The group G_{a} is the additive group of an algebraically closed field k. In this
chapter we give some classification results about compatibleG_{a}-actions on affine
T-varieties. More precisely, we give a full classification of compatibleG_{a}-actions in two
cases: for toric varieties, and for T-varieties of complexity 1. In general complexity,
we give a classification of compatibleG_{a}-actions whose general orbits are contained
in the closures of the general orbits of the T-action. Finally, we show that in all
these three cases the ring of invariants of aG_{a}-action is finitely generated.

2.1. Locally nilpotent derivations and G_{a}-actions

Any affine k-algebra A can be regarded as a vector space over the base field k.

A derivation ∂:A→A is a linear morphism satisfying the Leibniz rule

∂(aa^{′}) =a∂(a^{′}) +a^{′}∂(a), for all a, a^{′}∈A .

Definition 2.1.1. A derivation onAis calledlocally nilpotent (LND for short)
if for everya∈A there existsn∈Z_{≥0} such that ∂^{n}(a) = 0. Theadditive group G_{a}
is defined as the algebraic varietyA^{1} ≃kendowed with the group structure induced
by the addition onk.

Given an LND∂ on A, the map

φ_{∂}:G_{a}×A→A, (t, a)7→e^{t∂}a:=

X∞ i=0

t^{i}∂^{i}a
i!

defines aG_{a}-action on X= SpecA. Conversely, given aG_{a}-action φon X, the map

∂_{φ}:A→A, a7→ d

dt(φ^{∗}(a))

t=0

defines an LND onA. The following well known lemma shows that these maps are mutually inverse.

Lemma 2.1.2. The maps defined above are mutually inverse and so there is a
bijective correspondence between LNDs on A andG_{a}-actions on X= SpecA.

Proof. See [Fre06, Section 1.5].

Remark 2.1.3. Under the above correspondence, the kernel ker∂ corresponds
to the ring of invariantsk[X]^{G}^{a} of the corresponding G_{a}-action.

In the following lemma we collect some well known facts about LNDs over a field of characteristic 0 not necessarily algebraically closed, needed for later purposes, see e.g., [ML; Fre06].

Lemma2.1.4. Let A be a finitely generated normal domain over a field of
char-acteristic 0. If ∂ and ∂^{′} are two LNDs on A, then the following hold:

49

(i) ker∂ is a normal subdomain of codimension 1.

(ii) ker∂ is factorially closed i.e.,ab∈ker∂⇒a, b∈ker∂.

(iii) If a∈A is invertible, thena∈ker∂.

(iv) If ker∂= ker∂^{′}, then there exist a, a^{′} ∈ker∂ such that a∂=a^{′}∂^{′}.
(v) If a∈ker∂, then a∂ is again an LND.

(vi) If ∂(a)∈(a) for some a∈A, then a∈ker∂.

(vii) The field extension Frac(ker∂)⊆FracA is purely transcendental of degree 1.

The following definition is motivated by Lemma 2.1.4 (iv).

Definition2.1.5. We say that two LNDs∂and∂^{′} onAareequivalent if ker∂=
ker∂^{′}. Geometrically this means that the general orbits of the associatedG_{a}-actions
on X= SpecA coincide.

Let as before M and N be dual lattices. Let k(Y) be the field of rational
functions of an algebraic variety Y. We consider a finitely generated effectively
M-graded domain^{1}

A= M

m∈ωM

A_{m}χ^{m}, where A_{m}⊆k(Y). (3)

A derivation∂onAis calledhomogeneous if it sends homogeneous elements into
homogeneous elements. Hence ∂ sends homogeneous pieces of A into homogeneous
pieces. AG_{a}-action on an affine T-variety is called compatible if the corresponding
LND is homogeneous. In geometric terms, a G_{a}-action is compatible if and only if
it is normalized by the torusT. Let

M_{∂}={m∈ω_{M} |∂(A_{m}χ^{m})6= 0} .

The action of∂ on homogeneous pieces of A defines a map ∂_{M} :M_{∂} →ω_{M} i.e.,

∂(Amχ^{m}) ⊆ A_{∂}_{M}_{(m)}χ^{m}. By Leibniz rule, for homogeneous elements a ∈ Amχ^{m}\
ker∂ and a^{′} ∈A_{m}^{′}χ^{m}^{′}\ker∂ we have

∂(aa^{′}) =a∂(a^{′}) +a^{′}∂(a)∈A_{∂(m+m}^{′}_{)},
and so

∂M(m+m^{′}) =m+∂M(m^{′}) =m^{′}+∂M(m).

