $X=\mathbb{P}^{r}$. Then consider

$\varphi:X-B\to
W,\,[s:t_{1}:\dots:t_{r}]\mapsto[s^{m-|a_{0}|}t^{a_{0}}:\cdots:s^{m-|a_{q}|}t^{a_{q}}]$

where $m=\max_{i}|a_{i}|$ and $B=V(s^{m-|a_{i}|}t^{a_{i}}\mid 0\leq i\leq q)$
the base locus. Taking $s=1$, we see that the above $\varphi$ is an extension
of a torus embedding and, in particular, is dominant. Let $\mathbb{G}_{m}^{r}$
act on $X=\mathbb{P}^{r}$ in the standard way and then, clearly, $\varphi$ is
$\mathbb{G}_{m}^{r}$-equivariant. Now, if $\varphi$ extends to some continuous
map $\widetilde{\varphi}:X\to W$, then $\widetilde{\varphi}$ is surjective (as
it has dense and closed image) and still equivariant but that implies that $W$
has the same or fewer number of orbits than $X=\mathbb{P}^{r}$, which can be
true but not always.

######  1.16 Remark.

The morphism $\varphi$ constructed in the first part of Proposition 1.14 can
be viewed as the restriction of a Veronese embedding followed by a projection;
i.e.,

$\textstyle{X-B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{\mathbb{P}^{N}-L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W}$

where the top map is given by the Veronese embedding
$\mathbb{P}^{n}\hookrightarrow\mathbb{P}^{N}$ and the vertical map on the
right is a projection $\mathbb{P}^{N}-L\to\mathbb{P}^{q}$, $L$ some linear
subspace. (This observation will become important in §3.)

The second part of the propostion says that, at least locally, _any_ morphism
$\varphi:X\to W$ is of that form. In other words, any morphism $\varphi:X\to
W$ is necessarily obtained by extending a torus embedding, when $f_{i}$ are
thought of as coordinates.

With an eye on applications, in this paper, we only consider a map to a quasi-
projective (possibly non-normal) toric variety. However, the “quasi-
projective” assumption can be dropped as follows. We first need to introduce a
definition. For our purpose, we say a variety $X$ is a not-necessarily-normal
toric variety or a non-normal toric variety if $X$ has an action of torus
$\mathbb{G}_{m}^{r}$, there is an open dense torus orbit on $X$ and there is
an open invariant affine chart on $X$.999This definition of a non-normal toric
variety coincides with the definition in the first author’s thesis. It is then
easy to see that for each open invariant subset $U\subset X$,
$\mathcal{O}_{X}(U)$ is the semigroup algebra of a subsemigroup of
$\mathbb{Z}^{r}$ so that the induced $\mathbb{G}_{m}^{r}$-action on $U$
coincides with that on $X$.

######  1.17 Lemma.

Let $X$ be a scheme over $k$ and $W$ an affine non-normal toric variety; i.e.,
$W$ is the Spec of the semigroup algebra of a semigroup
$S\subset(\mathbb{Z}^{r},+,0)$. Then there is a natural101010“natural” in the
sense in category theory. bijection

$\operatorname{Mor}_{k}(X,W)=\\{\text{homomorphisms
}\,S\to(\Gamma(X,\mathcal{O}_{X}),\cdot)\\}$

where the notation $(A,\cdot)$ means the ring $A$ is viewed as a unital
semigroup with respect to the multiplication.

###### Proof.

For any $k$-algebra $A$, we have
$\operatorname{Mor}_{k}(X,\operatorname{Spec}(A))=\operatorname{Hom}_{k}(A,\Gamma(X,\mathcal{O}_{X}))$.
If $A$ is the semigroup algebra of $S$, by the universal property of a
semigroup algebra, each ring homomorphism $A\to\Gamma(X,\mathcal{O}_{X})$
uniquely corresponds to a semigroup homomorphism
$S\to(\Gamma(X,\mathcal{O}_{X}),\cdot)$. ∎

We note a curious feature that the right-hand side is independent of the base
field $k$. In fact, this is the reason why a non-normal toric variety is
sometimes thought of as an example of a beyond-a-scheme notion. We also note
that the right-hand side is independent of the additive structure of
$\Gamma(X,\mathcal{O}_{X})$; this explains the monomial nature of a map to a
toric variety.

In the non-affine case, we have:

######  1.18 Proposition.

Let $X$ be a scheme over $k$ and $W$ a non-normal toric variety. Then each
$k$-morphism $X\to W$ determines an open affine cover $U_{i}$ of $X$ together
semigroup homomorphisms

$S_{i}\to(\Gamma(U_{i},\mathcal{O}_{X}),\cdot)$

such that the induced maps $U_{i}\to\operatorname{Spec}k[S_{i}]$ agree on
overlaps. Conversely, such an open affine cover determines a $k$-morphism
$X\to W$. The resulting bijection between the set of $k$-morphisms $X\to W$
and the set111111To be precise, the set of equivalence classes of open affine
covers with semigroup homomorphisms. of open affine covers with semigroup
homomorphisms is natural in $X,W$.

###### Proof.

Let $\varphi:X\to W$ be a morphism. Choose an open affine cover $U_{i}$ of $X$
such that for each $i$, there is an open affine invariant subset
$V_{i}=\operatorname{Spec}k[S_{i}]$ of $W$ with $\varphi(U_{i})\subset V_{i}$.
Then by the early lemma, $\varphi:U_{i}\to V_{i}$ is given by
$S_{i}\to(\Gamma(U_{i},\mathcal{O}_{X}),\cdot)$. The converse is clear. ∎

##  2\. Toric degenerations

A degeneration is a classical technique in algebraic geometry, especially in
enumerative algebraic geometry. The underlying idea behind the technique is
the following: in order to count the number of solutions in an algebraic
system, the coefficients of the system can be _perturbed_ , without affecting
the count. In modern terms, a degeneration of a variety $X$ is a flat
surjective morphism (i.e., a faithfully flat morphism) whose general fiber is
$X$; the flatness implies, for example, all the fibers share the same Hilbert
function (in particular, the arithmetic genus agrees).

###  2.1. Flatness

A toric degeneration is a degeneration whose special fiber is a possibly non-
normal toric variety. Note, since a normal toric variety has arithmetic genus
$0$ (since it is rational), it is almost a mandate here that we do not require
the special fiber is normal. A toric degeneration has a number of
applications, a famous one among which is an application to mirror symmetry
[BCFKvS00].

This section gives a brief discussion of a toric degeneration; most of the
materials here are review but some results do not seem to appear elsewhere.

######  2.1 Definition.

Given an algebraic variety $X$, a (flat) abstract degeneration of $X$ is a
flat surjective morphism of schemes:

$\xi:\mathfrak{X}\to\mathbb{A}^{1}$

together with an isomorphism over $\mathbb{A}^{1}-0$:

---
$\textstyle{\mathfrak{X}-\xi^{-1}(0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi}$$\textstyle{\sim}$$\textstyle{X\times(\mathbb{A}^{1}-0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{pr}_{2}}$$\textstyle{\mathbb{A}^{1}-0.}$

The scheme $\mathfrak{X}$ is called the total space of the degeneration and
$\xi^{-1}(0)$ called the special fiber. Finally, an abstract toric
degeneration is a degeneration whose special fiber is a variety admitting an
action by the torus $\mathbb{G}_{m}^{\dim X}$ with a dense orbit and open
invariant affine charts.

######  2.2 Remark.

Let $\xi:\mathfrak{X}\to\mathbb{A}^{1}=\operatorname{Spec}k[t]$ be a toric
degeneration of a variety $X$. Let $\mathcal{L}$ be an invertible
$\mathcal{O}_{\mathfrak{X}}$-module that is flat over $\mathbb{A}^{1}$ (i.e.,
$\Gamma(U,\mathcal{L})$ is a flat k[t]-module for any open subset $U$). If
$\mathcal{L}_{0}:=\mathcal{L}|_{\xi^{-1}(0)}$ is very ample, then, for the
vector space $V=\Gamma(\xi^{-1}(0),\mathcal{L}_{0})$, by definition, there is
a closed immersion

$\xi^{-1}(0)\hookrightarrow\mathbb{P}(V^{*})$

such that $\mathcal{L}_{0}=\mathcal{O}_{\mathbb{P}(V^{*})}(1)|_{\xi^{-1}(0)}$.
Since $\operatorname{Sym}^{n}(V)=\Gamma(\mathbb{P}(V^{*}),\mathcal{O}(n))$,
the above determines a graded ring homomorphism

$\textstyle\operatorname{Sym}(V)\to\bigoplus_{0}^{\infty}\Gamma(\xi^{-1}(0),\mathcal{L}_{0}^{\otimes
n})$

whose image $R_{\xi^{-1}(0)}$ is the homogeneous coordinate ring of
$\xi^{-1}(0)$.

Further, if $V$ is a $\mathbb{G}_{m}^{d}$-module and
$\xi^{-1}(0)\hookrightarrow\mathbb{P}(V^{*})$ is equivariant,121212Usually,
$\mathcal{L}_{0}$ is said to be linearized. then, as in the discussion
preceding Theorem 1.3, we see $R=R_{\xi^{-1}(0)}$ is a graded semigroup
algebra:

$R_{\xi^{-1}(0)}=k[S]=\bigoplus_{(n,u)\in S}k\cdot\chi^{(n,u)}$

of some finitely generated subsemigroup
$S\subset\mathbb{N}\times\mathbb{Z}^{d}$, the grading given by the
$\mathbb{N}$-factor.

We recall that ampleness is an open condition in the following sense:

######  2.3 Theorem ([Gro63] Theorem 4.7.1.131313This theorem in the
formulation weaker than what we need can also be found as Theorem 1.2.17. in
[Laz04].).

Let $S$ be the Spec of a Noetherian ring, $f:X\to S$ a proper morphism of
schemes (not necessarily flat) and $\mathcal{L}$ an invertible
$\mathcal{O}_{X}$-module.

If $\mathcal{L}|_{f^{-1}(s)}$ is ample for some $s$ in $S$, then there exists
an open neighborhood $U\subset S$ of $s$ such that $\mathcal{L}|_{f^{-1}(U)}$
is relatively ample over $U$.

Going back to the situation of Remark 2.2, we note, assuming $\xi$ is proper,
that the theorem implies that $\mathcal{L}$ is relatively ample over
$\mathbb{A}^{1}$ (since $\xi$ is trivial away from the special fiber); i.e.,
there is a closed immersion

$\mathfrak{X}\hookrightarrow\mathbb{P}^{N}\times\mathbb{A}^{1}$

such that $\xi$ coincides with the inclusion followed by the projection.

It is well-known that the flatness of a family of projective varieties is
equivalent to the constancy of Hilbert polynomial. We want to note this fact
remains true if “Hilbert polynomial” is replaced by “Hilbert function”.

######  2.4 Lemma.

Let $S$ be the Spec of a Noetherian integral domain and
$X\subset\mathbb{P}^{N}_{S}:=\mathbb{P}^{N}\times_{\mathbb{Z}}S$ a closed
subscheme. For each $s\in S$, let $R_{X_{s}}$ denote the homogeneous
coordinate ring of the fiber $X_{s}\subset\mathbb{P}^{N}_{\kappa(s)}$ over $s$
and then let $f(n,s)=\dim_{\kappa(s)}R_{X_{s},n}$, the Hilbert function of
$R_{X_{s}}$.

Then $X\to S$ is flat if and only if $f(n,s)$ is independent of $s$.

###### Proof after Exercise 20.14. in [Eis13].

This follows from showing the algebraic fact that, for a finite module $M$
over a Noetherian integral domain $A$, $M$ is flat if and only if $M$ is
projective if and only if
$\mu_{\mathfrak{p}}(M):=\dim_{\kappa(\mathfrak{p})}(M\otimes_{A}\kappa(\mathfrak{p}))$
= the minimum number of the generators of $M_{\mathfrak{p}}$ is independent of
prime ideals $\mathfrak{p}$ of $A$.

(Take $A=\Gamma(\mathcal{O}_{S})$ and $M=R_{n}$ to get the assertion, where
$R$ is the relative homogeneous coordinate ring of $X$.) ∎

###  2.2. Toric degenerations and valuations

In what follows we recall a construction of toric degenerations obtained by
certain valuations. The main result goes back to Anderson’s work [And13]. It
has since been shown that his construction is not a special case but rather
applies to many toric degenerations as we see in Theorem 2.6 due to [KMM22]
below.

Let $k$ be an algebraically closed field. Throughout this section we denote by
$R$ a positively (multi-)graded algebra and domain, that is
$R=\bigoplus_{w\in\mathbb{Z}^{m}_{\geq 0}}R_{w}$. Let $\Gamma$ be an abelian
group that is totally ordered by $<$. By a (Krull) valuation on $R$ we mean a
map $\nu:R\setminus\\{0\\}\to\Gamma$ that satisfies for all $a,b\in R$ and
$c\in k$

$\nu(ab)=\nu(a)+\nu(b),\quad\nu(a+b)\leq\max\\{\nu(a),\nu(b)\\},\quad\nu(ca)=\nu(a).$

If $\nu$ only satisfies $\nu(ab)\leq\nu(a)+\nu(b)$ it is called a
quasivaluation. Notice that the image of a valuation $\nu$ carries the
structure of an additive semigroup. It is therefore called the value semigroup
of $\nu$ and we denote it by $S(R,\nu)$. The rank of $\nu$ is defined as the
rank of the group completion of its semigroup inside $\Gamma$, $\nu$ is said
to have full rank if its rank coincides with the Krull dimension of $A$. Every
valuation induces a filtration on $A$ with filtered pieces for
$\gamma\in\Gamma$ defined by

$\mathcal{F}_{\nu,\gamma}:=\\{a\in R:\nu(a)\geq\gamma\\}\quad\left(\text{resp.
}\mathcal{F}_{\nu,>\gamma}:=\\{a\in R:\nu(a)>\gamma\\}\right).$

The associated graded algebra is
$\operatorname{gr}_{\nu}(R):=\bigoplus_{\gamma\in\Gamma}\mathcal{F}_{\nu,\gamma}/\mathcal{F}_{\nu,>\gamma}$.
There is a natural quotient map from $R$ to $\operatorname{gr}_{\nu}(R)$ given
by sending $f\in R$ to $\mathcal{F}_{\nu,\nu(a)}/\mathcal{F}_{\nu,>\nu(a)}$,
denote its image by $\hat{f}\in\operatorname{gr}_{\nu}(R)$. If the quotients
$F_{\nu,\gamma}/\mathcal{F}_{\nu,>\gamma}$ are at most one-dimensional, then
we say $\nu$ has one-dimensional leaves. This property is desirable as it
gives an identification

$\operatorname{gr}_{\nu}(R)\to k[S(R,\nu)],\quad\text{given
by}\quad\hat{f}_{\gamma}\mapsto\nu(f_{\gamma}),$

where $\hat{f}_{\gamma}\in\mathcal{F}_{\nu,\gamma}/\mathcal{F}_{\nu,>\gamma}$
is a generator and $f_{\gamma}\in R$ lies in the preimage of
$\hat{f}_{\gamma}$ under the quotient map
$\hat{}:R\to\operatorname{gr}_{\nu}(R)$. It is a consequence of Abhyankar’s
inequality that full-rank valuations have one-dimensional leaves.

An important definition is the notion of a Khovanskii basis: that is, a subset
$B$ of $A$ whose image in $\operatorname{gr}_{\nu}(R)$ is an algebra
generating set. It is not hard to see that if $B$ is a Khovanskii basis for
$\nu$ then the set $\\{\nu(b):b\in B\\}$ generates the value semigroup [KM19a,
Lemma 2.10].

A valuation is called homogeneous if it respects the grading on $R$; more
precisely, if $f\in R$ has homogeneous presentation $\sum_{i}f_{i}$ then
$\nu(f)=\max\\{\nu(f_{i})\\}$. A valuation is fully homogeneous if
$\nu(f)=(\deg(f),\nu^{\prime}(f))$, that is $S(R,\nu)\subset\mathbb{Z}_{\geq
0}^{m}\times\Gamma^{\prime}$. Any homogeneous valuation is obtained from a
fully homogeneous one by composing with an isomorphism of semigroups [IW20,
Remark 2.6]. So when studying homogeneous valuation we may without loss of
generality assume they are fully homogeneous. We incorporate this assumption
in the notation by writing
$\widehat{\nu}:A\setminus\\{0\\}\to\mathbb{Z}^{m}_{\geq
0}\times\Gamma^{\prime}$.

Given a fully homogeneous valuation
$\widehat{\nu}:A\setminus\\{0\\}\to\mathbb{Z}^{m}_{\geq
0}\times\Gamma^{\prime}$ we define its Newton–Okounkov cone as

(2)
$C(R,\widehat{\nu}):=\overline{\operatorname{Cone}(S(R,\widehat{\nu}))}=\overline{\operatorname{Cone}(\widehat{\nu}(f):f\in
A)},$

where the closure (in the Euclidean topology) is taken inside
$(\mathbb{Z}^{m}_{\geq
0}\times\Gamma^{\prime})\otimes_{\mathbb{Z}}\mathbb{R}$. Let
$\Gamma^{\prime}_{\mathbb{R}}=\Gamma^{\prime}\otimes_{\mathbb{Z}}\mathbb{R}$.
The Newton–Okounkov body of $\widehat{\nu}$ is then defined as the
intersection

(3)
$\Delta(R,\widehat{\nu}):=C(R,\widehat{\nu})\cap\\{(1,\dots,1)\\}\times\Gamma^{\prime}_{\mathbb{R}},$

where $(1,\dots,1)$ denotes the element whose entries are all one in
$\mathbb{Z}^{m}_{\geq 0}$. Later we will show that $\Delta(R,\widehat{\nu})$
is also the image of the moment map. We define the Rees algebra of
$\widehat{\nu}$ (or equivalently of $\nu$, where $\nu$ is the composition of
$\widehat{\nu}$ followed by the projection onto $\Gamma^{\prime}$) as

$\mathcal{R}_{\nu,R}:=\bigoplus_{i\geq 0}t^{i}\mathcal{F}_{\nu,i}\subset
R[t],$

where the $\mathbb{Z}$-filtration $\\{\mathcal{F}_{\nu,i}\\}_{i\in\mathbb{Z}}$
of $A$ is obtained from the $\Gamma$-filtration
$\\{\mathcal{F}_{\nu,\gamma}\\}_{\gamma\in\mathbb{Z}^{d}}$ by an appropriate
projection $\mathbb{Z}^{m}_{\geq 0}\times\Gamma^{\prime}\to\mathbb{Z}$, _i.e._
$\operatorname{gr}_{\nu}(R)\cong\bigoplus_{i\in\mathbb{Z}}\mathcal{F}_{\nu,i}/\mathcal{F}_{\nu,>i}$
[And13, Lemma 5.2].

######  2.5 Proposition (Proposition 5.1 in [And13]).

The Rees algebra $\mathcal{R}_{\nu,R}$ is a flat $k[t]$-algebra with
$\mathcal{R}_{\nu,R}/(t)\cong\operatorname{gr}_{\nu}(R)$ and
$\mathcal{R}_{\nu,R}[t^{-1}]\cong R[t,t^{-1}]$ as $k[t]$-modules.

Moreover, the isomorphisms in the above proposition hold for the _graded_
algebras (remember that $\widehat{\nu}$ is homogeneous). In particular,
$\phi:\operatorname{Proj}(\mathcal{R}_{\nu,R})\to\mathbb{A}^{1}$ is a toric
degeneration of $X$ with special fiber
$\phi^{-1}(0)=X_{0}=\operatorname{Proj}(\operatorname{gr}_{\nu}(R))$. While
$X_{0}$ is normal if and only if $S(R,\widehat{\nu})$ is saturated, Anderson
shows that its normalization is the projective toric variety associated with
the Newton–Okounkov polytope $\Delta(R,\widehat{\nu})$. In particular, for the
toric variety the Newton–Okounkov polytope coincides (up to scaling) with the
polytope $\Delta(1)$ defined in Definition 1.1.

Given the example of a toric degeneration induced by a valuation we may
formulate an _algebraic_ definition of toric degeneration. We summarize the
definition and its relation to valuations in the following result of Kaveh,
Manon and Murata:

######  2.6 Theorem (Theorem 1.11 in [KMM22]).

Let $R$ be a positively graded domain and let $\mathcal{R}$ be a finitely
generated positively graded $k[t]$-module and domain with the following
properties:

  * •

$\mathcal{R}[t^{-1}]\cong R[t,t^{-1}]$ as $k[t]$-modules and graded algebras;

  * •

the algebra $\mathcal{R}/(t)$ is a graded semigroup algebra $k[S]$ where
$S\subset\mathbb{N}\times\mathbb{Z}^{d}$;

  * •

the standard $\mathbb{G}_{m}$-action on $k[t]$ extends to an action on
$\mathcal{R}$ respecting its grading. Moreover the induced $k[t]$-action on
$\mathcal{R}/(t)=k[S]$ is via a homomorphism $S\to\mathbb{Z}$.

Then there is a full-rank valuation
$\hat{\nu}:A\setminus\\{0\\}\to\mathbb{N}\times\mathbb{Z}^{d}$ such that
$S=S(R,\nu)$.

###  2.3. Toric Gröbner degeneration

In the rest of the section we recall the relationship between toric
degenerations by valuations and Gröbner degeneration. We use the notation of
[Bos21] and [KM19a]. For background on Gröbner degenerations please consult
[Stu96, §1] and [Eis13, §15.8]. For background on the tropicalization of an
ideal we refer to [SS04, §2].

The following theorem is a consequence of the main result in [Bos21] based on
work of Kaveh–Manon [KM19a]. We include the proof for completeness.

######  2.7 Theorem.

Let $R$ be a graded $k$-algebra and domain of Krull-dimension $d+1$ and $k$ a
field of characteristic zero. Let
$\widehat{\nu}:R\setminus\\{0\\}\to\mathbb{N}\times\mathbb{Z}^{d}$ be a full
rank valuation with finitely generated value semigroup. Then there exists a
(graded) presentation of $R$

$R\cong k[x_{0},\dots,x_{n}]/J$

and a maximal cone $\tau$ in the tropicalization $\mathcal{T}(J)$ such that
$k[S(R,\widehat{\nu})]\cong k[x_{0},\dots,x_{n}]/\operatorname{in}_{\tau}(J)$.

###### Proof.

Consider a finite Khovanksii basis $b_{0},\dots,b_{n}$ for $R$ and
$\widehat{\nu}$. Define $M$ as the $(d+1)\times(n+1)$ integer matrix whose
columns are $\widehat{\nu}(b_{0}),\dots,\widehat{\nu}(b_{n})$. Notice that M
is of full rank $d+1\leq n+1$ as $\widehat{\nu}$ is of full rank. Define the
(graded) projection $\pi:k[x_{0},\dots,x_{n}]\to R$, by $x_{i}\mapsto b_{i}$
and let $J$ be its kernel. Notice that the grading on $k[x_{0},\dots,x_{n}]$
is not necessarily the standard grading but rather induced by the Khovanskii
basis elements in $R$: that is, $\deg(x_{i}):=\deg(b_{i})$. Then by [Bos21,
Theorem 1] the initial ideal $\operatorname{in}_{M}(J)$ is prime as the value
semigroup $S(R,\widehat{\nu})$ is generated by $\nu(b_{0}),\dots,\nu(b_{n})$.
The ideal $\operatorname{in}_{M}(J)$ is the initial ideal by means of _higher
Gröbner theory_ , see e.g. [KM19a, Appendix]. By definition is depends on a
total order on $\mathbb{N}\times\mathbb{Z}^{d}$, which we choose to be the
total order used by $\widehat{\nu}$. However, we may immediately restrict our
attention to the case of usual initial ideals as by [Bos21, Lemma 3] there
exists a weight vector $w\in\mathbb{Z}^{n+1}$ such that
$\operatorname{in}_{w}(J)=\operatorname{in}_{M}(J)$. It is left to show that

  1. (i)

$w\in\mathcal{T}(J)$;

  2. (ii)

$k[S(R,\widehat{\nu})]\cong k[x_{0},\dots,x_{n}]/\operatorname{in}_{\tau}(J)$,
where $w\in\tau^{\circ}$ (the open cone of $\mathcal{T}(J)$ containing $w$ in
its relative interior).

The first item follows from [Bos21, Corollary 3]. For the second consider the
quasi-valuation $\nu_{M}$ defined by M (as in Kaveh–Manon’s paper [KM19a], or
see [Bos21, Definition 5]). In this case $\nu_{M}$ is a valuation and by
[Bos21, Proposition 1] it satisfies $\widehat{\nu}=\nu_{M}$. Further, by
[Bos21, Equation 3.3] we have

$k[x_{0},\dots,x_{n}]/in_{w}(J)\cong\operatorname{gr}_{v}(R)\cong
k[S(R,\widehat{\nu})],$

where $\operatorname{gr}_{\nu}(R)$ is the associated graded algebra induced by
the filtration defined by $\widehat{\nu}$. ∎

The above result implies that we may assume without loss of generality that
our algebraic toric degeneration is a Gröbner degeneration. For simplicity we
assume $k=\mathbb{C}$. As $R$ is positively graded and $J$ is the kernel of a
graded homomorphism, we know that $J$ is homogeneous. This implies in
particular that the weight vector $w$ may be assumed to have only positive
integer entries. The total space defining the degeneration is embedded inside
$\mathbb{P}^{n}\times\mathbb{A}^{1}$ and defined by the ideal (see [Eis13,
Theorem 15.17])

$\hat{J}_{w}:=\left\langle\sum_{\alpha\in\mathbb{Z}^{n+1}_{\geq
0}}c_{\alpha}{\it x}^{\alpha}t^{w\cdot\alpha-\min\\{w\cdot\beta:\
c_{\beta}\not=0\\}}:\sum_{\alpha\in\mathbb{Z}^{n+1}_{\geq 0}}c_{\alpha}{\it
x}^{\alpha}\in J\right\rangle\subseteq k[t][x_{0},\dots,x_{n}],$

where $x_{0},\dots,x_{n}$ are the coordinates of $\mathbb{P}^{n}$ and $t$ is
the coordinate of $\mathbb{A}^{1}$.