Thus ∂_{M}(m)−m ∈ M is a constant function on M_{∂}. This leads to the following
definition.

Definition 2.1.6. Let ∂ be a nonzero homogeneous derivation on A. The
de-gree of ∂ is the lattice vector deg∂ defined by deg∂ = deg∂(f)−deg(f) for any
homogeneous elementf /∈ker∂. With this notation the map ∂_{M} :M_{∂} →ω_{M} is just
the translation by the vector deg∂.

We also say that an LND ∂ on A is negative if deg∂ /∈ ω_{M}, non-negative if
deg∂∈ω_{M}, and positive if∂ is non-negative and deg∂6= 0.

The following well known fact shows that any LND onAdecomposes into a sum of homogeneous derivations, some of which are locally nilpotent.

Lemma 2.1.7. Let A be a finitely generated normal M-graded domain. For any derivation∂ onAthere is a decomposition∂=P

e∈M∂_{e}, where∂_{e}is a homogeneous
derivation of degree e. Moreover, let ∆(∂) be the convex hull in M_{Q} of the set

1Recall our convention regardingM-graded algebras in Remark 1.3.10 (i).

2.1. LOCALLY NILPOTENT DERIVATIONS AND Ga-ACTIONS 51

{e ∈ M : ∂_{e} 6= 0}. Then ∆(∂) is a bounded polyhedron and for every vertex e of

∆(∂), ∂_{e} is locally nilpotent if ∂ is.

Proof. Lettinga1,· · ·, arbe a set of homogeneous generators ofAwe haveA≃
k^{[r]}/I, where k^{[r]}=k[x_{1},· · · , x_{r}] and I denotes the ideal of relations of a_{1},· · · , a_{r}.
TheM-grading and the derivation∂ can be lifted to anM-grading and a derivation

∂^{′} on k^{[r]}, respectively.

The proof of Proposition 3.4 in [Fre06] can be applied to anM-grading, proving
that ∂^{′} = P

e∈M∂^{′}_{e}, where ∂^{′}_{e} is a homogeneous derivation on k^{[r]}. Furthermore,
since ∂^{′}(I) ⊆ I and I is homogeneous, we have ∂_{e}^{′}(I) ⊆ I. Hence ∂_{e}^{′} induces a
homogeneous derivation ∂_{e} on A of degreee, proving the first assertion.

The algebra A being finitely generated, the set {e ∈ M : ∂e 6= 0} is finite and so ∆(∂) is a bounded polyhedron. Let e be a vertex of ∆(∂) and n ≥ 1. If ne = Pn

i=1m_{i} with m_{i} ∈ ∆(∂)∩M, then m_{i} = e ∀i. For a ∈ A_{m}χ^{m} this yields

∂^{n}_{e}(a) = (∂^{n}(a))_{m+ne}, where (∂^{n}(a))_{m+ne}stands for the summand of degree m+ne
in the homogeneous decomposition of ∂^{n}(a). Hence ∂e is locally nilpotent if ∂ is

so.

In the following lemma we extend Lemma 1.8 in [FZ05a] to more general grad-ings. This lemma shows that any LND ∂ on a normal domain can be extended as an LND to a cyclic ring extension defined by an element of ker∂.

Lemma2.1.8. LetA be a finitely generated normal domain and let∂ be an LND onA.

(i) Given a nonzero elementv∈ker∂ andd >0, we letA^{′} denote the
normaliza-tion of the cyclic ring extension A[u]⊇A in its fraction field, where u^{d}= v.

Then∂ extends in a unique way to an LND∂^{′} onA^{′}.

(ii) Moreover, if A is M-graded and ∂ and v are homogeneous, with degv =dm
for somem∈M, then A^{′} isM-graded as well, and u and∂^{′} are homogeneous
withdegu=m and deg∂^{′} = deg∂.

Proof. Actually (i) is contained in [FZ05a, Lemma 1.8] while the proof of (ii)

is similar and so we omit it.

Recall that A =L

m∈ωM A_{m}χ^{m}, where A_{m} ⊆k(Y), k(Y) is a field of rational
functions of an algebraic variety and FracA = k(Y)(M) ^{2}. The following lemma
provides some useful extension of a homogeneous LND∂ on A.