######  2.8 Remark.

Notice that the ideal $\hat{J}_{w}$ is homogeneous with respect to the grading
vector

$\hat{w}:=(w_{0},\dots,w_{n},-1)\in\mathbb{Z}^{n+2},$

where the entry $-1$ defines the degree of the variable $t$ and the degrees of
the variables $x_{i}$ are the corresponding entries of
$w=(w_{0},\dots,w_{n})$. Geometrically, this grading induces a
$\mathbb{G}_{m}$-action on
$\mathfrak{X}=V(\hat{J}_{v})\subset\mathbb{P}^{n}\times\mathbb{A}^{1}$:

$\displaystyle\mathfrak{X}\times\mathbb{G}_{m}$ $\displaystyle\to$
$\displaystyle\mathfrak{X}$
$\displaystyle\left(([x_{0}:\dots:x_{n}],t),z\right)$ $\displaystyle\mapsto$
$\displaystyle([z^{w_{0}}x_{0}:\dots:z^{w_{n}}x_{n}],z^{-1}t).$

Let $p\in\mathfrak{X}$ be a generic point with coordinates
$p=([x_{0}:\dots:x_{n}],t)$, i.e. $t\not=0$. Then the $\mathbb{G}_{m}$-orbit
of $p$ is by dimension reasons a curve in $\mathfrak{X}$, denote it by
$\mathbb{G}_{m}\cdot p$. Notice that $\mathbb{G}_{m}\cdot p$ does not
intersect the special fiber $\mathfrak{X}_{0}$ defined by $t=0$. This is a
particular case of the situation described in §0.3 of the introduction.

######  2.9 Convention.

We will assume without loss of generality that the image of our valuation is
in $\mathbb{Z}^{d}$ (see [BG09, Theorem 2.29]). Replacing $\mathbb{Z}^{d}$ by
a sublattice, we assume $\nu(k(X)^{*})=\mathbb{Z}^{d}$.

Given a subsemigroup $S\subset\mathbb{N}\times\mathbb{Z}^{r}$, let

$\Delta_{\mathbb{Q}}(S)=\\{0\\}\cup\\{a/n|(n,a)\in
S,n>0\\}\subset\mathbb{Q}^{r}.$

Then $\Delta_{\mathbb{Q}}(S)$ is a convex set in $\mathbb{Q}^{r}$ by the
following known141414At least, to the workers in the theory of the
Newton–Okounkov bodies. argument: given finitely many rational numbers
$r_{i}/s_{i}\geq 0$ that sum up to $1$ and $(n_{i},a_{i})\in S,n_{i}>0$, we
want to show the convex combination

$\sum_{i}\frac{r_{i}}{s_{i}}\frac{a_{i}}{n_{i}}$

is in $\Delta_{\mathbb{Q}}(S).$ For that, replacing $r_{i}$’s and $a_{i}$’s by
some positive integer multiples, we can assume $s_{1}=s_{2}=\cdots=s$ and
$n_{1}=n_{2}=\dots=n$ for some $s,n$. Then
$(sn,\sum_{i}r_{i}a_{i})=\sum_{i}r_{i}(n,a_{i})\in S$ and so the claim
follows.

The next lemma can be compared to [And13, Proposition 4.1.].

######  2.10 Lemma (Criterion for finite generation).

The semigroup $S$ is finitely generated if and only if
$\Delta_{\mathbb{Q}}(S)$ is a convex polytope in $\mathbb{Q}^{r}$.

###### Proof.

$(\Leftarrow)$ Clear. $(\Rightarrow)$ Let $C_{\mathbb{Q}}(S)$ be the convex
cone generated by $1\times\Delta_{\mathbb{Q}}(S)$ in $\mathbb{Q}^{1+r}$. It is
a convex polyhedral cone by assumption. It follows that the saturation
$\widetilde{S}=C_{\mathbb{Q}}(S)\cap\mathbb{Z}^{1+r}$ of $S$ is finitely
generated; thus, $k[\widetilde{S}]$ is a finitely generated $k$-algebra. Then,
by the Artin–Tate lemma [Eis13, Exercise 4.32], $k[S]$ is a finitely generated
algebra; thus, $S$ is finitely generated. ∎

######  2.11 Remark.

As is known, it can happen the Newton–Okounkov body is a polytope while the
semigroup to which the the Newton–Okounkov body is associated is not finitely
generated (see e.g. [IW20, Example 2.4]). The above lemma thus suggests to us
that this fact is pathological and that the conventional definition of the
Newton–Okounkov body is somehow _wrong_ ; taking closure introduces boundary
points that admit no interpretation.

######  2.12 Example (elliptic curve; cf. [KMM22] Example 1.6.).

Let $k=\mathbb{C}$ and $X\subset\mathbb{P}^{2}$ the elliptic curve given by
the equation $f=y^{2}z-x^{3}+xz^{2}$ with the homogeneous coordinate ring $R$.
Let $P=[0:1:0]$ and $\operatorname{ord}_{P}$ denote the order-of-vanishing
along $P$; i.e., the valuation corresponding to the discrete valuation ring
that is the local ring at $P$. In the affine chart $\\{y\neq 0\\}$, if $x=0$,
then $z=0$; thus, $\operatorname{ord}_{P}(x/y)=1$. Similarly,
$\operatorname{ord}_{P}(z/y)=3$. We let $\mathbb{G}_{m}$ act on
$\mathbb{P}^{1}$ so that $x,y,z$ has weights $1,0,3$. We have $t\cdot
f=t^{3}(y^{2}z-x^{3}+t^{4}xz^{2}).$ Thus,

$f_{t}=t^{-3}(t\cdot f)=y^{2}z-x^{3}+t^{4}xz^{2}.$

Then $V(f_{t})\subset\mathbb{P}^{2}\times\mathbb{A}^{1}$ is a toric
degeneration of $X$ to
$X^{\prime}=V(f_{t})\cap\\{t=0\\}=V(y^{2}z-x^{3})\subset\mathbb{P}^{2}$.

In the above example, a choice of the point $P$ is important; choosing other
points will not result in a toric degeneration. This is explained by the next
well-known fact.

######  2.13 Proposition.

Let $X\subset\mathbb{P}^{q}$ be a projective curve (smooth or not) with
homogeneous coordinate ring $R$ and $P$ a smooth point on $X$. Let
$\mathbb{G}_{m}$ act on $\mathbb{P}^{q}$ in such a way the homogeneous
coordinates $y_{i},0\leq i\leq q$ have weights
$a_{i}=\operatorname{ord}_{P}(y_{i}/y_{j})$ where $j$ is such that $y_{j}$
does not vanish at $P$.

Then $\mathfrak{X}=\overline{\mathbb{G}_{m}\cdot X}$ in
$\mathbb{P}^{N}\times\mathbb{A}^{1}$ is a toric degeneration for some Veronese
embedding $\mathbb{P}^{q}\hookrightarrow\mathbb{P}^{N}$ if and only if $P$ is
the support of a hypersurface on $X$.151515In the language of the first
author’s thesis, such a $P$ is called a good divisor relative to
$\operatorname{Spec}R$.

In the first author’s thesis, the above proposition in a more general
formulation is proved using commutative algebra. Here, we give a short proof
following [And13, Example 4.2.].

###### Sketch of proof.

We write $\nu(f)=\operatorname{ord}_{P}(f/y_{j}^{\deg(f)})$ for nonzero
homogeneous elements $f$ of $R$ of positive degree.

($\Rightarrow$) By Lemma 2.10, $\Delta_{\mathbb{Q}}(S)$ is a convex polytope
in $\mathbb{Q}$; i.e., a closed interval $[0,m]\cap\mathbb{Q}$. Thus, we can
find a homogeneous element $f$ of positive degree such that
$\nu(f)/\deg(f)=m$. Note $m=\deg(X)=\deg(\mathcal{O}_{X}(1))$. Let $H=V(f)$.
Then $\deg(H)=\deg(\mathcal{O}_{X}(1))\deg(f)=m\deg(f)$. On the other hand,
$H$ determines the effective Weil divisor $H^{\prime}$ and we can write
$H^{\prime}=\nu(f)P+\cdots$. Since $\nu(f)=\deg(H)$ and $H^{\prime}$ is
effective, we have $H^{\prime}=\nu(f)P$; that is, $\operatorname{Supp}(H)=P$.

($\Leftarrow$) $P=\operatorname{Supp}(H)$ for some hypersurface $H=V(f)$; that
is, $H=nP$ for some integer $n>0$. As above, we have $\deg(H)=n=\nu(f)$. Since
$\deg(H)=m\deg(f)$, $m=\nu(f)/\deg(f)$. Since the closure of
$\Delta_{\mathbb{Q}}(S)$ in $\mathbb{R}$ is an interval $[0,m]$, we have
$\Delta_{\mathbb{Q}}(S)=[0,m]\cap\mathbb{Q}$; i.e., $\Delta_{\mathbb{Q}}(S)$
is a polytope; hence, by Lemma 2.10, $S$ is finitely generated. ∎

######  2.14 Corollary.

A projective curve admits an abstract toric degeneration that is projective
(i.e., it can be embedded into $\mathbb{P}^{N}\times\mathbb{A}^{1}$ for some
$N$ in such a way that the structure map is the projection.)

###### Proof.

Let $P$ be a smooth point on $X$, which is an effective Cartier divisor on
$X$. Then for some integer $m>0$, the effective Cartier divisor $mP$ is very
ample161616If $X$ is normal, this is [Har77, Ch. IV, Corollary 3.2.]. If $X$
is not normal, the pre-image of $P$ under the normalization map is again a
point. Now use the fact that an invertible sheaf is ample if and only if its
pre-image under a finite surjective map is ample ([Har77, Ch. III, Exercise
5.7. (d)]). and thus we have a closed embedding
$X\hookrightarrow\mathbb{P}^{N}$ so that $\mathcal{O}_{X}(mP)$ is the pull-
back of $\mathcal{O}_{\mathbb{P}^{N}}(1)$. Then $P$ is the support of some
hypersurface $H$ in
$|\mathcal{O}_{X}(mP)|=\mathbb{P}(\Gamma(X,\mathcal{O}_{X}(mP)))$. ∎

Proposition 2.13 is a useful tool to furnish a counterexample to the existence
of a toric degeneration (preserving a power of a polarization). Precisely,
when the base field is _uncountable_ and $X$ is a smooth projective curve of
genus at least two, an argument with the Jacobian of $X$ shows that there is a
very ample divisor on $X$ such that, when $X$ is embedded into a projective
space using this very ample divisor, there is no point $P$ satisfying the
equivalent conditions of Proposition 2.13.

######  2.15 Remark.

We have recently learned that, for the purpose of constructing a toric
degeneration, the base field can be assumed to be countable (since a variety
is defined by finitely many polynomials; in particular, it is defined over a
countable subfield). So, there appears a loophole to the above accepted
argument for a counterexample.

On the question of the existence of a toric degeneration (with some control of
an ambient space), it is known that the curve case is the essential case.
Given Corollary 2.14, it is therefore reasonable to pose:

######  2.16 Problem.

Construct an abstract toric degeneration of a projective variety (if possible,
a degeneration that is projective).

###  2.4. A map to the special fiber

######  2.17 Example.

Let $X=V(y^{2}z=x^{3}-xz^{2})\subset\mathbb{P}^{2}$ be an elliptic curve.
Recall that there is a toric degeneration from $X$ to
$X^{\prime}=\operatorname{Proj}(k[S])\subset\mathbb{P}^{2}$, where
$S\subset\mathbb{N}\times\mathbb{Z}$ is generated by
$(1,0)=\widehat{\nu}(y),(1,1)=\widehat{\nu}(x),(1,3)=\widehat{\nu}(z)$.

We have $a_{0},a_{1},a_{2}=0,1,3,\,\Delta(1)=[0,3]$. Take
$h=y,u_{0}=y,u_{2}=z,m_{\varphi}=3$; so that $\nu(u_{2}/h)=3(a_{2}-a_{0})/3$.
Then $g_{0}=y^{3},g_{1}=y^{2}z,g_{2}=z^{3}$. Note $k[g_{0},g_{1},g_{2}]$ is
the semigroup algebra of the semigroup $3S$.

Explicitly, the map $\varphi$ is given as
$[x:y:z]\mapsto[y^{3}:y^{2}z:z^{3}].$

First, we note the following, which uses the key part of the proof of the
Noether normalization lemma.

######  2.18 Lemma.

Let $x_{1},\dots,x_{m}$ be homogeneous elements of $R$ of the same degree such
that $\nu(x_{i})<\nu(x_{m})$ for every $1\leq i<m$. Given constants
$\lambda_{i}\in k$, let $y_{i}=x_{i}-\lambda_{i}x_{m},1\leq i\leq m-1$. Then
$\nu(x_{i})=\nu(y_{i})$ and, if a choice of $\lambda_{i}$’s is general and
$m>d$, then $x_{m}$ is integral over $k[y_{1},\dots,y_{m-1}]$.

###### Proof.

Since $\nu(x_{m})>\nu(x_{i})$, we have $\nu(y_{i})=\nu(x_{i})$. Since
$x_{i}$’s are not algebraically independent over $k$, we can find a nonzero
polynomial $f$ in $m$ variables with coefficients in $k$ such that

$f(x_{1},\dots,x_{m})=0.$

It then implies

$f(y_{1}+\lambda_{1}x_{m},\dots,y_{m-1}+\lambda_{m-1}x_{m},x_{m})=0.$

If $\lambda_{i}$ are general, we see the leading term of the above equation is
some nonzero constant times some power of $x_{m}$; that is, $x_{m}$ satisfies
a monic polynomial over $k[y_{1},\dots,y_{m}]$. ∎

######  2.19 Proposition.

Relabeling $a_{i}$’s, assume $\rho_{0}=\\{1,2,\dots,d\\}$. For simplicity,
also assume $a_{0}=0$ and $a_{j}>a_{i}$ for each $0\leq i\leq d$ and $d+1\leq
j\leq q$. Then there exist homogeneous elements $u_{0},\dots,u_{d}$ of $R$ of
degree one such that $\nu(u_{i})=a_{i},\,0\leq i\leq d$, which implies $u_{i}$
are algebraically independent, and $R$ is finite over $k[u_{0},\dots,u_{d}]$.

###### Proof.

Let $x_{0},\dots,x_{q}$ be homogeneous elements of $R$ of degree one such that
$\nu(x_{i})=a_{i}$. We shall argue by induction on $q$. If $q=d$, then $R$ is
an $k$-algebra generated by $d+1$ elements when $d+1$ is the Krull dimension
of $R$; thus, $R$ is a polynomial ring and the assertion is clear.

If $q>d$, by the preceding lemma, we can find
$y_{i}=x_{i}-\lambda_{i}x_{q},\,0\leq i\leq q-1$ such that $R$ is finite over
$R^{\prime}=k[y_{0},\dots,y_{q-1}]$. By inductive hypothesis applied to
$R^{\prime}$, we have that $R^{\prime}_{1}$ contains $u_{0},\dots,u_{d}$ such
that $\nu(u_{i})=a_{i}$ and $R^{\prime}$ is finite over
$k[u_{0},\dots,u_{d}]$. ∎

######  2.20 Corollary.

Let $h=u_{0}$. Then

$\varphi:X_{h}\to X^{\prime}_{\chi^{(1,0)}}$

is finite.

###### Proof.

Since $R_{+}$ is the radical of the ideal generated by
$u_{0},u_{i},i\in\rho_{0}$, by (iii) of the graded Nakayama lemma (Lemma
1.11), $R$ is finite over $k[u_{i}\mid i\in\rho_{0}\cup\\{0\\}]$. Also, as we
noted at the end of Proposition 1.14, we have that $k[u_{i}/u_{0}\mid
i\in\rho_{0}]$ is finite over $R[u_{0}^{-1}]_{0}$. ∎

The following problem seems interesting:

######  2.21 Problem.

Given a morphism $\varphi:X\to W$, identify the pre-image $\varphi^{-1}(H)$ of
a hyperplace $H$ on $W$. Is it irreducible for a general $H$? (the Bertini-
type theorem).171717Can we give sense to the notion of “general”? Is it
possible that $\varphi^{-1}(H)$ degenerates to $H$ for some $H$?

###  2.5. Description in terms of Gröbner bases

In this section we treat our main question from an algebraic point of view and
show how to construct a map to a toric variety given a Gröbner toric
degeneration. Let us reformulate the question.

######  2.22 Problem.

Given a positively graded algebra and domain $R$ with a full-rank valuation
$\widehat{\nu}:R\setminus\\{0\\}\to\mathbb{N}\times\mathbb{Z}^{d}$ whose value
semigroup $S$ is finitely generated, can we always find an embedding
$k[S]\hookrightarrow R$ such that the image is generated by standard
monomials?

To approach this question, as in the proof of Theorem 2.7 we fix a (graded)
presentation of $R$ given by a finite Khovanskii basis:

$R\cong k[x_{0},\dots,x_{n}]/J,$

such that there exists a maximal prime cone $\tau\in\mathcal{T}(I)$ in the
tropicalization of $J$ such that

$k[x_{0}\dots,x_{n}]/\operatorname{in}_{\tau}(J)\cong k[S].$

As $J$ is homogeneous with respect to a positive grading by [BMN21, Lemma 2.9]
we may choose the valuation such that $S\subset\mathbb{Z}_{\geq 0}^{d}$.

######  2.23 Theorem.

There exists a maximal cone $C$ in the Gröbner fan of $J$ with standard
monomial basis $\mathbb{B}_{C}=\\{\bar{\it x}^{m}\in R:{\it
x}^{m}\not\in\operatorname{in}_{C}(I)\\}$ and an embedding

(4) $\displaystyle\eta:k[S]\hookrightarrow R,\quad\text{such
that}\quad\eta(\nu(b_{i}))\in\mathbb{B}_{C}\ \forall i$

and $\eta$ sends a homogeneous element of degree $m$ in $k[S]$ to a
homogeneous element of degree $Nm$ in $R$ for some $N\in\mathbb{N}$.

Notice that $k[S]$ and $R$ have the same Hilbert polynomial, so in particular
$\dim k[S]_{m}=\dim R_{m}$. Let $R^{\prime}$ be the image of $k[S]$ under
$\eta$. Regarding the graded pieces of $R^{\prime}$ we observe for all $m>0$:

(5) $\dim R_{m}=\dim k[S]_{m}=\dim R^{\prime}_{Nm}$

###### Proof.

Among the elements of the Khovanskii basis $b_{0},\dots,b_{n}$ we choose
$b_{i_{1}},\dots,b_{i_{d}}$ such that

  1. (i)

$\bar{x}_{i_{1}},\dots,\bar{x}_{i_{d}}\in k[S]\cong
k[x_{0},\dots,x_{n}]/\operatorname{in}_{\tau}(J)$ form a maximal algebraically
independent set, and

  2. (ii)

$R\cong k[x_{0},\dots,x_{n}]/J$ is integral over
$k[\bar{x}_{i_{1}},\dots,\bar{x}_{i_{d}}]$.

After a possibly necessary linear change of coordinates we may assume without
loss of generality that $b_{0}=\bar{x}_{0},\dots,b_{d}=\bar{x}_{d}$. In
particular, by (i) there exists no elements
$\operatorname{in}_{\tau}(f)\in\operatorname{in}_{\tau}(J)\cap
k[x_{0},\dots,x_{d}]$ and moreover no element $f\in J\cap
k[x_{0},\dots,x_{d}]$.

Let $<$ be the lexicographic term order on $k[x_{0},\dots,x_{n}]$ determined
by $x_{n}>\dots>x_{0}$. By the above assumptions on $x_{0},\dots,x_{d}$ we
have for all $f\in J$

$\operatorname{in}_{<}(\operatorname{in}_{\tau}(f))\not\in
k[x_{0},\dots,x_{d}].$

Let $C\in\operatorname{GF}(J)$ be the maximal cone associated with the term
order $<_{\tau}$. Then all monomials $x_{0}^{a_{0}}\cdots x_{d}^{a_{d}}$ are
standard monomials for
$\operatorname{in}_{C}(J)=\operatorname{in}_{<_{\tau}}(J)$.

We may assume that the semigroup $S\subset\mathbb{N}\times\mathbb{Z}^{d}$ is
graded with degree captured by the first component. Moreover, without loss of
generality, we assume $R$ and hence $S$ are generated in degree one (otherwise
we use a suitable Veronose embedding). To obtain an embedding $\eta:k[S]\to R$
which sends homogeneous prime ideals to homogeneous prime ideals we first
embed $S$ as a graded subsemigroup into the positive orthant with _total
degree_. Let $\widehat{\nu}(b_{i})=(1,a_{i1},\dots,a_{id})\in S$ be the
generators of $S$ for $1\leq i\leq n$ and set
$N:=\max\\{a_{i1}+\dots+a_{id}:0\leq i\leq n,1\leq j\leq d\\}$. We define

$\iota:S\hookrightarrow\mathbb{Z}^{d+1}_{\geq
0},\quad\text{by}\quad(1,a_{i1},\dots,a_{id})\mapsto(N,a_{i1},\dots,a_{id}).$

Notice that each element $(N,a_{i1},\dots,a_{id})$ can be written uniquely as

(6)
$\displaystyle(N,a_{i1},\dots,a_{id})=c_{i0}e_{0}+\sum_{j=1}^{d}c_{ij}(e_{0}+e_{j})\quad\text{with}\quad\sum_{j=0}^{d}c_{ij}=N,$

where $\\{e_{0},\dots,e_{d}\\}$ is the standard basis for $\mathbb{Z}^{d+1}$.
So $\iota(S)$ is a graded subsemigroup inside the semigroup of lattice points
inside the polyhedral simplicial cone $C(S)$ with rays
$\\{e_{0},e_{0}+e_{1},\dots,e_{0}+e_{d}\\}$. In particular, $\iota(S)$ is
graded by total degree by Equation (6). Lastly, we apply a unimodular map that
sends the simplicial cone to the positive orthant spanned by
$e_{0},\dots,e_{d}$. Using Equation (6) this yields the following graded
embedding of semigroups

(7) $\displaystyle\iota:S\hookrightarrow\mathbb{Z}^{d+1}_{\geq
0},\quad\text{by}\quad(1,a_{i1},\dots,a_{id})\mapsto(c_{i0},\dots,c_{id}).$

With respect to the standard basis of $\mathbb{Z}^{d}$ the map $\iota$ is
given by the matrix

$\left(\begin{matrix}N&-1&\dots&-1\\\ 0&1&\dots&0\\\ \vdots&&\ddots&\vdots\\\
0&0&\dots&1\end{matrix}\right)$

We use $\iota$ to define a homomorphism of algebras $\eta:k[S]\to R$ given by

$\widehat{\nu}(b_{i})\mapsto\bar{x}_{0}^{c_{i0}}\cdots\bar{x}_{d}^{c_{id}}.$

Notice that $\eta$ sends the vector space basis $S$ of $k[S]$ to elements of
the standard monomial basis $\mathbb{B}_{C}$. Hence, $\eta$ has trivial kernel
and is an embedding of algebras. ∎

We obtain the following result (compare to Proposition 1.14):

######  2.24 Corollary.

The embedding $\eta:k[S]\to R$ from Equation (4) induces a dominant map

$\text{\rm Proj}(R)\setminus V(R_{+}^{\prime}R)\to\text{\rm Proj}(S),$

where $R^{\prime}=\eta(k[S])\subset R$.

######  2.25 Example.

We illustrate Theorem 2.23 in our running example: the toric degeneration
$\mathfrak{X}=V(y^{2}z-x^{3}-txz^{2})\subset\mathbb{P}_{x:y:z}^{2}\times\mathbb{A}_{t}^{1}$
of the elliptic curve $X=V(y^{2}z-x^{3}-xz^{2})$ to the toric variety
$X^{\prime}=V(y^{2}z-x^{3})$. Algebraically, as described in Example 2.17, the
map $\varphi:X\to X^{\prime}$ corresponds to an embedding of the semigroup
algebra $R_{0}=k[x,y,z]/(y^{2}z-x^{3})$ into
$R=k[x,y,z]/(y^{2}z-x^{3}-xz^{2})$.