Lemma 2.1.9. For any homogeneous LND ∂ onA, the following hold:

(i) The derivation ∂ extends in a unique way to a homogeneous k-derivation on k(Y)[M].

(ii) If ∂(k(Y)) = 0 then the extension of ∂ as in (i) restricts to a homogeneous
locally nilpotentk(Y)-derivation on k(Y)[ω_{M}].

Proof. Since FracA = k(Y)(M), any f χ^{m}, f ∈ k(Y), can be written as

To show (ii), suppose that ∂(k(Y)) = 0. Assuming that f χ^{m} ∈k(Y)[ω_{M}], we
consider r >0 such that A_{rm} 6= 0. Lettingg ∈ A_{rm}, we havef^{r}χ^{rm} =f^{′}gχ^{rm} for
somef^{′}∈k(Y). Thus f^{r}χ^{rm} is nilpotent an so is f χ^{m}.
In the setting as in the previous lemma, the extension of ∂ to k(Y)[M] will be
still denoted by∂. Following [FZ05a] we use the next definition.

Definition 2.1.10. With A as in (3), a homogeneous LND ∂ on A or,
equiva-lently, a G_{a}-action on X = SpecA, is said to be offiber type if∂(k(Y)) = 0 and of
horizontal type otherwise.

Let A be a finitely generated domain and X = Spec A. In this setting, ∂ is
of fiber type if and only if the general orbits of the corresponding G_{a}-action are
contained in the closures of general orbits of theT-action given by the M-grading.

Otherwise, ∂ is of horizontal type.

2.2. Compatible G_{a}-actions on toric varieties

In this section we consider more generally toric varieties defined over a fieldkof characteristic 0, not necessarily algebraically closed.

Let as before M and N be dual lattices of rank n. We also let N_{Q} = N ⊗Q,
MQ=M⊗Q, and we consider the natural dualityMQ×NQ →Q, (m, p)7→ hm, pi.
Notation 2.2.1. Letρ∈N and e∈M be lattice vectors. We define∂_{ρ,e} as the
homogeneous derivation of degreeeon k[M] given by∂ρ,e(χ^{m}) =hm, ρi ·χ^{m+e}.

An easy computation shows that ∂_{ρ,e} is indeed a derivation. Let H_{ρ} be the
subspace ofMQ orthogonal toρ, andH_{ρ}^{+}be the halfspace ofMQgiven by h·, ρi ≥0.

The kernel ker∂_{ρ,e} is spanned by all characters χ^{m} with m ∈ M orthogonal to ρ,
i.e., ker∂_{ρ,e}=k[H_{ρ}∩M].

Let Nil(∂_{ρ,e}) be the subalgebra of k[M] where∂_{ρ,e} acts in a nilpotent way.
As-sume that he, ρi=−1. For every m∈H_{ρ}^{+}∩M, the character χ^{m} ∈Nil(∂_{ρ,e}) since

∂^{ℓ}_{ρ,e}(χ^{m}) = 0, whereℓ=hm, ρi+ 1. Thus, the derivation ∂_{ρ,e} restricted to the
sub-algebra k[H_{ρ}^{+}∩M] is a homogeneous LND. On the other hand, ∂_{ρ,e} is not locally
nilpotent ink[M], in fact for everym /∈H_{ρ}^{+}∩M the character χ^{m}∈/ Nil(∂ρ,e).

Remark2.2.2. If∂ρ,e stabilizes a subalgebraA⊆k[H_{ρ}^{+}∩M], then∂ρ,e|Ais also
a homogeneous LND on Aof degree eand ker(∂_{ρ,e}|A) =A∩k[H_{ρ}∩M].

For the rest of this section, we let σ be a pointed polyhedral cone in the vector space NQ with dual cone ω⊆MQ, and

A=k[ω_{M}] = M

m∈ωM

kχ^{m}

be the affine semigroup algebra of σ with the corresponding affine toric variety X= Spec A. Since the cone σ is pointed, ω is of full dimension and the subalgebra A⊆k[M] is effectively graded byM.

To every ray ρ ⊆ σ we can associate a facet τ ⊆ ω given by τ = ω∩ρ^{⊥}. As
usual, we denote a ray and its primitive vector by the same letterρ. Thusω⊆H_{ρ}^{+}
and τ ⊆H_{ρ}.