The Gröbner fan of $I=(x^{3}+xz^{2}-y^{2}z)\subset k[x,y,z]$ has three maximal
cones, so we have three standard monomial bases for $R=k[x,y,z]/I$. Let $\tau$
be the cone in the tropicalization of $I$ corresponding to our degeneration.
So, $\text{in}_{\tau}(x^{3}+xz^{2}-y^{2}z)=(x^{3}-y^{2}z)$ and there are two
adjacent maximal cones in the Gröbner fan that $\tau$ can be refined to.
Pictorially we have

$(y^{2}z)$$(x^{3})$$(x^{3}-y^{2}z)$

Consider $R_{0}=k[x,y,z]/(x^{3}-yz^{2})$ then for every choice of maximal
algebraically independent variables, monomials in these variables are standard
in either $\mathbb{B}_{(x^{3})}$ _or_ $\mathbb{B}_{(yz^{2})}$. More precisely:
the monomials of form $x^{a}y^{b}$ are standard in $\mathbb{B}_{(y^{2}z)}$ as
$c<1$, the monomials $x^{a}z^{c}$ are standard in $\mathbb{B}_{(y^{2}z)}$ as
$b<2$, and the monomials $y^{b}z^{c}$ are standard in $\mathbb{B}_{(x^{3})}$
as $a<3$. In particular, the conditions (i) and (ii) of the Proof of Theorem
2.23 are always satisfied.

Recall that $S=\langle(1,0),(1,1),(1,3)\rangle_{\geq 0}$ is graded by
$\deg(a,b)=a$, but in $R$ monomials $y^{a}z^{b}$ have degree $a+b$. Se we
embed $S$ into a semigroup $S^{\prime}$ graded by total degree corresponding
to the map $\iota$ in the proof (below x denotes a point in the semigroup
while
${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\rm
o}$ denotes a _missing_ point):

$S$1123xxxxo$\left(\begin{smallmatrix}3&1\\\
0&1\end{smallmatrix}\right)$123123xxxxoooooo$\left(\begin{smallmatrix}1&-1\\\
0&1\end{smallmatrix}\right)$123123xxxxoooooo$S^{\prime}$$y$$z$

Recall that $y^{a}z^{b}\in R$ are standard monomials. So we may embed

$k[S]\hookrightarrow R\quad\text{by}\quad(1,0)\mapsto y^{3},\quad(1,1)\mapsto
y^{2}z\quad\text{and}\quad(1,3)\mapsto z^{3}.$

which gives $\mathbb{P}^{2}\supset X\twoheadrightarrow
X_{0}\subset\mathbb{P}^{2}\quad\text{ by
}\quad[x:y:z]\mapsto[y^{2}z:y^{3}:z^{3}]$.

The results of this section naturally lead to an interesting problem:

######  2.26 Problem.

Given an injective ring homomorphism $k[S]\hookrightarrow R$ that is graded of
some degree, under which assumptions is it possible to extend the built-in
valuation on $k[S]$ to a valuation $\nu$ on $R$ so that $S$ is the image of
$R-0$ under $\widehat{\nu}=(\deg,\nu)$?

##  3\. Degeneration by projection

We first recall how to construct a degeneration by projection ([Har77][Ch.
III, Example 9.8.3.]). Given a projective space $\mathbb{P}^{N}$ with
homogeneous coordinates $x_{0},\dots,x_{N}$, we have a projection, for some
$N^{\prime}\leq N$,

$p:\mathbb{P}^{N}-V(x_{0},\dots,x_{N^{\prime}})\to\mathbb{P}^{N^{\prime}}\quad\text{given
by}\quad[c_{0}:\cdots:c_{N}]\mapsto[c_{0}:\cdots:c_{N^{\prime}}].$

We note that $p$ is the map corresponding to a choice of sections
$x_{0},\dots,x_{N^{\prime}}$ of $\mathcal{O}_{\mathbb{P}^{N}}(1)$ and we say
$p$ is the the projection given by $x_{0},\dots,x_{N^{\prime}}$. The locus
$V(x_{0},\dots,x_{N^{\prime}})$ is called the center of the projection $p$.
Algebraically, the above morphism is induced by taking Proj of the ring
homomorphism

$S^{\prime}=k[x_{0},\dots,x_{N^{\prime}}]\hookrightarrow
S=k[x_{0},\dots,x_{N}].$

In this notation, $V(S^{\prime}_{+}S)=V(x_{0},\dots,x_{N^{\prime}})$. Also,
note that we can identify $\mathbb{P}^{N^{\prime}}$ as a closed subvariety
$V(x_{N^{\prime}+1},\dots,x_{N})\subset\mathbb{P}^{N}$, which corresponds to
viewing $S^{\prime}$ as a quotient of $S$.

The next theorem states a fairly classical fact that the projection $p$
induces a degeneration. But the statement below is more precise than those we
are aware of.

######  3.1 Theorem (Degeneration by projection).

Let $\mathbb{G}_{m}$ act linearly on $\mathbb{P}^{N}$ in such a way
$x_{0},\dots,x_{N^{\prime}}$ have weights $0$ and
$x_{N^{\prime}+1},\dots,x_{N}$ have weights $-1$. Let $B=X\cap
V(S^{\prime}_{+}S)$.

Then for each closed subvariety $X\subset\mathbb{P}^{N}$, we have:

$\mathfrak{X}=\overline{\mathbb{G}_{m}\cdot
X}\subset\mathbb{P}^{N}\times\mathbb{A}^{1}$

is a flat degeneration of $X$ such that, for the limit $X^{\prime}$,

  1. (i)

$X^{\prime}\cap\mathbb{P}^{N^{\prime}}=\overline{p(X-B)}$ as a scheme; thus,
as a set, $X^{\prime}$ is the union of irreducible components of $X^{\prime}$
not lying in $\mathbb{P}^{N^{\prime}}$ and $\overline{p(X-B)}$,

  2. (ii)

if $B=\emptyset$, then $X^{\prime}=p(X)$ as a set and
$X^{\prime}\cap\mathbb{P}^{N^{\prime}}=p(X)$ as a scheme.

###### Proof.

That $\mathfrak{X}$ is a flat degeneration is a special case of a Gröbner
degeneration described, for example, in [Eis13, 15.10.5.]. Alternatively, the
same reasoning as in [Har77][Ch. III, Example 9.8.3.] shows that
$\mathfrak{X}$ is a flat degeneration.

Now, let $I\subset S$ be the ideal defining $X\subset\mathbb{P}^{N}$ and
$I^{\prime}\subset S$ the ideal defining the limit
$X^{\prime}\subset\mathbb{P}^{N}$.181818This ideal $I^{\prime}$ is analogous
to what Eisenbud calls $I_{\infty}$ in [Eis13, 15.10.5.]. Also, let
$\mathfrak{a}\subset S$ denote the defining ideal of
$\mathbb{P}^{N^{\prime}}\subset\mathbb{P}^{N}$; i.e., the kernel of $S\to
S^{\prime}$. Note $I^{\prime}\cdot
S^{\prime}=(I^{\prime}+\mathfrak{a})/\mathfrak{a}$ when we view $S^{\prime}$
as an $S$-module via the quotient map $S\to S^{\prime}$.

Given a nonzero homogeneous element $f$ in $S$, let $n(f)\geq 0$ be the
largest integer such that

$f\in f^{\prime}+\mathfrak{a}^{n(f)-1}+\cdots+S^{\prime}$

for some nonzero $f^{\prime}\in\mathfrak{a}^{n(f)}$. Then $t\cdot f\in
t^{-n(f)}f^{\prime}+(t^{-1}\mathfrak{a})^{n(f)-1}+\cdots+S^{\prime}$. Thus,
among the defining equations of $\mathfrak{X}$, $f_{t}:=t^{n(f)}(t\cdot f)$ is
in $f^{\prime}+tS[t]$. Then $I^{\prime}$ is generated by $f^{\prime}$ for each
$f$ in $I$. Note that if $n(f)=0$, then $f\in S^{\prime}$ and $f_{t}=f$; thus
$f^{\prime}=f$.

Then we shall now show:

$I^{\prime}\cdot S^{\prime}=I\cap S^{\prime}.$

It is not necessarily the case that $I^{\prime}$ is generated by $f^{\prime}$
for some homogeneous generators $f$ of $I$, but it is still the case that
$I^{\prime}$ is generated by $f^{\prime}$ for all homogeneous $f$ in $I$.
Thus, let $f\in I$ be a homogeneous element. If $n(f)>0$, then $f^{\prime}$ is
in $\mathfrak{a}^{n(f)}\subset\mathfrak{a}$. If $n(f)=0$, then, as noted
early, $f\in S^{\prime}$ and $f^{\prime}=f$. Thus, in any case,
$f^{\prime}\mod\mathfrak{a}$ is in $I\cap S^{\prime}$, proving the inclusion
$\subset$. Conversely, if $f\in I\cap S^{\prime}$, then $f=f^{\prime}$ and so
$f\in I^{\prime}$; thus, $f\in I^{\prime}\cdot S^{\prime}$, proving the claim.
Since $\overline{p(X-B)}=V(I\cap S^{\prime})$, the claim then implies (i).

Next, to prove (ii), assume $B=\emptyset$. Note that $p(X)$ is then closed
since $X$ is projective. We shall show $I^{\prime}$ contains some power of
$\mathfrak{a}$; i.e., $V(\sqrt{I^{\prime}})\subset\mathbb{P}^{N^{\prime}}$.
The assumption $B=\emptyset$ means that $\sqrt{I+S_{+}^{\prime}S}=S_{+}$.
Thus, for each $j$ such that $N^{\prime}+1\leq j\leq N$, we can write
$x_{j}^{n}=f+g$ where $n>0$, $f\in I$ and $g\in S^{\prime}_{+}S$ such that
$f,g$ are homogeneous of degree $n$. Put in another way, we have
$f=x_{j}^{n}-g.$ We note that $g$ cannot contain a term from
$\mathfrak{a}^{n}$, since $g$ has degree $n$ and $g\in S_{+}^{\prime}S$.
Hence, $f^{\prime}=x_{j}^{n}$ and thus $x_{j}^{n}\in I^{\prime}$. It follows
that $I^{\prime}$ contains some power of $\mathfrak{a}$. ∎

######  3.2 Corollary.

$X^{\prime}\cap\mathbb{P}^{N^{\prime}}$ is reduced.

###### Proof.

The defining ideal of $\overline{p(X-B)}$ is $I\cap S^{\prime}$, which is
prime; in particular, radical. ∎

######  3.3 Corollary.

If the dimension of an irreducible component of $X^{\prime}$ lying in
$\mathbb{P}^{N^{\prime}}$ is $<\dim X$, then $X^{\prime}$ coincides with
$\overline{p(X-B)}$ as a set (cf. Example 3.10.)

###### Proof.

This is because $\dim X^{\prime}=\dim X$. ∎

######  3.4 Corollary.

If $X\neq X^{\prime}$, then $X^{\prime}$ cannot be a variety (i.e., reduced
and irreducible).

###### Proof.

Indeed, $S^{\prime}\hookrightarrow S$ induces the grade-preserving ring
homomorphism $S^{\prime}/(I\cap S^{\prime})\hookrightarrow S/I$ where the two
rings are the homogeneous coordinate rings of $\overline{p(X-B)}$ and $X$,
respectively. If $X^{\prime}$ is a variety, then
$X^{\prime}=\overline{p(X-B)}$ as closed subschemes of
$\mathbb{P}^{N^{\prime}}$; thus as closed subschemes of $\mathbb{P}^{N}$. But,
as closed subschemes of $\mathbb{P}^{N}$, $X^{\prime}$ and $X$ have the same
Hilbert function (Lemma 2.4); hence, $S^{\prime}/(I\cap S^{\prime})=S/I$. ∎

######  3.5 Remark (GIT interpretation).

In the setup of the proposition, we have $S^{\prime}=S^{\mathbb{G}_{m}}$.
Thus, $\mathbb{P}^{N,ss}=\mathbb{P}^{N}-V(S^{\prime}_{+}S)$ is the semistable
locus with respect to the $\mathbb{G}_{m}$-action and the projection
$p:\mathbb{P}^{N,ss}\to\mathbb{P}^{N^{\prime}}$ is the GIT quotient map. Since
$X$ is not $\mathbb{G}_{m}$-invariant, the restriction of $p$ to $X$ is not a
GIT quotient map; in particular, $p(X-B)$ may not be closed. (But it would be
interesting if geometric invariant theory can be applied in some fashion to
the setup here.)

######  3.6 Remark (Geometric vertex decomposition).

Essentially the same result to above Theorem 3.1 in the affine setup appears
in §1.2. of [Knu07], which originally appeared in [KMY07].

To understand their method (called the geometric vertex decomposition), it is
helpful to have the following simple example in mind: let
$X=V(xy-1)\subset\mathbb{A}^{2}$. Then $V(xy-t)\subset\mathbb{A}^{3}$ is a
flat degeneration from $X$ to $X^{\prime}=V(xy)$. Then $X^{\prime}$ is the
union of the projection part $V(y)$, the closure of the projection of $X$ to
$V(y)$ and the cone part $V(x)$, the part that remains intact under the
degeneration (see Example 1.1. of [KMY07]). The observation of the authors of
[KMY07] is that this decomposition into the projection part and the cone part
generalizes to the limit of a degeneration by projection. Moreover, as they
observe, it is in fact the analog of the vertex decompsition of a simplicial
complex, the fact that a simplicial complex $\Delta$ is the pushout

$\delta\cup_{\lambda}\sigma$

of two subcomplexes, the one $\delta$ consisting of all the faces not
containing a given vertex (called the deletion of the vertex) and the other
$\sigma$ consisting of all the faces containing the given vertex (called the
star or cone), along the link $\lambda=\delta\cap\sigma$.

It is unclear to us how to formulate the preceding theorem in a way analogous
to the geometric vertex decomposition, which is affine in nature. However, it
can still be applied to open affine charts of $X$ as follows (alternatively,
it can be applied to the affine cone of $X$, which is straightforward and we
omit.)

######  3.7 Proposition.

In the setup of the preceding theorem and the notations in the proof, let
$X^{\prime}_{c}=V((I^{\prime}:\mathfrak{a}^{\infty}))$ where
$(I^{\prime}:\mathfrak{a}^{\infty})=\\{x\in S|\mathfrak{a}^{n}x\in
I^{\prime},n\gg 0\\}$ and we call it the cone part of $X^{\prime}$. Then, as a
set, $X^{\prime}_{c}$ is the same as the set of irreducible components of
$X^{\prime}$ not lying in $\mathbb{P}^{N^{\prime}}$ and thus, as a set,
$X^{\prime}$ is the union of $X^{\prime}_{c}$ and $\overline{p(X-B)}$.

Next assume $N=N^{\prime}+1$. Let $X_{i}=X\cap\\{x_{i}\neq 0\\}$ be the open
affine charts of $X$, which lie in $\mathbb{A}^{N}$. Let $\overline{X_{i}}$ be
the closure of $X_{i}$ in $\mathbb{A}^{N-1}\times\mathbb{P}^{1}$. Then
$X^{\prime}_{c}\cap\\{x_{i}\neq 0\\}$ is the fiber of
$\overline{X_{i}}\to\mathbb{P}^{1}$ over $\infty$ times $\mathbb{A}^{1}$ for
$i<N$.

###### Proof.

Let $I^{\prime}=\cap_{i}Q_{i}$ be a primary decomposition. Then it is easy to
see that $(I^{\prime}:\mathfrak{a}^{\infty})$ is the intersection of all
$Q_{i}$’s such that $\sqrt{Q_{i}}\not\supset\mathfrak{a}$. From this, the
first assertion follows.

To see the second assertion, replacing $X$ by $X_{i}$, we assume
$X\subset\mathbb{A}^{N}$. Given a polynomial $f$ in $S$, we write
$\widetilde{f}$ for the homogenization of $f$ with respect to variable $x_{N}$
using an extra variable $t$; so $\widetilde{f}$ is homogeneous in $x_{N},t$.
Then we can write $\widetilde{f}\in x_{N}^{n}\widetilde{f}_{0}+(t)$. Thus, the
fiber of $\widetilde{X_{i}}\to\mathbb{P}^{1}$ over $\infty$ is given by
$\widetilde{f}_{0}$ for each $f\in I$. Since $x_{N}^{n}\widetilde{f}_{0}$ is
in $I^{\prime}$, we have
$\widetilde{f}_{0}\in(I^{\prime}:\mathfrak{a}^{\infty})$. Conversely, if $g$
is $(I^{\prime}:\mathfrak{a}^{\infty})$, then $x_{N}^{n}g\in I^{\prime}$ and
then $x_{N}^{n}g=f_{1}^{\prime}+\dots+f_{n}^{\prime}$ for some $f_{i}$’s in
$I$ where $f_{i}^{\prime}$ are defined as in the proof of the previous
theorem. Hence, $g$ is in the defining ideal of the fiber of
$\overline{X_{i}}\to\mathbb{P}^{1}$ over $\infty$.

∎

The above proposition is perhaps more transparent if we look at the defining
equations of $\mathfrak{X}$; namely,

######  3.8 Corollary.

Suppose $\mathfrak{X}$ is given by the equations $x_{N}\,g_{i}+t\,h_{i}$ where
$g_{i}\in S$ and $h_{i}\in S[t]$. Then $X^{\prime}_{c}$ is given by $g_{i}$’s.

###### Proof.

This is essentially the same as what we proved at the end of the proof of the
previous proposition. ∎

We now look at concrete examples.

######  3.9 Example (cf. Example 1.1. in [KMY07]).

Let $X=V(I)\subset\mathbb{P}^{3},I=(xy-z^{2})$ and consider the projection
$p:\mathbb{P}^{3}-P\to\mathbb{P}^{2}$, where $P=[0:1:0]$, given by
$S^{\prime}=k[x,z]\hookrightarrow S$. Then $I\cap S^{\prime}=0$; i.e.,
$\overline{p(X-P)}=V(y)=\mathbb{P}^{1}$.

On the other hand, the degeneration $\mathfrak{X}$ is given by the equation
$xy-tz^{2}$ and thus $X^{\prime}=V(I^{\prime}),I^{\prime}=(xy)$. We have
$(I^{\prime}:y^{\infty})=(x)$. Hence, $X^{\prime}$ is the union of
$V((I^{\prime}:y^{\infty}))$ and $\overline{p(X-P)}$, not just as sets but as
schemes in this particular case.

######  3.10 Example (twisted cubic curve; cf. Example 9.8.3. and Exercise
9.2. in [Har77] Ch. III.).

Let $\mathbb{P}^{1}\hookrightarrow\mathbb{P}^{3}$ be the Veronese embedding;
i.e., $\mathbb{P}^{3}$ has homogeneous coordinates parametrized by monomials
of degree $3$ in $x,y$, the homogeneous coordinates on $\mathbb{P}^{1}$ and
the embedding is given by:

$k[u_{3},u_{2},u_{1},u_{0}]\to k[x,y],\,u_{3}\mapsto x^{3},u_{2}\mapsto
x^{2}y,\dots.$

where we wrote $u_{a}=u_{x^{a}y^{3-a}}$. The image of $\mathbb{P}^{1}$ in
$\mathbb{P}^{3}$ is defined by the kernel $I$ of this ring homomorphism;
explicitly, $I$ is generated by three quadratic polynomials
$u_{2}^{2}-u_{3}u_{1},u_{1}^{2}-u_{2}u_{0},u_{2}u_{1}-u_{3}u_{0}$.

Let $X$ denote the image of $\mathbb{P}^{1}$ in $\mathbb{P}^{3}$, called a
rational normal curve or a twisted cubic curve. Let

$p:\mathbb{P}^{3}-P\to\mathbb{P}^{2}=V(u_{1}),$

where $P=V(u_{3},u_{2},u_{0})$, be the projection given by forgetting $u_{1}$.
We have $u_{2}^{3}-u_{3}^{2}u_{0}\in I\cap S^{\prime}$. Thus, $p(X)$ is
$V(u_{2}^{3}-u_{3}^{2}u_{0})$, a cuspidal cubic curve. On the other hand,
perhaps by computing a Gröbner basis, we see that the defining equations of
the degeneration $\mathfrak{X}$ are

$tu_{2}^{2}=u_{3}u_{1},u_{1}^{2}=t^{2}u_{2}u_{0},u_{2}u_{1}=tu_{3}u_{0},u_{2}^{3}=u_{3}^{2}u_{0}.$

Thus, $I^{\prime}$ is generated by
$u_{3}u_{1},u_{1}^{2},u_{2}u_{1},u_{2}^{3}-u_{3}^{2}u_{0}$. Since
$\mathfrak{a}=(u_{1})$, we have $I^{\prime}\cdot
S^{\prime}=(u_{2}^{3}-u_{3}^{2}u_{0})\cdot S^{\prime}=I\cap S^{\prime}$; so
the equality in the claim in the proof of the proposition holds.

Note that $X=\mathbb{P}^{1}$ has arithmetic genus zero, while the cuspidal
cubic curve has arithmetic genus 1. Since $X,X^{\prime}$ share the same
arithmetic genus, $X^{\prime}$ cannot be reduced, just as [Har77, Ch. III.,
Example 9.8.4.] notes for a similar example. Since $u_{1}^{2}\in I^{\prime}$,
we have $1\in(I^{\prime}:\mathfrak{a}^{\infty})$; i.e., $X^{\prime}_{c}$ is
empty. However, $X^{\prime}$ does not agree with $p(X-B)$.

Now, given a projective variety $X$, suppose we are given a toric degeneration
from $X$ to $X^{\prime}$ embedded in $\mathbb{P}^{n}\times\mathbb{A}^{1}$.
Then, by the method in §2.5 or Proposition 1.14 (when it applies), we find a
morphism $\varphi:X-B\to X^{\prime}$. Now, by construction, $\varphi$ is given
by sections $g_{0},\dots,g_{q}$ of $\mathcal{O}_{X}(m)$, some $m>0$, such that
each $g_{i}$ is a monomial in homogeneous coordinates on $\mathbb{P}^{n}$
restricted to $X$. We recall that the $m$-th Veronese embedding
$\mathbb{P}^{n}\hookrightarrow\mathbb{P}^{N},N=\binom{n+m}{m}-1$ is given by
all the monomials of degree $m$ in homogeneous coordinates on
$\mathbb{P}^{n}$. Hence, $\varphi$ factors as the $m$-th Veronese embedding
restricted to $X-B$ followed by a projection.

######  3.11 Example (elliptic curve).

Let $X=V(y^{2}z-x^{3}+xz^{2})\subset\mathbb{P}^{2}$ be an elliptic curve,
which has a toric degeneration to $X^{\prime}=V(y^{2}z-x^{3})$ inside
$\mathbb{P}^{2}$. Then, under the Veronese embedding
$v:\mathbb{P}^{2}\hookrightarrow\mathbb{P}^{9}$,
$v(X)\subset\mathbb{P}^{9}=\operatorname{Proj}S$ is given by the ideal $I$
generated by $u_{y^{2}z}-u_{x^{3}}+u_{xz^{2}}$ and the kernel of
$S=k[u_{x^{3}},u_{x^{2}y},\textrm{etc.}]\to k[x,y,z],\,u_{x^{3}}\mapsto
x^{3},\textrm{etc.}$ Then the map $\varphi:X\to
X^{\prime},\,[x:y:z]\mapsto[y^{3}:z^{3}:y^{2}z]$ is the Veronese embedding $v$
followed by the projection $p$ given by
$S^{\prime}=k[u_{y^{2}z},u_{y^{3}},u_{z^{3}}]\hookrightarrow S$. Note that
$u_{y^{2}z}^{3}-u_{y^{3}}^{2}u_{z^{3}}\in I\cap S^{\prime}$ and thus the image
of $v(X)$ under $p$ is $V(u_{y^{2}z}^{3}-u_{y^{3}}^{2}u_{z^{3}})$ in
$\mathbb{P}^{2}$.

On the other hand, the limit $X^{\prime\prime}$ of the degeneration by
projection of $X$ is given by
$u_{y^{2}z}^{3}=u_{y^{3}}^{2}u_{z^{3}},u_{x^{3}}=u_{xz^{2}}$, etc. Perhaps
using the computer (Macaulay2), we see that the radical of the defining ideal
of $X^{\prime\prime}$ is generated by
$u_{y^{2}z}^{3}-u_{y^{3}}^{2}u_{z^{3}},u_{x^{3}},u_{xz^{2}}$, etc; that is,
$X^{\prime\prime}$ coincides with $X^{\prime}=\varphi(X)$ as a set. In fact,
it is necessarily that $X^{\prime\prime}$ is not reduced by the following
reasoning. By Theorem 3.1, we have $X^{\prime}=p(v(X))\subset
X^{\prime\prime}$. Here $X^{\prime}$ and $X^{\prime\prime}$ have different
Hilbert polynomial and thus the inclusion cannot be the equality.

The above example exhibits the following general fact.

######  3.12 Theorem.

Suppose we are given a toric degeneration $\mathfrak{X}$ of a projective
variety $X$ to $X^{\prime}$ that is embedded in
$\mathbb{P}^{q}\times\mathbb{A}^{1}$. Let $\varphi:X-B\to X^{\prime}$ be a
dominant morphism that is the restriction of a Veronese embedded
$v:\mathbb{P}^{q}\hookrightarrow\mathbb{P}^{N}$ followed by projection
$p:\mathbb{P}^{N}-V(S^{\prime}_{+}S)\to\mathbb{P}^{q}$ (e.g., $\varphi$ is one
constructed in §2.5 as discussed before Example 3.11). Also, let
$X^{\prime\prime}$ be the limit of the degeneration of $v(X)$ by the
projection $p$. Assume $\dim X>0$.