2.2. COMPATIBLE Ga-ACTIONS ON TORIC VARIETIES 53

ρ

σρ

σ⊆NQ

τ

σ^{∨}_{ρ}

σ^{∨}⊆MQ

{h·, ρi=−1} Sρ

Figure 5. The setS_{ρ} and the coneσ_{ρ}

Definition 2.2.3. Letσ_{ρ} be the cone spanned by all the rays ofσ exceptρ, so
thatω =σ_{ρ}^{∨}∩H_{ρ}^{+}. We also let

S_{ρ}=σ_{ρ}^{∨}∩ {e∈M | he, ρi=−1}.

This definition is illustrated in Figure 5, where ρ ⊆ N_{Q} is pointing upwards.

Alternatively, we can defineS_{ρ}as the set of lattice vectorsm∈Msuch thathρ, mi=

−1 andhρ^{′}, mi ≥0 for every other ray ρ^{′} ⊆σ.

Lemma 2.2.4. Let e∈M. Then e∈S_{ρ} if and only if
(i) e /∈ω_{M}, and

(ii) m+e∈ω_{M},∀m∈ω_{M} \τ_{M}.

Proof. Assume first that e ∈ S_{ρ}. Then (i) is evident. To show (ii), we let
m∈ω_{M} \τ_{M}. Thenm+e∈H_{ρ}^{+} becausehm+e, ρi=hm, ρi −1. Alsom∈ω ⊆σ_{ρ}^{∨}
yieldingm+e∈σ_{ρ}^{∨}. Thusm+e∈ω =σ_{ρ}^{∨}∩H_{ρ}^{+}.

To show the converse, we let e∈M be such that (i) and (ii) hold. Letting ρ_{i},
i= 1,· · ·, ℓ be all the rays ofσ exceptρ, form∈ω_{M} \τ_{M} we have

hm+e, ρ_{i}i=hm, ρ_{i}i+he, ρ_{i}i ≥0, ∀i∈ {1,· · · , ℓ}.

Ifm∈ρ^{⊥}_{i} ∩ω_{M} thenhm, ρ_{i}i= 0 and sohe, ρ_{i}i ≥0∀i. Thuse∈σ^{∨}_{ρ}. Sincee∈σ^{∨}_{ρ}\ω,
he, ρi is negative. We havehe, ρi=−1, otherwise m+e /∈ ω for any m∈ω_{M} such

thathm, ρi= 1. This yieldse∈S_{ρ}.

Remark 2.2.5. Since ρ /∈ σ_{ρ} we have S_{ρ} 6= ∅. Furthermore, by the previous
lemma,e+m∈S_{ρ} whenever e∈S_{ρ} and m∈τ_{M}.

In the following lemma we provide a translation of Lemma 2.2.4 from the lan-guage of convex geometry to that of affine semigroup algebras.

Lemma 2.2.6. For every pair (ρ, e), where ρ is a ray of σ and e is a lattice
vector in Sρ, the homogeneous derivation ∂ρ,e restricts to a homogeneous LND on
A=k[ω_{M}]with kernel ker∂_{ρ,e}=k[τ_{M}]and deg∂_{ρ,e}=e.

Proof. Ifσ ={0}, thenσ has no, so the statement is trivial. We assume in the
sequel that σ has at least one ray ρ. By Lemma 2.2.4 ∂_{ρ,e} stabilizes A. Hence by
Remark 2.2.2 (2),∂_{ρ,e} is a homogeneous LND onA with kernelk[τ_{M}] and of degree

e.

The following theorem completes our classification, cf. [Dem70, Prop. 11] and [Oda88, Section 3.4].

Theorem2.2.7. If∂6= 0 is a homogeneous LND onA, then∂=λ∂_{ρ,e} for some
ray ρ on σ, some lattice vector e∈S_{ρ}, and someλ∈k^{∗}.

Proof. The kernel ker∂ is a subsemigroup subalgebra of A of codimension 1.

Since ker∂is factorially closed (see Lemma 2.1.4), it follows that ker∂=k[ω_{M}∩H]

for a certain codimension 1 subspaceH of M_{Q}.

If ω∩H is not a facet of ω, then H divides the cone ω into two pieces. Since the action of ∂ on characters is a translation by a constant vector deg∂, only the characters from one of these pieces can reach H in a finite number of iterations of

∂, which contradicts the fact that ∂ is locally nilpotent.