Then $X^{\prime\prime}$ is a variety (i.e., reduced and irreducible) if and
only if $v$ is the identity; i.e., $q=N$. In particular, except for some
trivial cases, $\mathfrak{X}$ cannot be given as a degeneration by projection
(in the sense of Theorem 3.1).

However, we still have that $X^{\prime}$ coincides as a scheme with
$X^{\prime\prime}\cap\mathbb{P}^{q}$.

###### Proof.

This is seen exactly as in Example 3.11. Indeed, since $X^{\prime}$ is reduced
and $X^{\prime\prime}_{red}\subset X^{\prime\prime}$, by Theorem 3.1, we have:
as closed schemes of $\mathbb{P}^{N}$,

$X^{\prime}=\overline{p(v(X-B))}\subset X^{\prime\prime}.$

Note that $X^{\prime\prime}$ have the same Hilbert polynomial as $v(X)$. Thus,
if $X^{\prime\prime}$ is a variety, then $X^{\prime}=X^{\prime\prime}$ for
dimension reason and so $X^{\prime}$ and $v(X)$ have the same Hilbert
polynomial, which cannot happen unless $v$ is the identity. Conversely, if
$X^{\prime}$ and $X^{\prime\prime}$ have the same Hilbert polynomial, then the
above inclusion is the equality. The remaining assertions also follow from
Theorem 3.1. ∎

As we mention in the introduction, the theorem in particular gives a negative
answer to the question of Dolgachev and Kaveh as to whether a projective toric
degeneration can be given as a degeneration by projection, at least in the
formulation given here.

##  4\. Collapsing a family to the special fiber

The purpose of this section is to construct a collapsing map from a proper
family to the special fiber in the classical topology, assuming the base is
small. When restricted to a general fiber, such a map is then a map from the
general fiber to the special fiber. The existence of such maps is due to §7 of
[Gor81] and §6. of [GM83] (see also Remark 4.4). But our construction is
slightly simpler; we also provide proofs that cannot be found in literature
and we state more precise results. In the next section, we will give a sketch
of how these results can be used to extend the results of [HK15] outside the
regular locus.

Unless stated otherwise, the topology in this section refers to the classical
topology. The precise result we prove in this section is:

######  4.1 Theorem.

Let $p:\mathfrak{X}\to\mathbb{A}^{1}$ be a surjective proper191919A proper
morphism in the sense in the scheme theory, which is then a proper map in the
classical topology. morphism from a complex algebraic variety. Assume that
$\mathfrak{X}$ is a closed subvariety of a smooth complex algebraic variety
$M$ and that $p$ is the restriction of a proper morphism $M\to\mathbb{A}^{1}$;
e.g., $M=\mathbb{P}^{n}\times\mathbb{A}^{1}$ and $p$ the projection.

Then, for some $0\neq t\in\mathbb{A}^{1}$, there exists a surjective
continuous map

$\varphi:p^{-1}(t)\to W$

such that for each stratum $A$ of $W$, the pre-image $\varphi^{-1}(A)$ has a
natural reducible-variety structure so that $\varphi:\varphi^{-1}(A)\to A$ is
a finite étale morphism.

######  4.2 Remark.

The morphism $p$ in the theorem is not assumed to be flat but it is actually
flat, since $\mathfrak{X}$ is a variety and a torsion-free module over a PID
is flat202020Proof: A torsion-free finitely generated module over a PID is
free; thus flat and the inductive limit of such modules is flat. (cf. [Har77,
Ch. III, Proposition 9.7.]). Also, if $p$ is trivial away from the special
fiber $W=p^{-1}(0)$, then the statement holds for any $t\neq 0$ since it holds
for some $t\neq 0$.

######  4.3 Remark.

The above theorem cannot hold for the Zariski topology in general since if
there were a surjective map that is continuous in the Zariski topology for an
arbitrary flat degeneration of a variety, the image has to be irreducible;
i.e. the limit is irreducible, which is usually not true. We do not know if
the theorem of the above type holds in the Zariski topology _if_ the limit of
a degeneration is irreducible.

Also, the map $\varphi$ itself cannot be a covering since a normal toric
variety whose fan contains a cone of dimension equal to the rank of the
lattice (e.g., a complete normal toric variety) is simply connected in the
classical topology. Considering a stratification thus allows us to have a non-
trivial covering.

In the case of a toric degeneration, the stratification will be taken to be
the orbit decomposition (so a stratum is an orbit); see Remark 4.5. The
construction of [GM83] is fairly intuitive; given a (Whitney) stratification
on $W=p^{-1}(0)$, there is a system of tubular neighborhoods parameterized by
the stratification, a fact due to Thom and Mather ([Mat12]). Collapsing each
tubular neighborhood then gives, via a partition of unity type argument, a map
that collapses a neighborhood of $V$. Restricting that map to the general
fiber gives the theorem. The reason why such a map is étale on strata is
because a tubular neighborhood has a conical structure.

######  4.4 Remark (on the literature).

According to the discussion in [Mat17], it is a folklore result that a map
$\varphi$ as in Theorem 4.1 exists, perhaps with less properties (for example,
apparently it is known to Deligne) at least when a general fiber is smooth. In
fact, when the general fibers are smooth, the comments in [Mat17] suggest that
a retraction from a neighborhood of the special fiber to the special fiber can
be constructed using a flow, even when a special fiber is non-smooth. One
suggestion there is to consider a flow on the resolution of singularities of
the family $\mathfrak{X}$ and then compose the flow with the map from the
resolution to $\mathfrak{X}$. Meanwhile, Harada and Kaveh in [HK15] give the
same result using the uniform continuity to do a continuous extension. When
$p$ is semistable (meaning the special fiber is a divisor with normal
crossings) and the general fibers are smooth, the existence of such a
collapsing map is a result of Clemens in [Cle77].

The key difference with the construction of MacPherson and Goresky and that of
us is that they and we do not use a limit or continuity argument and that the
map respects the local structure of a family; these are key to extending the
results of [HK15] outside the regular locus.

######  4.5 Remark (Whitney stratification).

For the definition of a Whitney stratification, we refer to [GM88, Part 1. Ch.
1.] as well as [Mat12]. It is known that a subvariety of a smooth complex
algebraic variety $M$ admits a Whitney stratification into smooth complex
algebraic subvarieties of $M$ (see e.g., [GM88, Part 1, Theorem 1.7.]). It is
also known that the decomposition into finitely many orbits of an algebraic
group on a complex algebraic variety gives a Whitney stratification ([Mac90]
Theorem 7.6.) Thus, in particular, in the case when $p$ is a toric
degeneration the strata in Theorem 4.1 can be taken to be torus-orbits.

We need some notations and terminology. Given two strata $A,B$, we write
$A\leq B$ if $A$ lies in the closure of $B$ and $A<B$ if $A\leq B$ and $A\neq
B$. Two strata are said to be incomparable if neither $A\leq B$ nor $B\leq A$
holds. By a tubular neighborhood of a submanifold $A$ in $M$, we mean an
smooth embedding $\psi_{A}:E\to M$ from a vector bundle $E$ over $A$ to $M$
that is the identity on $A$ and the image of $\psi_{A}$ is an (open)
neighborhood of $A$ in $M$.

Given such a $\psi_{A}$, we write $\mathfrak{T}_{A}$ for the image of
$\psi_{A}$ and $\pi_{A}$ for the composition
$\mathfrak{T}_{A}\overset{\psi_{A}^{-1}}{\to}E\overset{\xi}{\to}A$ where $\xi$
is the natural projection. Note that $\mathbb{R}$ acts on $E$ by scalar
multiplication on each fiber. Hence, $\pi_{A}$ is a strong deformation retract
of $A\subset\mathfrak{T}_{A}$ (i.e., there is a homotopy from $\pi_{A}$ to the
identity on $\mathfrak{T}_{A}$ such that the homotopy is the identity on $A$).
We also put an inner product on each fiber of $E$ such that the inner products
vary smoothly across fibers and then write
$\rho_{A}:\mathfrak{T}_{A}\to\mathbb{R}$ given as
$\rho_{A}(x)=|\psi_{A}^{-1}(x)|^{2}$ where $|\cdot|$ denotes the norm on a
fiber.

In his notes [Mat12], Mather constructs a system of tubular neighborhoods
inductively so that each tubular neighborhood of a stratum is compatible with
those smaller strata. Precisely, by induction on the strata, he constructs the
system $\\{\psi_{A}\mid A\\}$ of tubular neighborhooods of $A$ such that, for
strata $A<B$, we have

$\pi_{A}\circ\pi_{B}=\pi_{A},\rho_{A}\circ\pi_{B}=\rho_{A}$

where the equations are defined. Shrinking $\mathfrak{T}_{A}$’s, we assume
$\mathfrak{T}_{A},\mathfrak{T}_{B}$ are disjoint if $A,B$ are incomparable.

######  4.6 Remark (Conically smooth stratified spaces).

Recently, the notion of a conically smooth stratified space has been
introduced by Ayala, Francis and Tanaka [AFT17]. The above system of tubular
neighborhoods as well as the conical structures induced by it (Proposition 4.8
below) then may be thought of as the structure of a conically smooth
stratified space; this is claimed to be proved by [NV23] (the proof of [NV23,
Theorem 3.4.] is somehow unclear but Proposition 4.8 and Remark 4.10 here can
be used as a fix.)

We need the following variant of Thom’s second isotopy lemma ([Mat12,
Proposition 11.2.]):

######  4.7 Lemma.

Fix a stratum $A$ of $W$. Let $S$ be a Whitney-stratified locally closed
subset of $\mathfrak{T}_{A}$ and $f:S\to N$ the restriction of a smooth map
$M\to N$, $N$ a smooth manifold. Assume that $f$ is a proper submersion on
each stratum. Then for each point $y$ in $N$, there exists a neighborhood $U$
of $y$ in $N$ and a homeomorphism $H$ such that the diagram

---
$\textstyle{f^{-1}(y)\times
U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{A}\times\operatorname{id}}$$\scriptstyle{\sim}$$\textstyle{H}$$\textstyle{f^{-1}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\pi_{A},f)}$$\textstyle{A\times
U}$

commutes, $H$ preserves the strata and the restriction of it to each stratum
is a diffeomorphism onto the image.

Also, if $N=\mathbb{R}^{n}$, then $U$ can be taken to be $\mathbb{R}^{n}$.

###### Proof.

The lemma can be proven by adapting the proof of [Mat73, Theorem 8.6.]. But
here we will give a more direct proof. Since the statement is local, it is
enough to prove the case $N=\mathbb{R}^{n}$ with $U=\mathbb{R}^{n}$. For the
notational clarity, we shall also assume $n=1$ (but the proof for the general
case is the same).

By Thom’s first isotopy lemma [Mat12, Proposition 11.1.], we have a
trivialization $H$:

---
$\textstyle{f^{-1}(y)\times
U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\sim}$$\textstyle{H}$$\textstyle{f^{-1}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{U.}$

Thus, we only need to verify that the asserted diagram commutes with the above
$H$; i.e., $\pi_{A}(H(x,t))$ is independent of $t$. In the proof of the first
isotopy lemma, the trivialization $H$ is given by the flow $\varphi_{t}$
generated by a set-theoretic vector field $\eta$ smooth on each
stratum212121By a _set-theoretic vector field_ , we mean a not-necessarily-
smooth-nor-continuous section of a cotangent bundle.; namely,
$H(x,t)=\varphi_{t}(x)$. Now, by construction, $\eta$ is tangent everywhere to
the fibers of $\rho_{A}$. Thus,

$\frac{d}{dt}(\pi_{A}\circ\varphi_{t})=d\pi_{A}(\eta\circ\varphi_{t})=0.$

That is to say, $\pi_{A}\circ\varphi_{t}=\pi_{A}\circ H(\cdot,t)$ is
independent of $t$. This proves the claim. ∎

The next proposition is Theorem 8.3. of [Mat73] and it establishes the conical
structure of a Whitney stratification (in fact, given this result, some
authors define a stratified space as a space that has a stratification with
the conical structure).

######  4.8 Proposition (Conical structure).

Given a stratum $A$ of $W$, consider

$l_{A}:T_{A}\to\mathbb{R},\quad\text{given by}\quad
x\mapsto\frac{\rho_{A}(x)^{1/2}}{(\epsilon\circ\pi_{A})(x)},$

where $\epsilon$ is a positive smooth function on $A$. If $\epsilon$ is small
enough, then $(\pi_{A},l_{A}):l_{A}^{-1}((0,1])\to A\times(0,1]$ is trivial
over $\mathbb{R}$ as a family over $A$ in the sense that there is a
homeomorphism $H_{A}$ over $A\times(0,1]$:

---
$\textstyle{l_{A}^{-1}(1)\times(0,1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{A}\times\operatorname{id}}$$\scriptstyle{\sim}$$\textstyle{H_{A}}$$\textstyle{l_{A}^{-1}((0,1])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\pi_{A},l_{A})}$$\textstyle{A\times(0,1]}$

such that $H_{A}$ preserves the strata and the restriction of it to each
stratum is a diffeomorphism onto the image. (Cf. Remark 4.11.)

###### Proof.

By Lemma 7.3. and the remark preceding it in [Mat12], we can choose
$\epsilon>0$ such that $(\pi_{A},l_{A})$ is proper and also, for each stratum
$B$ on $W$, it is a submersion on $l_{A}^{-1}([0,1])\cap B$. Then
$l_{A}^{-1}((0,1])$ has a Whitney stratification induced from that on $W$ (see
the proof [Mat73, Theorem 8.3.]); namely, the strata are
$l_{A}^{-1}((0,1])\cap B$, $B$ strata of $W$ and $l_{A}^{-1}([0,1])-W$. The
assertion then follows from Lemma 4.7 applied to
$f=l_{A}:S=l_{A}^{-1}((0,1])\to\mathbb{R}$. ∎

The reason why the above gives a conical structure is as follows.

######  4.9 Corollary.

$H_{A}$ identifies $l_{A}^{-1}([0,1])$ as the mapping cylinder of
$\pi_{A}:l_{A}^{-1}(1)\to A$; i.e., there is a push-out diagram:

$\textstyle{l_{A}^{-1}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\operatorname{id},0)}$$\scriptstyle{\pi_{A}}$$\textstyle{l_{A}^{-1}(1)\times[0,1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{l_{A}^{-1}([0,1])}$

where the right vertical map is a unique extension of $H_{A}$ (which we shall
also denote by $H_{A}$).

###### Proof.

Let us extend $H_{A}$ by setting $H_{A}(x,0)=\pi_{A}(x)$. We shall show that
$H_{A}$ is then continuous.

Since the problem is near $A$, we can assume $M$ is a coordinate chart such
that $\pi_{A}$ is the orthogonal projection onto $A$ and
$\rho_{A}(x)=|x-\pi_{A}(x)|^{2}$. Then, for $x=H_{A}(x_{0},s)$, since
$\pi_{A}(x)=\pi_{A}(x_{0})$, we have:

$|H_{A}(x_{0},s)-H_{A}(x_{0},0)|=|x-\pi_{A}(x)|=\rho_{A}(x)^{1/2}\leq
C\,l_{A}(x)=Cs$

for some constant $C$. Also, with $x=H_{A}(x_{0},t),y=H_{A}(y_{0},t)$,

$\displaystyle|H_{A}(x_{0},t)-H_{A}(y_{0},t)|$
$\displaystyle\leq|x-\pi_{A}(x)|+|\pi_{A}(x_{0})-\pi_{A}(y_{0})|+|\pi_{A}(y)-y|$
$\displaystyle\leq 2Ct+|\pi_{A}(x_{0})-\pi_{A}(y_{0})|.$

The continuity of $H_{A}$ now follows easily. ∎

In fact, we can do further trivialization

######  4.10 Remark.

By the submersion theorem applied to $f=\pi_{A}|_{l_{A}^{-1}(1)}$, for each
point $a\in A$, we can find a neighborhood $U$ of $a$ in $M$ and a
diffeomorphism over $A$:

$(U\cap A)\times L\simeq l_{A}^{-1}(1)\cap\pi_{A}^{-1}(U\cap A)$

where $L=\pi_{A}^{-1}(a)\cap l_{A}^{-1}(1)$, called the link of $a$. For
simplicity, replacing $A$ by $U\cap A$, assume $A\subset U$. Then the above
means $A\times L\simeq l_{A}^{-1}(1)$, which, together with
$H_{A}:l_{A}^{-1}((0,1])\simeq l_{A}^{-1}(1)\times(0,1]$, gives the strata-
preserving $A$-homeomorphism:

$l_{A}^{-1}([0,1])\simeq A\times CL,\,x\mapsto(\pi_{A}(x),l_{A}(x)u(x))$

where $u:l_{A}^{-1}((0,1])\to l_{A}^{-1}(1)\to L$. In general, the above
trivialization holds locally.

######  4.11 Remark.

As noted after Theorem 8.3. of [Mat73], the trivialization $H_{A}$ in the
above proposition cannot be taken to be smooth in general. However, since $W$
is a complex algebraic variety, we can (and did) assume the stratification on
$W$ is complex-algebraic. It is known that Thom’s first and second isotopy
lemmas hold for Nash manifolds222222Nash manifolds are semialgebraic real-
analytic manifolds. ([CS92], [Esc01]); in particular, we can assume the
trivialization $H_{A}$ as well as the trivialization in Remark 4.10 are Nash
and so smooth.

Alternatively, it is also known from Hironaka’s works that we can find the
trivializations like the above that are real analytic.

Using the above Proposition 4.8, we shall define an extension of $\pi_{A}$
from some subset of the tubular neighborhood $\mathfrak{T}_{A}$. First, by
Whitney’s theorem (which says a closed subset of a manifold is the zero set of
a smooth function), choose a smooth real-valued function $\delta$ on $M$ such
that $\delta^{-1}(0)=W$ and $0\leq\delta\leq 1/4$ identically.232323If
$W=f^{-1}(0)$ for some smooth function $f$, take $\delta$ to be a multiple of
$p\circ f$ for some smooth bounded non-negative function $p$ on $\mathbb{R}$
such that $p^{-1}(0)=0$.

For a stratum $A$ of $W$ and a real-valued function $p$, we write
$A(p)=\\{x\in\mathfrak{T}_{A}\mid l_{A}(x)^{2}\leq p(x)\\}$. Then let
$\lambda_{A}$ be a smooth function on $M-W=\\{\delta>0\\}$ such that
$\lambda_{A}=1$ on $A(\delta)$, $\lambda_{A}<1$ on
$\mathfrak{T}_{A}-A(\delta)$ and $\operatorname{Supp}(\lambda_{A})\subset
A(2\delta)$.

Then define $\widetilde{\pi}_{A}:M\to M$ by:

$\widetilde{\pi}_{A}(x):=\left\\{\begin{matrix}H_{A}(x_{0},(1-\lambda_{A}(x))s),&\text{if
}x\in A(2\delta)\cap\\{\delta>0\\}\text{ and }x=H_{A}(x_{0},s)\\\
x,&\text{otherwise}.\end{matrix}\right.$

Note $\widetilde{\pi}_{A}$ is then an extension of $\pi_{A}|_{A(\delta)}$.
Also,

######  4.12 Lemma.

$\widetilde{\pi}_{A}:M\to M$ is a continuous map and
$\widetilde{\pi}_{A}:M-\overline{A}\to M$ is a smooth map. However,
$\widetilde{\pi}_{A}:M\to M$ is not smooth (not even continuously
differentiable.)

###### Proof.

Since $H_{A}$ is smooth away from $W$ by construction, the smoothness is clear
away from $W$. Thus, we only need to show that $\widetilde{\pi}_{A}$ is smooth
on some neighborhood of each point $x$ of $W$. Since $\widetilde{\pi}_{A}$ is
smooth (in fact the identity) outside the tubular neighborhood
$\mathfrak{T}_{A}$ and $A$ is closed in $\mathfrak{T}_{A}$, we can assume $A$
is closed.

If $x$ is in $W-A$, then let $U\subset\mathfrak{T}_{A}$ be a neighborhood of
$x$ that is disjoint from $A$. Then we have $U\subset\\{l_{A}\geq c\\}$ for
some constant $c>0$. Since $\delta(x)=0$, shrinking $U$, we can assume
$U\subset\\{\delta\leq c/2\\}$. But then $l_{A}/\delta\geq 2$ on $U$. Thus,
$\lambda_{A}=0$ on $U$ and so $\widetilde{\pi}_{A}$ is the identity on $U$, in
particular smooth.

Next, we shall show $\widetilde{\pi}_{A}$ is continuous. For that, given the
above, we only need to show $\widetilde{\pi}_{A}$ is continuous at each point
$x$ of $A$. We write $x=H_{A}(x_{0},0)$. For any $y$ near $x$, if $y$ is in
$\\{\delta>0\\}\cap A(2\delta)$, then we write $y=H_{A}(y_{0},s)$ and then
$\widetilde{\pi}_{A}(y)=H_{A}(y_{0},(1-\lambda_{A}(y))s)$. Here,
$|1-\lambda_{A}(y)|\leq 1$ and thus $(y_{0},(1-\lambda_{A}(y))s)\to(x_{0},0)$
as $(y_{0},s)\to(x_{0},0)$. Otherwise, $\widetilde{\pi}_{A}(y)=y$. In any
case, we have $\widetilde{\pi}_{A}(y)\to x=\widetilde{\pi}_{A}(x)$ as $y\to
x$. (See Lemma 4.13 for the stronger result.)

Finally, we show that the derivative $\widetilde{\pi}_{A}^{\prime}$, if it
exists at all, is not continuous. For simplicity, via the trivialization
$H_{A}$, view $\widetilde{\pi}_{A}$ as a map defined on $(0,1]\times
l_{A}^{-1}(1)$. The Jacobian matrix of it in the local coordinates
$(y_{1},\dots,y_{n},s)$ is then

$\begin{bmatrix}1&&0\\\ \frac{\partial\lambda_{A}}{\partial
y}s&&1-\lambda_{A}(y,s)-\frac{\partial\lambda_{A}}{\partial s}s\end{bmatrix}$

where $\frac{\partial\lambda_{A}}{\partial y}s$ denotes the matrix consisting
of partial derivatives. Here, $\lambda_{A}(y,s)$ does not go to $0$ as $s\to
0$. That is, the derivative $\widetilde{\pi}_{A}^{\prime}$ represented by the
Jacobian matrix is not continuous. ∎

In the next section, we will need the following stronger continuity property.

######  4.13 Lemma.

With an appropriate choices of $\lambda_{A}$, we have that
$\widetilde{\pi}_{A}:M\to M$ is locally Lipschitz continuous;242424In the
paragraph just prior to [Tes12, Theorem 1.3.], “locally Lipschitz continuous”
is defined so that the estimate holds on each compact set but it is known and
is easy to see that that condition is equivalent to the condition given here.
i.e., each point of $M$ has a neighborhood $U$ in some coordinate chart
together with a constant $C$ such that for all $x,y\in U$,

$|\widetilde{\pi}_{A}(x)-\widetilde{\pi}_{A}(y)|\leq C|x-y|$

where $|\cdot|$ denotes the Euclidean length coming from the coordinate chart.

###### Proof.

It is enough to prove the assertion for a neighborhood $U$ of a point in $A$
such that $\overline{U}$ is compact and contained in some coordinate chart.
Shrinking $U$, we shall assume $U\subset l_{A}^{-1}([0,1]).$

Let $x=H_{A}(x_{0},s),y=H_{A}(y_{0},t)$ in $U\cap\\{\delta>0\\}$. We have:

$|\lambda_{A}(x)s-\lambda_{A}(y)t|\leq
s|\lambda_{A}(x)-\lambda_{A}(y)|+|s-t|.$

Take $\lambda_{A}=q\circ(l_{A}/\delta)$ where $q$ is a smooth function with
$q=1$ on $[0,1]$, $<1$ on $(1,2]$ and $=0$ on $[2,\infty)$. By the mean value
inequality,

$\displaystyle s|\lambda_{A}(x)-\lambda_{A}(y)|$
$\displaystyle\leq\sup|q^{\prime}|\,s\,|s/\delta(x)-t/\delta(y)|$
$\displaystyle\leq\sup|q^{\prime}|\left(\frac{s}{\delta(x)}|s-t|+\frac{st}{\delta(x)\delta(y)}|\delta(x)-\delta(y)|\right).$

Now, we can assume $\lambda_{A}(x),\lambda_{A}(y)$ are not both zero; without
loss of generality, assume $x\in\operatorname{Supp(\lambda_{A})}\subset
A(2\delta)$. Then $s/\delta(x)\leq 2$. If $y$ is also in
$\operatorname{Supp}(\lambda_{A})$, then the above is bounded by
$\sup|q^{\prime}|\,(2|s-t|+4|\delta(x)-\delta(y)|).$

Assume $y\not\in\operatorname{Supp}(\lambda_{A})$. Then let $c=t/2$ so that
$t/c=2$. Since $\lambda_{A}(y)=0=q(t/c)$, the same analysis as above gives

$s|\lambda_{A}(x)-\lambda_{A}(y)|\leq\sup|q^{\prime}|\,s\,|s/\delta(x)-t/c|.$

Here, since $s\delta(y)\leq sc\leq t\delta(x)$,

$\displaystyle s\left|\frac{s}{\delta(x)}-\frac{t}{c}\right|$