In the case whereω∩H =τ is a facet ofω, we letρ be the ray dual toτ. Since

∂ is an homogeneous LND, the translation by e= deg∂ maps (ω_{M} \τ_{M}) into ω_{M}.
So by Lemma 2.2.4,e∈S_{ρ} and∂ =λ∂_{ρ,e}, as required.

Remark2.2.8. In [Dem70] a similar result is proved for smooth, not necessarily affine, toric varieties. Inloc. cit. the elements in the set

R= [

ρ⊆σ

−Sρ

are called the roots of σ.

From our classification we obtain the following corollaries.

Corollary 2.2.9. A homogeneous LND ∂ on a toric variety is uniquely deter-mined, up to a constant factor, by its degree.

Proof. By Theorem 2.2.7 we have ∂ =λ∂_{ρ,e} where e= deg∂. We claim that
the ρ is uniquely determined by e. Indeed, the sets S_{ρ} and S_{ρ}^{′} are disjoint for

ρ6=ρ^{′}.

Corollary 2.2.10. Every homogeneous LND ∂ on a toric variety X is of fiber type and negative.

Proof. The first claim is evident because T acts with an open orbit. By
Theorem 2.2.7, any LND on a toric variety is of the form λ∂ρ,e. Its degree is
deg∂_{ρ,e}=e∈S_{ρ} and S_{ρ}∩ω=∅, so∂ is negative.

Corollary 2.2.11. Two homogeneous LNDs ∂ =λ∂_{ρ,e} and ∂^{′} =λ^{′}∂_{ρ}^{′}_{,e}^{′} on A
are equivalent if and only if ρ =ρ^{′}. In particular, there is only a finite number of
pairwise non-equivalent homogeneous LNDs on A.

Proof. The first assertion follows from the description of ker∂_{ρ,e} in Lemma
2.2.6 and the second one from the fact that σ, being polyhedral, has only a finite

number of rays.

Example 2.2.12. With N =Z^{3} we let σ be the cone in NQ having rays ρ1 =
(1,0,0),ρ_{2} = (0,1,0),ρ_{3} = (1,0,1), andρ_{4}= (0,1,1). The dual coneω⊆M_{Q} =Q^{3}
is spanned by the lattice vectors u_{1} = (1,0,0), u_{2} = (0,1,0), u_{3} = (0,0,1), and
u_{4} = (1,1,−1). Furthermore, these elements satisfy the relation u_{1}+u_{2} =u_{3}+u_{4}
and the algebraA=k[ωM] is generated by xi =χ^{u}^{i},i= 1, . . . ,4. Thus

A≃k[x_{1}, x_{2}, x_{3}, x_{4}]/(x_{1}x_{2}−x_{3}x_{4}). (4)
Corollary 2.2.11 shows that there are four non-equivalent homogeneous LNDs
on Acorresponding to the rays ρi⊆σ. By a routine calculation we obtain

S_{ρ}_{1} ={(−1, b, c)∈M |b≥0, c≥1}, S_{ρ}_{2} ={(a,−1, c)∈M |a≥0, c≥1},

2.3. COMPATIBLE Ga-ACTIONS ON T-VARIETIES OF COMPLEXITY 1 55

Finally, under the isomorphism of (4) the four homogeneous LNDs on A are given
by
2.3. Compatible G_{a}-actions on T-varieties of complexity 1

In this section we give a complete classification of homogeneous LNDs on T-varieties of complexity 1 over an algebraically closed field k of characteristic 0. In the first part we treat the case of a homogeneous LNDs of fiber type, while in the second one we deal with the more delicate case of homogeneous LNDs of horizontal type.

Lettingk(C) be the function field of C, we consider the affine variety X= Spec A, where

We also fix a homogeneous LND ∂ on A. In this context, we can interpret Definitions 2.1.6 and 2.1.10 as follows.

Lemma 2.3.1. With the notation as above, let ∂ be a homogeneous LND on A.

Then the following hold.

(i) If ∂ is of fiber type, then∂ is negative andker∂=L

m∈τM A_{m}χ^{m}, where τ is
a facet of ω.

(ii) Assuming further that A is non-elliptic, ∂ is of fiber type if and only if ∂ is negative.

Proof. To prove (i) we let ∂ be a homogeneous LND of fiber type on A. By
Lemma 2.1.9 we can extend ∂to a homogeneous LND ¯∂ on ¯A=k(C)[ω_{M}] which is
an affine semigroup algebra overk(C). Since ∂(k(C)) = 0, ¯∂ is a locally nilpotent
k(C)-derivation. It follows from Corollary 2.2.10 that deg∂ = deg ¯∂ /∈ω_{M}, so ∂ is
negative.

Furthermore, Lemma 2.2.6 and Theorem 2.2.7 show that ker ¯∂ = k(C)[τ_{M}],
whereτ is a facet ofω. Thus

ker∂=A∩ker ¯∂= M

m∈τM

(A_{m}∩k(C))χ^{m} = M

m∈τM

A_{m}χ^{m},
which proves (i).

To prove (ii) we assume further that A is non-elliptic. Let ∂ be a negative
homogeneous LND on A. Let ¯∂ be the extension of ∂ to k(C)[M] as in Lemma
2.1.9. Since ∂ is negative, ∂(A_{0}) ⊆ A_{deg}_{∂} = 0. Since A is non-elliptic we have
k(C) = Frac A_{0}, so ¯∂(k(C)) = 0 and∂ is of fiber type.

Remark 2.3.2. In the elliptic case, the second assertion in Lemma 2.3.1 does
not hold, in general. Consider for instance the elliptick-domainA=k[x, y] graded
via degx = degy = 1. Then the partial derivative ∂_{x} is a negative homogeneous
LND of horizontal type onA.

2.3.1. Homogeneous LNDs of fiber type. In this subsection we consider
a homogeneous LND ∂ on A of fiber type. Let as before ¯A = k(C)[ω_{M}] be the
affine semigroup k(C)-algebra with cone σ ∈ NQ over the field k(C). By Lemma
2.1.9, ∂ can be extended to a homogeneous locally nilpotentk(C)-derivation on ¯A.

To classify homogeneous LNDs of fiber type, we will rely on the classification of homogeneous LNDs on affine semigroup algebras from the previous section.

Ifσ has no ray then σ= 0 and ω=MQ. By Lemma 2.3.1 in this case there are
no homogeneous LND of fiber type. So we may assume in the sequel that σ has at
least one ray, sayρ. Letτ be its dual facet, and letS_{ρ}be as defined in Lemma 2.2.4.

Lemma 2.3.3. For any e∈Sρ,
D_{e}:= X

z∈C

m∈ωmaxM\τM

(h_{z}(m)−h_{z}(m+e))·z
is a well defined Q-divisor on C.

Proof. By Lemma 2.2.4, for all m ∈ω_{M} \τ_{M}, m+e is contained in ω_{M} and
thush_{z}(m) and h_{z}(m+e) are well defined. Recall that for anyz∈C, the function
h_{z} is concave and piecewise linear onω. Thus the above maximum is achieved by
one of the linear pieces ofhz i.e., by one of the maximal cones in the normal quasifan
Λ(h_{z}) (see Definition 1.5.3).

For everyz∈C, we let{δ_{1,z},· · · , δ_{ℓ}_{z}_{,z}} be the set of all maximal cones in Λ(h_{z})
andg_{r,z}, r∈ {1,· · ·, ℓ_{z}}be the linear extension ofh_{z}|δr,z toMQ. Since the maximum
is achieved by one of the linear pieces we have

m∈ωmaxM\τM

(hz(m)−hz(m+e)) = max

r∈{1,···,ℓz}(−gr,z(e)).

Sincegr,z(e)∈Q∀(r, z),De is indeed a Q-divisor.

Remark 2.3.4. An alternative description ofD_{e} is as follows. Let the notation
be as in the preceding proof. Since τ is a facet ofω, it is contained as a face in one
and only one maximal coneδr,z. We may assume thatτ ⊆δ1,z. By the concavity of
h_{z} we have g_{1,z}(e)≤g_{r,z}(e) ∀r and so

D_{e}=−X

z∈C

g_{1,z}(e)·z .

2.3. COMPATIBLE Ga-ACTIONS ON T-VARIETIES OF COMPLEXITY 1 57

Notation 2.3.5. We let

Φ_{e}=H^{0}(C,OC(−D_{e})), and Φ^{∗}_{e} = Φ_{e}\ {0}.
We need the following lemma.

Lemma2.3.6. Letρ∈σ be a ray,τ ∈ωbe its dual facet ande∈S_{ρ}. Ifϕ∈k(C),
thenϕ∈Φ_{e} if and only if ϕA_{m} ⊆A_{m+e} for anym∈ω_{M} \τ_{M}.

Proof. Ifϕ∈Φe, then for everym∈ωM \τM,
div(ϕ)≥D_{e} ≥X

z∈C

(h_{z}(m)−h_{z}(m+e))·z=D(m)−D(m+e).

If f ∈ ϕA_{m} then div(f) +D(m) ≥ div(ϕ) and so div(f) +D(m+e) ≥ 0. Thus
ϕA_{m}⊆A_{m+e}.

To prove the converse, we let ϕ ∈ k(C) be such that ϕA_{m} ⊆ A_{m+e} for any
m∈ω_{M}\τ_{M}. With the notation of Remark 2.3.4, we letm∈M be a lattice vector
such that D(m) is an integral divisor, andm and m+ebelong to rel.int(δ1,z), for
any z∈C.

For everyz∈SuppD, we let fz ∈Am be a rational function such that
ord_{z}(f_{z}) =−h_{z}(m) =−g_{1,z}(m).

By our assumptionϕf_{z}∈A_{m+e} and so

ord_{z}(ϕf_{z})≥ −h_{z}(m+e) =−g_{1,z}(m+e).

This yields ord_{z}(ϕ) ≥ −g_{1,z}(m+e) +g_{1,z}(m) =−g_{1,z}(e) and so ϕ∈ Φ_{e}. This

proves the lemma.

There is a natural way to associate to a nonzero functionϕ∈Φ^{∗}_{e} a homogeneous
LND of fiber type onA. More precisely we have the following lemma.

Lemma 2.3.7. To any triple (ρ, e, ϕ), where ρ is a ray of σ, e∈ Sρ is a lattice
vector, and ϕ ∈ Φ^{∗}_{e} is a nonzero function, we can associate a homogeneous LND

∂ρ,e,ϕ onA=A[C,D]with kernel
ker∂_{ρ,e,ϕ}= M

m∈τM

A_{m}χ^{m}, and deg∂_{ρ,e,ϕ} =e .

Proof. Letting ¯A = k(C)[ω_{M}], we consider the k(C)-LND ∂_{ρ,e} on ¯A as in
Lemma 2.2.6. Sinceϕ∈k(C),ϕ∂ρ,e is again an k(C)-LND on ¯A.

We claim that ϕ∂_{ρ,e} stabilizesA ⊆A. Indeed, let¯ f ∈A_{m} ⊆k(C) be a
homo-geneous element so that div f +D(m) ≥ 0. If m ∈ τ_{M}, then ϕ∂ρ,e(f χ^{m}) = 0. If
m∈ω_{M} \τ_{M}, then

ϕ∂_{ρ,e}(f χ^{m}) =ϕf ∂_{ρ,e}(χ^{m}) =m_{0}ϕf χ^{m+e},

wherem0 :=hm, ρi ∈ Z_{>0}. By Lemma 2.3.6ϕf χ^{m+e}∈A and so does m0ϕf χ^{m+e},
proving the claim.

Finally∂ρ,e,ϕ:=ϕ∂ρ,e|A is an homogeneous LND onA with kernel
ker∂_{ρ,e,ϕ}= M

m∈τM

A_{m}χ^{m},

as desired.

The following theorem gives the converse of Lemma 2.3.7 and so completes our classification of homogeneous LNDs of fiber type on T-varieties.

Theorem 2.3.8. Every nonzero homogeneous LND ∂ of fiber type on A =
Corollary 2.3.9. Let as beforeX = Spec A be a T-variety of complexity 1,∂
be a homogeneous LND of fiber type on A, and let f χ^{m} ∈ A\ker∂ be a

0f ∈k(C) is also uniquely determined by our data.

Corollary 2.3.10. Two homogeneous LND∂=∂_{ρ,e,ϕ} and ∂^{′} =∂_{ρ}^{′}_{,e}^{′}_{,ϕ}^{′} of fiber
type on Aare equivalent if and only ifρ=ρ^{′}. In particular, there is a finite number
of pairwise non-equivalent LNDs of fiber type on A.

Proof. The first assertion follows from the description of ker∂_{ρ,e,ϕ} in Lemma
2.3.7. The second one follows from the fact that σ has a finite number of rays.

Given a rayρ⊆σande∈S_{ρ}, it might happen that Φ^{∗}_{e}=∅, so that there exist no
homogeneous LND∂ of fiber type on Awith deg∂=eand ker∂=L

m∈τM Amχ^{m}.
In the following lemma we give a criterion for the existence of e ∈ S_{ρ} such that
dim Φe is nonzero.

Lemma 2.3.11. Let A =A[C,D], and let ρ ⊆σ be a ray dual to a facetτ ⊆ω.

There exists e∈S_{ρ} such that dim Φ_{e} is positive if and only if the curve C is affine
or C is projective andhdegD|τ 6≡0.

Proof. If C is affine, then for any Z-divisor D the sheaf OC(D) is generated
by the global sections. It follows in this case that dim Φ_{e} >0.

Let furtherCbe a projective curve of genusg. If deg⌊−D_{e}⌋<0 then dim Φ_{e}= 0.

On the other hand, by the Riemann-Roch theorem dim Φ_{e}>0 if deg⌊−D_{e}⌋ ≥g(see
Lemma 1.2 in [Har77, Chapter IV]).

Letting h = h_{deg}D = P

z∈Chz, with the notation of Remark 2.3.4 we have h|τ = P

z∈Cg_{1,z} and deg(−D_{e}) = P

z∈Cg_{1,z}(e). By the definition of proper
σ-polyhedral divisor,h(m)>0 for anym in the relative interior of ω.

If h|τ ≡ 0 then by the linearity of g_{1,z} we obtain that deg(−D_{e}) < 0, so
deg⌊−D_{e}⌋<0 and dim Φ_{e}= 0.

Ifh|τ 6≡0 then by the concavity ofh,h(m)>0 for allm in the relative interior
of τ. By Remark 2.3.4, deg(−D_{e}) is linear on eand so, according to Remark 2.2.5,
we can choose a suitablee∈Sρ so that deg⌊−De⌋ ≥g. Hence dim Φe>0.

We can now deduce the following corollary.

Corollary 2.3.12. Let A = A[C,D], and let ρ ⊆ σ be a ray dual to a facet τ ⊆ ω. There exists a homogeneous LND of fiber type ∂ on A such that ker∂ = L

m∈τM A_{m}χ^{m} if and only if C is affine or C is projective andρ∩degD=∅.

2.3. COMPATIBLE Ga-ACTIONS ON T-VARIETIES OF COMPLEXITY 1 59

Proof. Since ρ∩degD=∅ is equivalent to h_{deg}D|τ 6≡0, the corollary follows

from Theorem 2.3.8 and Lemma 2.3.11.

Remark 2.3.13. By Corollaries 2.3.10 and 2.3.12, the equivalence classes of LNDs of fiber type on A=A[C,D] are in one to one correspondence with the rays ρ⊆σ ifC is affine and with raysρ⊆σ such thatρ∩degD=∅ ifC is projective.

2.3.2. Homogeneous LNDs of horizontal type. LetA=A[C,D], whereD is a properσ-polyhedral divisor on a smooth curve C. We consider a homogeneous LND∂of horizontal type onA. We also denote by∂its extension to a homogeneous k-derivation on k(C)[M], where k(C) is the field of rational functions of C (see Lemma 2.1.9 (i)).

The existence of a homogeneous LND of horizontal type imposes strong restric-tions on C, as we show in the next lemma.

Lemma 2.3.14. If there exists a homogeneous LND ∂ of horizontal type on A=
A[C,D], thenC≃P^{1} in the case whereA is elliptic andC ≃A^{1} in the case whereA
is non-elliptic. In the latter caseA_{m} is a freeA_{0}-module of rank 1 for everym∈ω_{M}
and so

Am=ϕmA0 for some ϕm∈Am such that div(ϕm) +⌊D(m)⌋= 0.
Proof. Let π : X = SpecA 99K C be the rational quotient for the T-action
given by the inclusionπ^{∗}:k(C)֒→K= FracA. SinceX is normal, the
indetermi-nacy locusX_{0} of π has codimension greater than 1, and so the general orbits of the

Am=ϕmA0 for some ϕm∈Am such that div(ϕm) +⌊D(m)⌋= 0.
Proof. Let π : X = SpecA 99K C be the rational quotient for the T-action
given by the inclusionπ^{∗}:k(C)֒→K= FracA. SinceX is normal, the
indetermi-nacy locusX_{0} of π has codimension greater than 1, and so the general orbits of the